subdivision schemes and their applications to solve differential...

Subdivision Schemes and theirApplications to Solve Differential

Equations

By

Syeda Tehmina Ejaz

Ph. D. DISSERTATION

Session 2012-2015

DEPARTMENT OF MATHEMATICS

The Islamia University of Bahawalpur

Bahawalpur 63100, PAKISTAN

2016


Equations

By

Syeda Tehmina Ejaz

Supervised By

Prof. Dr. Ghulam Mustafa

Department of Mathematics



2016


Equations

By

Syeda Tehmina Ejaz

A dissertation submitted to the department of Mathematics,


in the partial fulfillment for the degree of

Doctor of Philosophy

in

Mathematics

Supervised By

Prof. Dr. Ghulam Mustafa

Department of Mathematics



2016

Scanned by CamScanner

Dedication

I WOULD LIKE TO DEDICATE MY THESIS TO MY

Beloved Fatherand

Sweet Mother

WHO ALWAYS PICKED ME UP ON TIME AND

ENCOURAGED ME TO GO ON EVERY ADVENTURE

ESPECIALLY THIS ONE

iv

Acknowledgments

I offer all the praises and deepest gratitude to Almighty ALLAH, the most

gracious, the most merciful and to His Holy Prophet Muhammad (Peace be

upon him), a teacher of the whole humanity and a source of inspiration and

guidance throughout my life.

I owe a scholarly debt of gratitude to Prof. Dr. Ghulam Mustafa, my super-

visor & Chairman Department of Mathematics, whose charisma, skill and con-

cern surpassed all understanding. This task would not have been accomplished

without his brilliant and devoted supervision. I extend my deepest thanks and

felicitation for his monumentally scholarly enterprise and giving me the chance

to make an enchanting voyage into the conglomerates of the present study.

I am grateful to my respectable teacher Prof. Dr. Tahir Mahmood, Ex-Chairman

Department of Mathematics for his care and cooperation throughout my stud-

ies.

Of course, I am grateful to my beloved parents, brothers and sisters for their

infinite patience and love; unconditional support. I owe everything I accom-

plished, indeed, without them this work would never have come into existence

(literally).

I am deeply indebted to my friends and well wishers especially, Nargis Khan

and Madiha Sana, who profusely lauded and encouraged me during this re-

search project. Their encouragement stimulated me to undertake this long and

onerous journey into the realms of our skewed culture of research.

v

I acknowledge that this research work is supported by Indigenous Ph. D 5000

Fellowship Program and National Research Program for Universities (NRPU)

Project No. 3183 of Higher Education Commission (HEC) of Pakistan.

Syeda Tehmina Ejaz

vi

Abstract

Subdivision schemes are important for the generation of smooth curves and

surfaces through an iterative process from a finite set of points. The subdivision

schemes have been considered well-regarded in many fields of computational

sciences. In this dissertation, we have used subdivision schemes for the numer-

ical solution of different types of boundary value problems. In literature three

methods such as spline based methods, finite difference methods and finite el-

ement methods are commonly used to find the numerical solution of boundary

value problems. Subdivision based algorithms for the numerical solution of

second order boundary value problems have also been used in the literature.

In this dissertation, we develop subdivision based collocation algorithms for

the numerical solution of linear and non linear boundary value problems of or-

der three and four. Subdivision based collocation algorithms for the solution

of second and third order singularly perturbed boundary value problems are

also presented in this dissertation. These algorithms are developed by using ba-

sis functions of subdivision schemes. Convergence analysis of these collocation

algorithms are also discussed. Accuracy and efficiency of the developed algo-

rithms are shown through comparison with the existing numerical algorithms.

vii

Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Approval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

1 Introduction 1

1.1 Computer aided geometric design . . . . . . . . . . . . . . . . . . 1

1.1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Numerical methods for BVPs . . . . . . . . . . . . . . . . . 5

1.2.2 Contribution of this dissertation . . . . . . . . . . . . . . . 13

1.3 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Outlines of the dissertation . . . . . . . . . . . . . . . . . . . . . . 17

2 Numerical Solution of Two Point Boundary Value Problems by Inter-

polating Subdivision Schemes 19

2.1 Interpolating schemes for curve design . . . . . . . . . . . . . . . 20

viii

2.1.1 8-point interpolating scheme . . . . . . . . . . . . . . . . . 21

2.2 Numerical interpolating collocation algorithm . . . . . . . . . . . 24

2.2.1 The collocation algorithm . . . . . . . . . . . . . . . . . . . 26

2.2.2 Adjustment of boundary conditions . . . . . . . . . . . . . 29

2.2.3 Existence of the solution . . . . . . . . . . . . . . . . . . . . 33

2.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Numerical examples and discussions . . . . . . . . . . . . . . . . . 38

2.5 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . 41

3 Subdivision Schemes Based Collocation Algorithms for the Solution

of Fourth Order Boundary Value Problems 45

3.1 Basic properties of the schemes . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Subdivision matrices . . . . . . . . . . . . . . . . . . . . . 47

3.1.2 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Description of numerical algorithms . . . . . . . . . . . . . . . . . 53

3.2.1 Collocation algorithms . . . . . . . . . . . . . . . . . . . . . 53

3.2.2 Interpolating collocation algorithm . . . . . . . . . . . . . . 56

3.2.3 Approximating collocation algorithm . . . . . . . . . . . . 57

3.2.4 Boundary conditions at end points . . . . . . . . . . . . . 61

3.2.5 Approximation of derivative boundary conditions . . . . 62

3.2.6 Adjustment of boundary conditions . . . . . . . . . . . . . 63

3.2.7 Stable systems of linear equations . . . . . . . . . . . . . . 65

3.2.8 Stable system for interpolating collocation algorithm . . . 65

3.2.9 Stable system for approximating collocation algorithm . . 68


3.2.11 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3 Numerical examples and comparison . . . . . . . . . . . . . . . . 74

3.3.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 74

3.3.2 Comparison: . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

ix

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 A Subdivision Based Iterative Collocations Algorithm for Nonlinear

Third Order Boundary Value Problems 85

4.1 Existence and uniqueness of the solution . . . . . . . . . . . . . . 86

4.2 Subdivision scheme and basis function . . . . . . . . . . . . . . . . 87

4.2.1 Interpolating subdivision scheme . . . . . . . . . . . . . . 87

4.2.2 Basis function and their derivatives . . . . . . . . . . . . . 87

4.3 Subdivision based iterative algorithm . . . . . . . . . . . . . . . . 89


4.3.2 Unstable nonlinear system . . . . . . . . . . . . . . . . . . 92

4.3.3 Stable nonlinear system . . . . . . . . . . . . . . . . . . . . 92

4.3.4 Approximated boundary condition . . . . . . . . . . . . . 93

4.3.5 Imposed boundary conditions . . . . . . . . . . . . . . . . 93

4.3.6 Non-singularity of a matrix . . . . . . . . . . . . . . . . . . 95

4.3.7 Iterative algorithm and its convergence . . . . . . . . . . . 97

4.3.8 Iterative algorithm based on basis function . . . . . . . . . 97

4.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.5 Examples, comparison and conclusion . . . . . . . . . . . . . . . . 104

4.5.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 104

4.5.2 Comparison and discussion . . . . . . . . . . . . . . . . . . 107

4.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5 A Numerical Approach Based on Subdivision Schemes for Solving

Nonlinear Fourth Order Boundary Value Problems 122

5.1 Basis functions and their derivatives . . . . . . . . . . . . . . . . . 123

5.1.1 Interpolating subdivision scheme . . . . . . . . . . . . . . 123

5.1.2 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2 Description of iterative numerical algorithm . . . . . . . . . . . . 125

x


5.2.2 Boundary conditions at end points . . . . . . . . . . . . . 130

5.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 130

5.2.4 Extrapolation method . . . . . . . . . . . . . . . . . . . . . 131

5.2.5 Non-singularity of a matrix . . . . . . . . . . . . . . . . . . 134

5.2.6 Iterative algorithm and its convergence . . . . . . . . . . . 134

5.2.7 Iterative algorithm based on basis function . . . . . . . . . 135

5.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.4 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . 142

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6 Subdivision Based Collocation Algorithm for Singularly Perturbed Bound-

ary Value Problems 147

6.1 Derivation of numerical algorithm . . . . . . . . . . . . . . . . . . 148

6.1.1 First and second derivatives of Ψ(x) . . . . . . . . . . . . . 149

6.1.2 The subdivision based collocation algorithm . . . . . . . . 149

6.1.3 Singularly perturbed linear system of equations . . . . . . 152

6.1.4 Compelled conditions . . . . . . . . . . . . . . . . . . . . . 153

6.1.5 Stable singularly perturbed linear system of equations . . 156


6.2 Convergence of the method . . . . . . . . . . . . . . . . . . . . . . 158

6.3 Numerical examples and discussions . . . . . . . . . . . . . . . . . 161

6.3.1 Results and discussion . . . . . . . . . . . . . . . . . . . . . 162

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7 A Subdivision Collocation Algorithm for Solving Two point Boundary

value Problems of Order Three 180

7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.1.1 Third order singularly perturbed BVP . . . . . . . . . . . . 181

xi

7.1.2 Subdivision scheme and derivatives . . . . . . . . . . . . . 181

7.2 Subdivision collocation algorithm . . . . . . . . . . . . . . . . . . 182

7.2.1 Singularly perturbed system . . . . . . . . . . . . . . . . . 184

7.2.2 Approximation of derivative conditions . . . . . . . . . . . 185

7.2.3 Necessitated conditions . . . . . . . . . . . . . . . . . . . . 186

7.2.4 Stable linear system of equations . . . . . . . . . . . . . . . 187

7.3 Convergence of collocation algorithm . . . . . . . . . . . . . . . . 188

7.4 Numerical results and discussion . . . . . . . . . . . . . . . . . . . 191

7.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8 Comparison, Conclusions, Limitations and Future Work 203

8.1 Comparison and Conclusion . . . . . . . . . . . . . . . . . . . . . . 203

8.1.1 Comparison with existing methods . . . . . . . . . . . . . 204

8.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.2 Limitations of algorithms . . . . . . . . . . . . . . . . . . . . . . . 205

8.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Bibliography 207

Publications of Syeda Tehmina Ejaz 222

xii

List of Tables

2.1 Determinants of the matrices: . . . . . . . . . . . . . . . . . . . . . 34

2.2 Solutions and error estimation of Example 2.4.1: . . . . . . . . . . 40

2.3 Solutions and error estimation of Example 2.4.2: . . . . . . . . . . 42

3.1 Eigenvalues and eigenvectors of the matrix S1 . . . . . . . . . . . 50

3.2 Eigenvalues and eigenvectors of the matrix S2 . . . . . . . . . . . 51

3.3 Derivatives of ϕ at cardinal data by (3.8) . . . . . . . . . . . . . . . 54

3.4 Derivatives of Φ at cardinal data by (3.11) . . . . . . . . . . . . . . 55

3.5 Determinants of the matrices . . . . . . . . . . . . . . . . . . . . . 73

3.6 Numerical results of Example 3.3.1 . . . . . . . . . . . . . . . . . . 78



3.9 Maximum absolute errors of Examples 3.3.1, 3.3.2 and 3.3.3 . . . . 83

3.10 Comparison of Example 3.3.1 with different methods . . . . . . . 84

4.1 Numerical results of Example 4.5.1: h = 10−1 . . . . . . . . . . . . 109

4.2 Numerical results of Example 4.5.2: h = 10−1 . . . . . . . . . . . . 110

4.3 Numerical results of Example 4.5.3 : h = 10−1 . . . . . . . . . . . . 111

4.4 Numerical results of Example 4.5.4 : k = 0 and h = 10−1 . . . . . . 112

4.5 Numerical results of Example 4.5.4 : k = 12

and h = 10−1 . . . . . . 113

4.6 Numerical results of Example 4.5.4 : k = 2 and h = 10−1 . . . . . . 114


xiii


6.1 First and second derivatives of Ψ . . . . . . . . . . . . . . . . . . . 149

6.2 Maximum absolute errors of Example 6.3.1 . . . . . . . . . . . . . 164












7.1 Maximum absolute errors for N = 10 of Example 7.4.1 . . . . . . 193



7.4 Maximum absolute errors for N = 10 of Example 7.4.2 . . . . . . 196



xiv

List of Figures

1.1 (a) represents primal binary and (b) represents dual binary scheme. . . 16

2.1 Interpolatory basis function ϕ3(i) is shown in (a), first derivative of

ϕ3(i) i-e. ϕ′3(i) shown in (b), second derivative of ϕ3(i) i-e. ϕ′′

3(i) shown

in (c) and third derivative of ϕ3(i) i-e. ϕ′′′3 (i) shown in (d) respectively . 25

2.2 Comparison between analytic and approximating solutions Example 2.4.1. 43

2.3 Comparison between analytic and approximating solutions of Example

2.4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Comparison between analytic and approximate solutions of Example

3.3.1 obtained by interpolating and approximating collocation algorithm-

s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81



s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81



s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xv

4.1 Comparison of the analytic and approximate solution of Example 4.5.1

by proposed algorithm and Caglar et al. (1999). In this figure solid line

shows exact solution, dotted lines show approximate solution by pro-

posed algorithm and dashed lines show the solution obtained by Caglar

et al. (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115


by proposed algorithm and Hasan (2012). In this figure solid line shows

exact solution, dotted lines show approximate solution by proposed al-

gorithm and dashed lines show the solution obtained by Hasan (2012). . 116


by proposed algorithm. In this figure solid line shows exact solution and

dashed lines show approximate solution by proposed algorithm. . . . . . 117

4.4 Comparison between exact and approximate solutions of Example 4.5.4

for k = 0. Solid line represents exact solution and dash line represents

approximate solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 118







4.7 Approximate solutions of Examples 4.5.1, 4.5.2 and 4.5.3 at different

step sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.1 Graphical representation of basis functions is shown in figure (a), and

first, second, third and fourth derivatives of basis function are shown in

figure (b), (c), (d) and (e). . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2 Comparison of the analytic and approximate solution of Example 5.4.1. 144

5.3 Comparison of the analytic and approximate solution of Example 5.4.2. 146

xvi

6.1 Physical behavior of analytic and approximate solutions of Example

6.3.1 for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c)

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174


6.3.1 for N = 32 and ε = 2−25. . . . . . . . . . . . . . . . . . . . . . 175


6.3.1 for N = 32 and ε = (2−20)2. . . . . . . . . . . . . . . . . . . . . 175



respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176


6.3.2 for N = 16 and ε = 10−8. . . . . . . . . . . . . . . . . . . . . . 177


6.3.2 for N = 32 and ε = 10−9. . . . . . . . . . . . . . . . . . . . . . 177



respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178


6.3.3 for N = 16 and ε = 10−5. . . . . . . . . . . . . . . . . . . . . . 179


6.3.3 for N = 16 and ε = 10−8. . . . . . . . . . . . . . . . . . . . . . 179

7.1 Comparability of analytic and approximate solutions of Example 7.4.1

for N = 100 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 193


for N = 200 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 196


for N = 100 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 199

xvii


for N = 200 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 199


for N = 250 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 200


for N = 300 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 200


for N = 250 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 201


for N = 300 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 201

xviii

Chapter 1

Introduction

This chapter provides a brief introduction to the subject and historical develop-

ments in the fields of computer aided geometric design, subdivision schemes

and different methods for the numerical solutions of boundary value problems.

1.1 Computer aided geometric design

Computer Aided Geometric Design (CAGD) is a branch of computational math-

ematics that studies methods and algorithms for the mathematical description

of shapes. It is concerned with construction and representation of free form

curves and surfaces given by a set of points using polynomial, rational piece-

wise polynomial or piecewise rational methods. This branch is closely related

to several other branches, such as geometric modeling. For example NURBS ob-

ject represent the fundamental structures of modern computer system used in

the aircraft and car industry, or data fitting (interpolation, approximation of set

of points). CAGD is the basis for modern design in most branches of industry,

from naval and aeronautic to textile industry.

One of the main issues, in designing a geometric modeler is selecting a math-

ematical representation for curves and surfaces. Selecting a particular repre-

1

sentation is important because later manipulations and analysis depend greatly

on the specific representation. Different techniques have been used for curve

and surface designing. Subdivision schemes are the latest among these and are

being used, most commonly, in geometric modeling.

Subdivision schemes generate smooth curves and surfaces in an efficient way

from the discrete set of control points. This is consistent and efficient iterative

algorithm to be used for modeling of curve and surfaces. During the past two

decades, much research has been undertaken to construct and analyze new sub-

division algorithms for curves/surfaces. Moreover, a number of mathematical

and numerical procedures have been established to improve geometric analysis

of subdivision surfaces. Subdivision schemes can be classified into two impor-

tant branches approximating and interpolating ones. Approximating scheme

means that the limit curve approximates the initial polygon and that after sub-

division, only the newly generated control points are in the limit curve. While

interpolating scheme means that after subdivision, the control points of the o-

riginal control polygon and the newly generated control points both lie on the

limit curve. Following are the advantages and disadvantages of interpolating

and approximating subdivision schemes in the field of geometric modeling:

• Interpolating schemes are more useful for engineering applications, espe-

cially the schemes with the shape control but approximating schemes do

not satisfy the shape control property.

• Interpolating subdivision schemes have the drawback that in order to cre-

ate smoother curves, it is necessary to enlarge the support of the mask.

The designers in geometric modeling require subdivision schemes to have

their masks with a possibly smaller support and to create good smooth

curves.

2

• Approximating schemes yield smoother curves with smaller support as

compare to the interpolating schemes.

In modern era, subdivision techniques can be consider as an integral part of

image processing, graphic visualization, engineering and medical science etc. It

can also be used to develop an iterative algorithms for the solution of different

types of boundary value problems.

1.1.1 Literature review

It was de Rham (1956) who initiated the idea of subdivision techniques. This

idea was further proceeded by Dyn et al. (1987). He introduced a family of

schemes with the complexity of size four. This family was index by a tension

parameter to control the shape of curves. The number of points inserted be-

tween two consecutive levels of subdivision rules is called arity of the scheme.

Even-ary (odd-ary) schemes insert the even (odd) number of points between t-

wo consecutive levels of iterations. A succinct review of higher arity even-point

and odd-point schemes is presented below.

Even point binary and ternary interpolating symmetric subdivision schemes

are introduced by Ko et. al. (2007). Mustafa and Khan (2009) introduced a

new 4-point C3 quaternary approximating subdivision scheme. Even- and odd

-schemes for curve design were offered by Lian (2009). He introduced 2m-point

non-parametric interpolating schemes. A ternary even symmetric 2n- point

subdivision scheme was introduced by Zheng et al. (2009b). The generaliza-

tion of B-splines into p-ary subdivision schemes was introduced by Zheng et al.

(2009c). The unification of existing even-point interpolating and approximat-

ing schemes was offered by Mustafa and Rehman (2010). They offered general

formula to generate the mask of (2m + 4)-point even-ary schemes. Mustafa et.

al. (2014b) offered two families of (2n)-point and (2n − 1)-point p-ary interpo-

lating subdivision schemes, for any integers n > 2 and p > 3. These schemes

3

were originated from Lagrange polynomial. The ternary and three point uni-

variate schemes were offered by Hassan and Dodgson (2003). A 4-point ternary

interpolating subdivision scheme was introduced by Hassan et al. (2002). Lian

(2008) introduced two a-ary schemes with complexity 3 and 4. A family of non-

parametric interpolating odd-ary schemes with complexity (2m + 1) for curve

design was offered by Lian (2009). A family of (2n − 1)-point ternary interpo-

latory subdivision schemes was also introduced by Zheng et al. (2009a). The

generation of mask of (2n − 1)-point interpolating as well as approximating

schemes by an explicit formula was offered by Aslam et al. (2011). Mustafa et

al. (2012) also introduced an explicit formula which help out to find the mask of

odd-points n-ary interpolating schemes.

Different techniques have been used for the construction of subdivision schemes.

Baccou and Liandrat (2013) developed a new type of binary interpolatory sub-

division scheme through a stochastic approach. Their construction combines

position dependent multi scale approximation and kriging theory.

Zheng et al. (2014) presented class of convergent binary subdivision schemes

with high continuity based on eigenvalues of their difference matrix and the re-

lation between subdivision schemes and difference schemes. Rehan and Siddiqi

(2015) presented a combined binary 6-point interpolating and approximating

subdivision scheme with tension parameters.

Deng and Ma (2016) presented an efficient algorithm for the construction of

binary and ternary subdivision schemes with polynomial reproduction proper-

ty. Si et al. (2016) presented a new binary scheme called penalized Lagrange.

Their construction is based on an original reformulation for the construction of

the coefficients of the mask associated to the classical 4-points Lagrange inter-

polatory subdivision scheme. The main purpose of their work is to introduced

a new approach that allows to transform locally an interpolatory scheme into a

non-interpolatory once.

4

Deng et al. (2016) presented the m−ary 2N−point Deslauriers and Dubuc

(1989) subdivision scheme (DDSS) using a series of repeated local operations.

Nowadays subdivision schemes are constructed by using statistical techniques.

Dyn et al. (2015) presented univariate subdivision schemes based on least square

minimization to deal noisy data. Mustafa et al. (2015) used l1-regression to con-

struct subdivision schemes. Their schemes gives best fit to any type of data with

and without added noise and outliers in high dimensional spaces.

1.2 Boundary value problems

A boundary value problem is a problem (BVP), typically an ordinary differen-

tial equation or a partial differential equation, which has values assigned on the

physical boundary of the domain in which the problem is specified. Bound-

ary value problems arise in several branches of physics and engineering. For

the duration of the past period there has been a significant progress of interest

in problems associated with system of linear and nonlinear ordinary differential

equations with split boundary conditions. All over engineering and applied sci-

ence, we are challenged with two point higher order boundary value problems

that cannot be solved by analytic methods. With this interest in finding the so-

lutions of linear and nonlinear boundary value problems, different algorithms

have become an increasing need for researchers.

1.2.1 Numerical methods for BVPs

In modern development of Mathematics there are so many research problems

occurs in the form of differential equation with some conditions and researchers

work on them day by day. Numerical solutions of various problems described

by differential equations involving parameters have become increasingly com-

plex. Therefore we require the use of asymptotic methods. Several numerical

5

methods have been introduced to find the solution of boundary value problems

such as spline based method, finite difference method, finite element method

etc.

The theory of spline functions is a very interesting and active field of approx-

imation theory and boundary value problems, when numerical characteristics

are considered. Many researchers have used spline functions to construct the

algorithms for the solutions of boundary value problems. There are different

types of spline functions such as linear, quadratic, cubic, quartic, quintic, sextic,

septic, octic, nonic etc. known as polynomial spline function.

Spline functions used in the context of boundary value problems was first s-

tudied by Bickley (1968) for the solution of linear two point second order bound-

ary value problems. After that Ablasiny and Hoskins (1969), Fyfe (1969, 1970),

Sakai (1971) developed spline based method both for the linear and nonlinear

two point boundary value problems.

Caglar et al. (1999) presented a collocation algorithm based on fourth degree

B-spline for the numerical solution of third order linear and nonlinear boundary

value problems. They tested their algorithm on linear and nonlinear boundary

value problems and showed that their algorithm achieved first order accuracy.

A numerical approach based on quartic spline known as quartic spline inter-

polation to approximate the solution of second order boundary value presented

by Hamid et al. (2012). They tested their approach on several examples and

the results were compared with those of cubic B-spline and extended cubic B-

spline. They observed that their approach gives more accurate numerical results

comparative to the others.

Pandey (2016) constructed an algorithm by using quartic splines for the solu-

tion of third order boundary value problems. He also test the proposed algo-

rithm on some problems and illustrated numerical results to show the efficiency

of algorithm.

6

El-Danaf (2008) developed quartic non polynomial spline method for the nu-

merical solution of third order two point boundary value problems. They shown

that their method gives approximations better than those produced by other s-

pline methods. Convergence analysis of the method is discussed through stan-

dard procedures.

Islam and Shirin (2011) presented a method for the numerical solution of

linear and nonlinear differential equation with Dirichlet, Neumann and Robin

boundary conditions. They approximated the solution by Galerkin approxima-

tion method using bernoulli polynomials.

Rahman et al. (2012) solved numerically a second order linear boundary val-

ue problem, by the technique of Galerkin method. For this, they derived a sim-

ple and efficient matrix formulation using Hermite polynomials as trial func-

tions. The proposed method is tested on several numerical examples of sec-

ond order linear boundary value problems with Neumann and Cauchy types

boundary conditions. Wazwaz (2000, 2001a) displayed Adomian strategy to

take care of boundary value problems with Dirichlet and Neumann condition-

s. He (2001b) likewise introduced a dependable calculation for getting posi-

tive answers for nonlinear boundary value problems. Wazwaz (2001c) has fur-

ther defended the legitimacy of utilizing the disintegration strategy when mixed

boundary conditions were utilized to acquire explode arrangements. Wazwaz

(2001d) introduced numerical aftereffects of fifth order boundary value prob-

lems by utilizing the disintegration strategy. He additionally contrasted mis-

takes acquired by their technique and the errors got by utilizing the 6th degree

B-spline strategy. Their outcomes demonstrates that the deterioration technique

was more precise and simple than B-spline strategy.

Duan and Rach (2011) proposed a new modified recursion scheme for the

resolution of multi-order and multi-point boundary value problems for nonlin-

ear ordinary and partial differential equations by the Adomian decomposition

7

method.

Hasan (2012) presented a new modification of the Adomian decomposition

method to overcome difficulties occurred in the standard Adomian decompo-

sition method for solving three point boundary value problems, named as the

modified Adomian decomposition method. They developed their technique

for the numerical solution of linear and nonlinear third order boundary value

problems. They showed the efficiency of their modified method by illustrating

numerical examples.

Abdullah et al. (2013) presented a method for the solution of third order non-

linear boundary value problem using fourth order block method. Convergence

of their algorithm is checked by Newton’s method. Their method is simple,

efficient and numerical results shows that their method gives better results as

compared to others.

Kalyani and Rao (2013) have used non-polynomial spline functions to con-

struct a numerical algorithm to obtained the solution of second order boundary

value problem. They have shown that non-polynomial spline produced more

accurate results in comparison with the results obtained by the finite difference

method and B-spline method.

A detail discussion on the numerical and asymptotic study of some third or-

der ordinary differential equations relevant to draining and coating flows is giv-

en by Tuck and Schwartz (1990). Duffy and Wilson (1997) described the prop-

erties of the nonlinear third order differential equation y′′′ = y−2 relevant to

Tanner’s law. The solution of this type of problems first time given by Ford

(1992). Macroscopic thin liquid films are entities that are important in biophysic-

s, physics, and engineering, as well as in natural settings. Oron et al. (1997) re-

viewed, a unified mathematical theory that takes advantage of the disparity of

the length scales and is based on the asymptotic procedure of reduction of the

full set of governing equations and boundary conditions to a simplified, highly

8

nonlinear, evolution equation or to a set of equations.

Craster and Matar (2009) are reviewed these problems and exciting research

avenues in this thriving area of fluid mechanics. Numerical investigation of this

type of problem also presented by Momonait (2011), they solved the differential

equation from thin film flow by comparing two finite difference methods.

A mathematical analysis for boundary value problem which arises in the

model for flows draining down a dry vertical wall is presented by Wang and

Zhang (1998). They also discussed the existence and qualitative properties of

solutions. A new algorithm for solving the general nonlinear third order differ-

ential equation is developed by Bhrawy and Abd-Elhameed (2011) using shifted

Jacobi-Gauss collocation spectral method.

Yalcinbas et al. (2016) presented a numerical approach to approximate the

solution of nonlinear boundary value problems. Their technique is based on the

truncated Fermat series and its matrix representation with collocation points.

Their technique is easy to implement and produced accurate results.

El-Salam and Zaki (2010) developed a class of accurate methods based on

quartic non polynomial spline function at midknots for the numerical solution

of a fourth order two point boundary value problems associated with plate de-

flection theory.

Siddiqi and Akram (2008) used quintic spline to construct the algorithm for

the numerical solutions of the fourth order linear special case boundary value

problems. End conditions for the definition of spline are derived, consistent

with the fourth order boundary value problem. They also proved that their

method is a second order convergent.

Sakai and Usmani (1983) presented an algorithm for nonlinear fourth order

two point boundary value problems based on spline function. They developed

two algorithms of order two and four for the continuous approximation of the

solution of a nonlinear fourth order two point boundary value problems.

9

Gupta and Srivastava (2011) presented computational method using cubic B-

spline to solve fourth order boundary value problems. The proposed method

is first used for solution of fourth order boundary value problems and then

extended it for the solution of nonlinear and singular problems.

Usmani (1983) offered finite difference method of order two, four and six for

the numerical solution of fourth order linear boundary value problems. He

also described the sufficient conditions which guarantees the unique solution

of fourth order boundary value problems. He also compared his method with

shooting method and fourth order Runge Kutta method.

The above mention methods are mostly based on finite difference methods

and spline based methods. For finite differences methods, only discrete ap-

proximate values of the unknown y(x) can be obtained. We need further data

processing techniques to get accurate fitted curve to data. For spline interpo-

lation or approximation methods the unknown function y(x) is assumed to be

piecewise polynomial which in turn requires at least piecewise higher order d-

ifferentiability of the function f(x, y, u, v).

To overcome above disadvantages, Qu (1996a) presented a new technique for

the numerical differentiation and integration by using the interpolating subdi-

vision algorithm. The main advantage of his numerical formulae is that they

produce better numerical results if data comes from functions with fractal like

derivatives. Qu and Agarwal (1996b) introduced the subdivision based algo-

rithm for the solution of two point second order boundary value problems.

They used 6-point binary subdivision scheme to construct the algorithm for the

solution of second order linear boundary value problems. They also presented

the numerical implementation of their algorithm and obtained better results as

compared to the others. Qu and Agawal (1997a) formulated an iterative algo-

rithm for the solution of second order singular nonlinear two point boundary

value problems. Their method is basically a collocation method for nonlinear

10

second order two point boundary value problems with singularities at either

one or both of the boundary points. Qu and Agrwal (1997b) used 6-point bi-

nary subdivision scheme to construct an iterative scheme for solving nonlinear

two point boundary value problems. They presented their technique only for

second order boundary value problems. Qu and Agarwal (1998) also offered

a high accuracy algorithm based on subdivision scheme to compute numerical

solutions for two point boundary value problems of differential equations with

deviating arguments.

The numerical treatment of singularly perturbed problems is currently a field,

in which active research is going on these days. Singularly perturbed problem-

s in which the term containing the highest order derivative is multiplied by a

small parameter ε occur in a number of areas of applied mathematics, science

and engineering among them fluid mechanics (boundary layer problems) elas-

ticity (edge effort in shells) and quantum mechanics.

The solution of a singularly perturbed boundary value problem act like a

multi-scale character. The solution varies quickly near at thin transition lay-

er while away from the layer the solution performs regularly and varies slowly.

Therefore many obstacles are met in solving singularly perturbed boundary val-

ue problems using standard numerical methods. In recent development a large

number of methods for different purpose have been established to provide ac-

curate results.

Three standard approaches are common to solve numerically the singularly

perturbed boundary value problems in literature i.e. the finite element method,

the finite difference method and spline approximation method.

Aziz and Khan (2002) and Khan and Aziz (2005) solved second order sin-

gularly perturbed boundary value problems using cubic spline in compression

and in tension. The convergence of their method depends on the choice of the

parameter involved in the method. A numerical method for self adjoint sin-

11

gular perturbation problems based on quintic spline is presented by Bawa and

Natesan (2005). Kadalbajoo and patidar (2002) presented tension spline approx-

imation method, to solve the third order boundary value problems. Kadalbajoo

and Aggarwal (2005) presented fitted mesh B-spline collocation algorithm for

the solution of self adjoint singular perturbation problems. Tirmizi et al. (2008)

presented a generalized scheme based on non-polynomial spline functions for

the solution of singularly perturbed two point boundary value problems. A nu-

merical method based on uniform Haar wavelet for the solution of singularly

perturbed two point boundary value problems is presented by Pandit and Ku-

mar (2014). Kumar and Mehra (2009) designed a wavelet optimized difference

method for singularly perturbed problems. Their method is based on interpo-

lating wavelet transform. Lubuma and Patidar (2006) presented non standard

finite difference scheme for singularly perturbed problems. The finite difference

scheme based methods also presented by Niijima (1980a , 1980b).

Only few results are available for higher order singularly perturbed bound-

ary value problems. The class of third order singularly perturbed boundary

value problems has been solved by Kumar (2002). He used fourth order fi-

nite difference scheme based on non uniform mesh. Akram (2012) solved third

order self-adjoint singulary perturbed boundary problem by using fourth de-

gree spline. Cui and Geng (2008) presented a new numerical method for the

class of third order boundary value problems with a boundary layer at the left

of the underlying interval. Boundary value technique for the solution of class

of third order singulary perturbed boundary value problems is presented by

Valarmatht and Ramanujam (2002). Su-rang et al. (2001) presented a method to

solved singularly perturbed boundary value problem for quasi linear third or-

der ordinary differential equations involving two parameters. A uniform Haar

wavelets method is proposed by Haq et al. (2011) to find approximate solu-

tion of multi-point third order boundary value problems related to flow in a

12

wedge-shaped region, sandwich beam model, system of ordinary differential

equations, nonlinear third order initial value problems in thin film flow. To

check the performance of their method, they compared their method with the

finite difference method, Pade approximant, spline based methods, method of

superposition, method of chasing and method of adjoint operators. The main

advantage of their method is its efficiency and simple applicability as compared

to others.

1.2.2 Contribution of this dissertation

Subdivision based methods for the numerical solution of boundary value prob-

lems have also been used in the literature by Qu and Agarwal (1996, 1997a,

1997b). They developed subdivision based methods only for second order bound-

ary value problems. Higher order boundary value problems have not been

solved by subdivision methods. This motivates us to solve these type of prob-

lems by subdivision algorithms. The following problems have been solved in

this dissertation:

Mustafa and Ejaz (2014) solved linear third order boundary value problems

by using subdivision technique. They used 8-point binary subdivision scheme

to construct collection algorithm for the solution of linear third order boundary

value problems. They compared their technique with B-spline based method

introduced by Caglar et al. (1999).

Ejaz et al. (2015) solved two point fourth order linear boundary value prob-

lems by subdivision based algorithm. They construct two collocation algorithm-

s based on interpolating and approximating subdivision schemes for the solu-

tion of fourth order boundary value problems.

Ejaz and Mustafa (2016) presented an algorithm for the numerical solution of

nonlinear third order boundary value problems. Their algorithm is based on

eight point binary subdivision scheme. Proposed algorithm is stable, conver-

13

gent and give more accurate results than fourth degree B-spline algorithm.

Mustafa et al. (2017) presented an iterative collocation numerical approach

based on interpolating subdivision schemes for the solution of nonlinear fourth

order boundary value problems involving ordinary differential equations. Nu-

merical evidence suggested that their scheme converges to a smooth approxi-

mate solution of nonlinear fourth order boundary value problem. The conver-

gence of their approach is also discussed.

Mustafa and Ejaz presented subdivision based collocation algorithm for the

numerical solution of second order singularly perturbed boundary value prob-

lems. They also discussed convergence of their algorithm. They concluded

that their algorithm for the solutions of singularly perturbed boundary value

problems gives better results comparative to existing methods such as Aziz

and Khan (2002), Bawa and Natesan (2005), Khan and Aziz (2005), Kadalbajoo

and Aggarwal (2005) , Lubuma (2006), Miller (1979), Niijima (1980a), Niijima

(1980b), Kumar and Mehra (2002) and Pandit and Kumar (2014). (Article sub-

mitted )

Mustafa and Ejaz (2017) solved linear third order singularly perturbed bound-

ary value problems by subdivision based collocation algorithm. Convergence

properties of their algorithm is discussed. The comparison of the solutions ob-

tained by their algorithm with B-spline based method is also given.

1.3 Basic definitions

Definition 1.3.1. Parametric curve In a coordinate plane a curve is traced out

by the points (x, y) = (f(t), g(t)) as t varies, x and y are given as functions of a

variable t (called a parameter) by the equations

x = f(t), y = g(t),

which is called parametric equations.

14

Definition 1.3.2. Parametric continuity It is a concept applied to parametric

curve describing the smoothness of parametric value with distance along the

curve. Parametric continuity indicates the smoothness of the motion.

Definition 1.3.3. Order of parametric continuity

• C0: the segments meet at the nodes or joint points.

• C1: first derivatives are equal and continuous at joint points.

• C2: first and second derivatives are equal and continuous at joint points.

• Cn: first through n-th derivatives are equal and continuous at joint points.

Definition 1.3.4. Subdivision It means divide and rule. It is a set of subdivi-

sion rules called subdivision scheme for generating curves and surfaces as a

sequence of successively refined polygons.

Definition 1.3.5. Binary subdivision schemes A general form of univariate bi-

nary subdivision scheme S which maps a control polygon pk = {pki }i∈Z to a

refined polygon pk+1 = {pk+1i }i∈Z defined as

pk+12i+S =

∑j∈Z

a2j+Spki−j, S = 0, 1,

where the set {ai : i ∈ Z} of coefficient is called mask of the subdivision scheme.

The z-transform of the mask a of subdivision scheme can be given as

a(z) =∑i∈Z

aizi,

which is called the symbol or Laurent polynomial of the scheme.

Definition 1.3.6. Primal binary scheme The scheme which keeps or modifies

the old vertices and create new vertex at each old edge of the control polygon is

called primal binary scheme i.e. if edge maps into vertices then scheme is called

primal (see Figure 1.1(a)).

15

Definition 1.3.7. Dual binary scheme The scheme which creates two new ver-

tices at each old edge is called dual binary scheme i.e. if edges maps into edges

then scheme is called dual ( see Figure 1.1(b)).

(a) (b)

Figure 1.1: (a) represents primal binary and (b) represents dual binary scheme.

Definition 1.3.8. Approximating and interpolating subdivision scheme Approx-

imating scheme means that the limit curve approximates the initial polygon and

that after subdivision, only the newly generated control points are in the limit

curve. While interpolating scheme means that after subdivision, the control

points of the original control polygon and the newly generated control points

both lie on the limit curve.

Definition 1.3.9. Necessary condition of convergent A binary subdivision scheme,

with a mask a satisfies the necessary condition of convergence, if∑i∈Z

a2i =∑i∈Z

a2i+1 = 1.

Definition 1.3.10. Continuity of the scheme Continuity refers to the differentia-

bility of the limit curve/surface produced by subdivision process. Subdivision

16

schemes should be continuous of a certain order prior to construction i.e. Cm

continuity means that the first through m-th derivatives are equal and continues

at the joint points.

Definition 1.3.11. Degree of generation The generation degree of a subdivision

scheme is the maximum degree of polynomials that can potentially be generated

by the scheme, provided that the initial data is chosen correctly. Suppose p0 is

polynomial of degree d of initial data f 0i and symbol of the scheme is

a(z) = (1 + z + z2 + · · ·+ zn−1)d+1b(z),

then the limit curve of the refined data fki at any level k is polynomial of de-

gree d. So the condition is necessary and sufficient for the scheme being able to

generate polynomial of degree d.

Definition 1.3.12. Degree of reproduction A subdivision scheme Sa reproduces

polynomials of degree d if it is convergent and if S∞a f 0 = p for any polynomial

p ∈ πd and initial data f 0 = p(t0i ), i ∈ Z.

Definition 1.3.13. Approximation order A convergent subdivision scheme that

reproduces polynomial of degree d has an approximation order d+ 1.

1.4 Outlines of the dissertation

Rest of the dissertation is organized as follows:

In Chapter 2: A numerical interpolating collocation algorithm is formulated,

based on 8-point binary interpolating subdivision schemes for the generation of

curves, to solve the two point third order boundary value problems. Numerical

examples are given to illustrate the algorithm and its convergence.

In Chapter 3: We present two collocation algorithms based on interpolating and

approximating subdivision schemes for the solution of fourth order boundary

17

value problems. Main purpose of this chapter is to explore and seek the appli-

cations of interpolating and approximating subdivision schemes in the field of

boundary value problems along with intrinsic comparison of the results obtain

by subdivision based algorithms.

In Chapter 4: We construct an iterative algorithm for the numerical solution of

non linear third order boundary value problems. This algorithm is based on

eight point binary subdivision scheme. Proposed algorithm is stable, conver-

gent and give more accurate results than fourth degree B-spline algorithm.

In Chapter 5: We present an iterative collocation algorithm based on interpo-

lating subdivision schemes for the solution of non linear fourth order bound-

ary value problems. The convergence of the algorithm is also discussed. It is

proved that the iterative algorithm converges to a smooth approximate solution

provided the boundary value problem is well posed and algorithm is applied

appropriately.

In Chapter 6: Singularly perturbed second order boundary value problems fre-

quently arise in the various field of science and engineering. In this chapter,

we introduce subdivision based collocation algorithm for the solutions of these

types of problems. Numerical results show that the suggested algorithm for the

solutions of singularly perturbed boundary value problems is batter than spline

based methods, finite difference methods and Haar wavelet methods.

In Chapter 7: We propose an algorithm for the numerical solution of self ad-

joint singularly perturbed third order boundary value problems. Convergence

of the subdivision collocation algorithm is also discussed. Numerical examples

are presented to illustrate the proposed algorithm.

In Chapter 8: We present the comparison, conclusion, limitations of proposed

algorithms and future research directions.

18

Chapter 2

Numerical Solution of Two Point

Boundary Value Problems by

Interpolating Subdivision Schemes

In this chapter, we formulate the subdivision based collocation algorithm for the

approximate solution of two point boundary value problems of order three. Our

formulated subdivision based collocation algorithm treats the following types

of boundary value problems:

y′′′(x) = a(x)y(x) + b(x), 0 6 x 6 1

y(0) = α1, y(1) = α2, y′(0) = 0.(2.1)

where a(x) and b(x) are continuous and a(x) > 0 on [0, 1]. In Section 2.1, we

rewrite general form of interpolating subdivision scheme for curve design pre-

sented by Qu (1994) and some related results. The 8-point binary interpolating

scheme and derivatives of its basis function have been also discussed in this sec-

tion. In Section 2.2, a numerical interpolating algorithm of collocation to solve

(2.1) is formulated and its boundary treatments are discussed. In Section 2.3,

approximation properties of this algorithm are given. In Section 2.4, numerical

examples are presented.

19

2.1 Interpolating schemes for curve design

A general compact form of symmetric univariate binary interpolating subdi-

vision scheme presented by Qu (1994) which maps polygon pk = {pki }i∈Z to a

refined polygon pk+1 = {pk+1i }i∈Z is defined by

pk+12i = pki ,

pk+12i+1 =

n∑j=0

Ln,j

(pki−j + pki+j+1

),

(2.2)

where n is called the degree of the scheme and the constants are given by

Ln,j =((2n+ 1)!!)2

2(4n)(2n+ 1)!

(−1)j

(2j + 1)

2n+ 1

n− j

, j = 0, 1, 2, · · · , n, (2.3)

where

2n+ 1

n− j

denotes the binomial coefficient.

The boundary treatments are necessary to produce smooth curve segments by

scheme (2.2). Normally higher order approximation formulae should be used

near the ends of the segments and thus Lagrange formulae of order 2n + 1 are

recommended.

Remark 2.1.1. Let ϕ(x) be the limit curve generated from the cardinal data {pi =

(i, δ0)T}, that is, ϕ(x) is the fundamental solution of the subdivision scheme (2.2),

then

ϕ(i) =

1, i = 0,

0, i = 0.(2.4)

Furthermore, ϕ(x) satisfies the following two-scale equation

ϕ(x) = ϕn(x) = ϕ(2x) +n∑

j=−n

Ln,|j|ϕ(2x− 2j + 1), x ∈ R. (2.5)

20

Lemma 2.1.1. ( Qu (1994), Qu and Agarwal (1995)). The support of the fundamental

solution ϕn(x) to scheme (2.2) is finite. Explicitly, supp ϕn(x) = (−2n− 1, 2n+ 1).

Lemma 2.1.2. ( Qu and Agarwal (1996)). Given a square matrix A of order n, let the

normalized left and right (generalized) eigenvectors of A be denoted by {ηi, ξi}. Then

for any vector f ∈ Rn, then there exist following Fourier expansion

f =n∑

i=1

(fTηi)ξi.

Lemma 2.1.3. ( Qu and Agarwal (1996)). Suppose F (t), t ∈ R is a regular and C2n+2

curve in Rm, m ≥ 2. Let P (t), t ∈ R be the limit curve generated by (2.2) from the

initial data Pi = F (ih), i ∈ Z, 0 < h < 1. Then, on any finite interval [a, b], following

estimates hold

||F (ht)− p(t)||∞ ≤ M2n+2(F )

(2n+ 2)!h2n+2 = O(h2n+2), (2.6)

and

||hjF j(ht)− pj(t)||∞ = O(h2n+2−j), j = 0, 1, 2, · · · , n+ 2

2, (2.7)

where the number M2n+2(F ) depends only on the derivatives of F (t) and n.

2.1.1 8-point interpolating scheme

For n = 3 by (2.2) and (2.3), we have the following 8-point binary interpolating

subdivision scheme for curve design

pk+12i = pki

pk+12i+1 =

12252048

(pki + pki+1

)− 245

2048

(pki−1 + pki+2

)+ 49

2048

(pki−2 + pki+3

)− 5

2048

(pki−3 + pki+4

).

(2.8)

21

This scheme is C3-continuous by Dyn (2002) and reproduce polynomial curve

of degree seven by Conti and Hormann (2011). The local subdivision matrix of

(2.8) is denoted and defined by

S =

0 0 0 1 0 0 0 0 0 0 0 0 0

L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0

0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0

0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3

0 0 0 0 0 0 0 0 0 1 0 0 0

where L3,0 = 1225

2048, L3,1 = − 245

2048, L3,2 = 49

2048and L3,3 = − 5

2048. The some of its

eigenvalues are

λ = 1, 12, 14, 18, 116, 132, 164, 1128

.

For an eigenvalue λ, the eigenvectors ξ and η that satisfies Sξ = λξ and ηST = ηλ

are called right and left eigenvectors of the matrix S respectively. Some of the

22

normalized left and right eigenvectors corresponding to eigenvalues are

ξ0 = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T ,

η0 = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)T ,

ξ1 = (−6,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, 6)T ,

η1 =(

15946360

)(−5, 1024, 13225,−199680, 1141695,−4715520, 0, 4715520,−1141695,

199680,−13225,−1024, 5)T ,

ξ2 = (36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36)T ,

η2 =(

134546860

)(275,−28160,−182613, 2607616,−12053651, 45634048,−71955030,

45634048,−12053651, 2607616,−182613,−28160, 275)T ,

ξ3 = (−216,−125,−64,−27,−8,−1, 0, 1, 8, 27, 64, 125, 216)T ,

η3 =(

115039360

)(225,−11520, 10952, 476928,−3047987, 4677632, 0,−4677632,

3047987,−476928,−10952, 11520,−225)T .

Since ξTi ηj = 1 for i = j and 0 otherwise then by using Lemmas 2.1.1 and 2.1.2,

we get following result.

Lemma 2.1.4. The fundamental solution (Cardinal basis) ϕ(x) is thrice continuously

differentiable and supported on (−7, 7) and its derivatives at integers are given by

ϕ′(i) = 2sign(i)eT|i|η1, ϕ′′(i) = 22eT|i|η2, ϕ′′′(i) = 23sign(i)eT|i|η3, −6 ≤ i ≤ 6,

where

e0 = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)T , e1 = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0)T ,

e2 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)T , e3 = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ,

e4 = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T , e5 = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ,

e6 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ,

23

and ϕ′(0) = 0, ϕ′(±1) = ∓78592

49553, ϕ′(±2) = ± 76113

198212,

ϕ′(±3) = ∓ 332849553

, ϕ′(±4) = ± 2645594636

, ϕ′(±5) = ± 256743295

,

ϕ′(±6) = ∓ 1594636

,

ϕ′′(0) = −342643

41124, ϕ′′(±1) = 5704256

1079505, ϕ′′(±2) = −12053651

8636040,

ϕ′′(±3) = 3259521079505

, ϕ′′(±4) = − 608712878680

, ϕ′′(±5) = − 704215901

,

ϕ′′(±6) = 551727208

,

ϕ′′′(0) = 0, ϕ′′′(±1) = ∓292352

117495, ϕ′′′(±2) = ±3047987

1879920,

ϕ′′′(±3) = ∓ 331213055

, ϕ′′′(±4) = ∓ 1369234990

, ϕ′′′(±5) = ± 162611

,

ϕ′′′(±6) = ∓ 541776

.

(2.9)

The graphical representations of the basis limit function defined on cardinal

data and its derivatives up to order three for n = 3 are shown in Figure 2.1. Fig-

ure 2.1(a) represents the basis limit function defined in (2.4). Graphical repre-

sentations of first, second and third derivatives of basis limit functions obtained

from (2.2) for n = 3 are shown in Figures 2.1(b), 2.1(c) and 2.1(d) at i = 0, 1,−1,

respectively. The numeric values of first, second and third derivative are given

in Lemma 2.1.4.

2.2 Numerical interpolating collocation algorithm

In this section, first we formulate a numerical interpolating collocation algorith-

m for linear third order two point boundary value problems. Then we settle

down the boundary conditions to get unique solution.

24

Figure 2.1: Interpolatory basis function ϕ3(i) is shown in (a), first derivative of ϕ3(i)

i-e. ϕ′3(i) shown in (b), second derivative of ϕ3(i) i-e. ϕ′′

3(i) shown in (c) and third

derivative of ϕ3(i) i-e. ϕ′′′3 (i) shown in (d) respectively

25

2.2.1 The collocation algorithm

Let N be a positive integer (N ≥ 6), h = 1/N and xi = i/N = ih, i = 0, 1, 2, · · ·N ,

and set ai = a(xi), bi = b(xi). Let

Z(x) =N+6∑i=−6

ziϕ(x−xi

h

), 0 ≤ x ≤ 1, (2.10)

be the approximate solution to (2.1) where {zi} are the unknown to be deter-

mined by (2.1). The collocation algorithm together with the boundary condi-

tions to be discussed, is given by setting

Z ′′′(xj) = a(xj)Z(xj) + b(xj), j = 0, 1, 2, · · · , N, (2.11)

where

Z ′(xj) =1h

N+6∑i=−6

ziϕ′ (xj−xi

h

),

Z ′′(xj) =1h2

N+6∑i=−6

ziϕ′′ (xj−xi

h

),

Z ′′′(xj) =1h3

N+6∑i=−6

ziϕ′′′ (xj−xi

h

).

(2.12)

Using (2.10) and (2.12) in (2.11), we get following N + 1 system of equationsN+6∑i=−6

ziϕ′′′ (xj−xi

h

)− h3aj

N+6∑i=−6

ziϕ(xj−xi

h

)= h3bj, j = 0, 1, 2, · · · , N. (2.13)

Now we simplify the above system in the following theorems.

Theorem 2.2.1. For j = 0 by (2.13), we get

z−6ϕ′′′6 + z−5ϕ

′′′5 + z−4ϕ

′′′4 + z−3ϕ

′′′3 + z−2ϕ

′′′2 + z−1ϕ

′′′1 + z0q0 + z1ϕ

′′′−1 + z2ϕ

′′′−2

+z3ϕ′′′−3 + z4ϕ

′′′−4 + z5ϕ

′′′−5 + z6ϕ

′′′−6 = h3b0, (2.14)

where ϕ′′′j = ϕ′′′(j) and q0 = ϕ′′′

0 − a0h3.

Proof. Substituting j = 0 in (2.13), we get{z−6ϕ

′′′(x0−x−6

h) + z−5ϕ

′′′(x0−x−5

h) + · · ·+ zN+5ϕ

′′′ (x0−xN+5

h

)+ zN+6ϕ

′′′ (x0−xN+6

h

)}−a0h

3{z−6ϕ(

x0−x−6

h) + z−5ϕ(

x0−x−5

h) + · · ·+ zN+5ϕ(

x0−xN+5

h) + zN+6ϕ(

x0−xN+6

h)}

= h3b0.

26

For xj = jh, j = 0, 1, 2 · · ·N , this implies

z−6ϕ′′′(6) + z−5ϕ

′′′(5) + · · ·+ zN+5ϕ′′′(−N − 5) + zN+6ϕ

′′′(−N − 6)

−a0h3 {z−6ϕ(6) + z−5ϕ(5) + · · ·+ zN+5ϕ(−N − 5) + zN+6ϕ(−N − 6)} = h3b0.

Since the support of basis function ϕ(x) is (−7, 7), ϕ′(x), ϕ′′(x) and ϕ′′′(x) are zero

out side the interval (−7, 7), also by (2.4) and (2.9), we get

z−6ϕ′′′(6) + z−5ϕ

′′′(5) + z−4ϕ′′′(4) + z−3ϕ

′′′(3) + z−2ϕ′′′(2) + z−1ϕ

′′′(1) + z0ϕ′′′(0)

+z1ϕ′′′(−1) + z2ϕ

′′′(−2) + z3ϕ′′′(−3) + z4ϕ

′′′(−4) + z5ϕ′′′(−5) + z6ϕ

′′′(−6)

−a0h3z0ϕ(0) = h3b0.

If ϕ′′′j = ϕ′′′(j), then

z−6ϕ′′′6 + z−5ϕ

′′′5 + z−4ϕ

′′′4 + z−3ϕ

′′′3 + z−2ϕ

′′′2 + z−1ϕ

′′′1 + z0(ϕ

′′′0 − a0h

3) + z1ϕ′′′−1

+z2ϕ′′′−2 + z3ϕ

′′′−3 + z4ϕ

′′′−4 + z5ϕ

′′′−5 + z6ϕ

′′′−6 = h3b0.

For q0 = ϕ′′′0 − a0h

3, we get (2.14). This completes the proof.

Theorem 2.2.2. For j = 1, 2, 3, · · · , N the system (2.13) is equivalent to

z−6ϕ′′′j+6 + z−5ϕ

′′′j+5 + · · ·+ z0ϕ

′′′j + z1(ϕ

′′′j−1 − ajh

3ϕj−1) + z2(ϕ′′′j−2 − ajh

3ϕj−2)

+ · · ·+ zN−1(ϕ′′′j−N+1 − ajh

3ϕj−N+1) + zN(ϕ′′′j−N − ajh

3ϕj−N) + zN+1ϕ′′′j−N−1

+ · · ·+ zN+6ϕ′′′j−N−6 = h3bj. (2.15)

Proof. By expanding equation (2.13), we get

z−6ϕ′′′(

xj−x−6

h) + z−5ϕ

′′′(xj−x−5

h) + · · ·+ zN+5ϕ

′′′(xj−xN+5

h) + zN+6ϕ

′′′(xj−xN+6

h)

−ajh3{z−6ϕ(

xj−x−6

h) + z−5ϕ(

xj−x−5

h) + · · ·+ zN+5ϕ(

xj−xN+5

h) + zN+6ϕ(

xj−xN+6

h)}

= h3bj.

27

For xj = jh, j = 1, 2 · · ·N , we get

z−6ϕ′′′(j + 6) + z−5ϕ

′′′(j + 5) + · · ·+ zN+5ϕ′′′(j −N − 5) + zN+6ϕ

′′′((j −N − 6)

−ajh3 {z−6ϕ(j + 6) + z−5ϕ(j + 5) + · · ·+ zN+5ϕ(j −N − 5) + zN+6ϕ(j −N − 6)}

= h3bj.

If ϕ′′′j = ϕ′′′(j), for j = 1, 2 · · ·N then

z−6(ϕ′′′j+6 − ajh

3ϕj+6) + z−5(ϕ′′′j+5 − ajh

3ϕj+5) + · · ·+ zN+5(ϕ′′′j−N−5 − ajh

3ϕj−N−5)

+zN+6(ϕ′′′j−N−6 − ajh

3ϕj−N−6) = h3bj.

Since ϕ′(x), ϕ′′(x) and ϕ′′′(x) are zero out side the interval (−7, 7) then by (2.4)

and (2.9), we get (2.15).

From (2.14) and (2.15), we get following un-determine system of (N + 1) e-

quations with (N + 13) unknowns {zi}

AZ = D, (2.16)

where the matrices A, Z, and D of orders (N + 1)× (N + 13), (N + 13)× 1 and

(N + 1)× 1 respectively are given by

A =

ϕ′′′6 ϕ′′′

5 ϕ′′′4 ϕ′′′

3 ϕ′′′2 ϕ′′′

1 q0 ϕ′′′−1 ϕ′′′

−2 ϕ′′′−3 ϕ′′′

−4 ϕ′′′−5 ϕ′′′

−6

0 ϕ′′′6 ϕ′′′

5 ϕ′′′4 ϕ′′′

3 ϕ′′′2 ϕ′′′

1 q1 ϕ′′′−1 ϕ′′′

−2 ϕ′′′−3 ϕ′′′

−4 ϕ′′′−5

0 0 ϕ′′′6 ϕ′′′

5 ϕ′′′4 ϕ′′′

3 ϕ′′′2 ϕ′′′

1 q2 ϕ′′′−1 ϕ′′′

−2 ϕ′′′−3 ϕ′′′

−4

0 0 0 ϕ′′′6 ϕ′′′

5 ϕ′′′4 ϕ′′′

3 ϕ′′′2 ϕ′′′

1 q3 ϕ′′′−1 ϕ′′′

−2 ϕ′′′−3

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

28

· · · 0 0 0

· · · 0 0 0

· · · 0 0 0

· · · 0 0 0

· · · · · · · · · · · ·

· · · ϕ′′′−5 ϕ′′′

−6 0

· · · ϕ′′′−4 ϕ′′′

−5 ϕ′′′−6

, (2.17)

Z = (z−6, z−5, z−4, z−3, z−2, · · · , zN+6)T , D = (b0h

3, b1h3, b2h

3, b3h3, · · · , bNh3)T ,

where ϕ′′′j = ϕ′′′(j) and qj = ϕ′′′

0 − ajh3.

2.2.2 Adjustment of boundary conditions

The order of the coefficient matrix (2.17) is (N + 1) × (N + 13). In order to get

unique solution of the system, we need twelve more conditions. Here we con-

sider only two different cases. In coming section we will show that the approx-

imate solution can be improved by adjusting different boundary conditions.

Case:-1 If we assume z′0 = 0 (equivalently z0 = yr = finite) then two conditions

can be achieved by using following given boundary conditions i.e.

z0 = yr, z′0 = 0, zN = yl. (2.18)

Still we need ten more conditions to get stable system. Since subdivision scheme

reproduces seven degree polynomials, we define boundary conditions of order

eight for solution of (2.16). For simplicity only the left end points are discussed

and the values of right end points zN+1, zN+2, zN+3, zN+4, zN+5 can be treated

similarly.

29

The values z−5, z−4, z−3, z−2, z−1 can be determined by the septic polynomial q(x)

interpolating at (xi, zi), 0 ≤ i ≤ 7. Precisely, we have

z−i = q(−xi), i = 1, 2, 3, 4, 5,

where

q(xi) =8∑

j=1

8

j

(−1)j+1Z(xi−j).

Since by (2.10) Z(xi) = zi for i = 1, 2, 3, 4, 5 then by replacing xi by −xi, we have

q(−xi) =8∑

j=1

8

j

(−1)j+1z−i+j.

Hence the following boundary conditions can be employed at the left end

8∑j=0

8

j

(−1)jz−i+j = 0, i = 5, 4, 3, 2, 1. (2.19)

Similarly for the right end, we can define zi = q(−xi), i = N + 1, N + 2, N + 3,

N + 4, N + 5 and

q(xi) =8∑

j=1

8

j

(−1)j+1zi−j.

So we have the following boundary conditions at the right end

8∑j=0

8

j

(−1)jzi−j = 0, i = N + 1, N + 2, N + 3, N + 4, N + 5. (2.20)

Finally, we get a following new system of (N + 13) linear equations with (N +

13) unknowns {zi}, in which N + 1 equations are obtained from (2.13), two

equations from boundary conditions (2.18) and ten from boundary conditions

(2.19) and (2.20)

BZ = R, (2.21)

30

where the coefficients matrix B = (BT0 , A

T , BT1 )

T , A is defined by (2.17), B0 and

B1 formed by (2.18), (2.19) and (2.20)

B0 =

0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0 · · · 0 0

0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 · · · 0 0

0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 · · · 0 0

0 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 · · · 0 0

0 0 0 0 0 1 −8 28 −56 70 −56 28 −8 1 · · · 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 · · · 0 0

,

where the first five rows of B0 come from (2.19) and the sixth row comes from

(2.18) at z0 = yr. Consider

B1 =

0 0 · · · 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 · · · 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0 0

0 0 · · · 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0

0 0 · · · 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0

0 0 · · · 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0

0 0 · · · 0 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0

,

where first row of B1 comes from (2.18) at zN = yl remaining rows come from

(2.20) and the matrices Z and R are defined below

Z = (z−6, z−5, · · · zN+5, zN+6)T ,

R = (0, 0, 0, 0, 0, yr, DT , yl, 0, 0, 0, 0, 0)

T .

Case:-2 In this case we express the given boundary condition z′0 = 0 in the follow-

ing way:

By using (2.12) we have

Z ′(xj) =1

h

N+6∑i=−6

ziϕ′(xj − xi

h

).

31

As we define earlier xj = jh if we put j = 0 we get x0 = 0, the above equation

can be written as

Z ′(0) =1

h

N+6∑i=−6

ziϕ′(−i).

Since by boundary condition z′0 = Z ′(0) = 0, so

N+6∑i=−6

ziϕ′(−i) = 0.

By using (2.17) we can express above equation as

1

594636z−6 +

256

743295z−5 +

2645

594636z−4 −

3328

49553z−3 +

76113

198212z−2 −

78592

49553z−1

+78592

49553z1 −

76113

198212z2 +

3328

49553z3 −

2645

594636z4 −

256

743295z5 −

1

594636z6 = 0.

(2.22)

Finally, we get a following new system of (N+13) linear equations with (N+13)

unknowns {zi}, in which N+1 equations are obtained from (2.14) and (2.15), two

equations from boundary conditions (2.18) and ten from boundary conditions

(2.19) for i = 5, 4, 3, 2, (2.20) and (2.22)

BZ = R, (2.23)

where the coefficients matrix B = (BT0 , A

T , BT1 )

T , A is defined by (2.17), B0 and

B1 formed by (2.18), (2.19), (2.20) and (2.22) are defined below

B0 =

0 1 −8 28 −56 70 −56 28 −8

0 0 1 −8 28 −56 70 −56 28

0 0 0 1 −8 28 −56 70 −56

0 0 0 0 1 −8 28 −56 70

1594636

256743295

2645594636 − 3328

4955376113198212 −78592

49553 0 7859249553

76113198212

0 0 0 0 0 0 1 0 0

32

1 0 0 0 0 · · · 0 0

−8 1 0 0 0 · · · 0 0

28 −8 1 0 0 · · · 0 0

−56 28 −8 1 0 · · · 0 0

332849553

− 2645594636

− 256743295

− 1594636

0 · · · 0 0

0 0 0 0 0 · · · 0 0

,

in B first four rows come from (2.19) for i = 5, 4, 3, 2, fifth row comes from (2.22)

and the last row comes from (2.18). The matrix B1 is same as defined in Case 1

and the matrices Z and R are defined below

Z = (z−6, z−5, · · · zN+5, zN+6)T ,

R = (0, 0, 0, 0, y′(0), yr, D

T , yl, 0, 0, 0, 0, 0)T .

The non-singularity of the coefficients matrix B has been discussed in next sec-

tion.

2.2.3 Existence of the solution

In this section, we discuss the non-singularity of the coefficients matrix B. We

observe that the coefficients matrix B is neither symmetric nor diagonally dom-

inant. However it can be shown that B is a non-singular. Since B is almost a

band matrix with half band width 7, numerical complexity for solving the linear

system using Gaussian elimination is about 49(N +9) multiplications. For large

N , the matrix is almost symmetric except the first and last six rows and columns

due to the boundary conditions. Therefore we first consider the symmetric part

33

of it i.e. square band matrix C of order N + 3 defined as

C =

ϕ′′′1 ϕ′′′

0 ϕ′′′−1 ϕ′′′

−2 ϕ′′′−3 ϕ′′′

−4 ϕ′′′−5 ϕ′′′

−6 · · · 0 0 0

ϕ′′′2 ϕ′′′

1 ϕ′′′0 ϕ′′′

−1 ϕ′′′−2 ϕ′′′

−3 ϕ′′′−4 ϕ′′′

−5 · · · 0 0 0

ϕ′′′3 ϕ′′′

2 ϕ′′′1 ϕ′′′

0 ϕ′′′−1 ϕ′′′

−2 ϕ′′′−3 ϕ′′′

−4 · · · 0 0 0

ϕ′′′4 ϕ′′′

3 ϕ′′′2 ϕ′′′

1 ϕ′′′0 ϕ′′′

−1 ϕ′′′−2 ϕ′′′

−3 · · · 0 0 0

0 0 0 0 0 0 0 0 · · · ϕ′′′4 ϕ′′′

3 ϕ′′′2

0 0 0 0 0 0 0 0 · · · ϕ′′′3 ϕ′′′

2 ϕ′′′1

.

It can be shown that C is always non-singular for each value of N . However,

B is non singular for N 6 1000. We have checked the non-singularity of matrix

B by different methods. In first method we observe the determinants of matrix

B increase as N increases and it is not zero for N 6 1000. The determinants of

B at some values of N are shown in Table 2.1. In second method we observe

that for N 6 1000, the eigenvalues of matrix B are non-zero so by Strang (2011)

matrix B is non-singular. However for N > 1000 matrix B may or may not be

non-singular. Therefore we claim that system (2.21) and (2.23) are stable for

N 6 1000.

Table 2.1: Determinants of the matrices:

N C B

10 −8667/56 1/1048870018371741

50 -177183 1/4981270309

100 -552709050 1/1964492

500 -5.033491471916955×1036 4.728852755761116×1021

1000 -4.477989536166907×1071 42069711017699999×1040

34

2.3 Error estimation

In this section, we discuss the approximation properties of the numerical in-

terpolating collocation algorithm. Since the scheme (2.8) reproduce polynomial

curve of degree seven so by Dyn (2002) scheme has approximation order eight.

So the collocation algorithm (2.10) with septic precision treatments at the end-

points has the power of approximation O(h2). Here we present our main result

for error estimation.

Proposition 2.3.1. Suppose the exact solution y(x) ∈ C8[0, 1] and {zi} are obtained

by solving either (2.21) or (2.23) with 8th order boundary condition at the end points,

then

||err(x)||∞ = ||Zj − yj||∞ = O(h2−j), j = 0, 1, 2, 3. (2.24)

here j denotes the order of derivative.

Proof. Since the order of approximation of subdivision scheme (2.8) is eight so

by eigenvector η3, we can write for smooth function y(x) and small h as

y′′′(xj) =23

15039360h3 {225y(xj − 6h)− 11520y(xj − 5h) + 10952y(xj − 4h)

+476928y(xj − 3h)− 3047987y(xj − 2h) + 4677632y(xj − h)− 4677632y(xj + h)

+3047987y(xj + 2h)− 476928y(xj + 3h)− 10952y(xj + 4h) + 11520y(xj + 5h)

−225y(xj + 6h)}+O(h8).

This can be written as

y′′′(xj) =23

15039360h3 {225yj−6 − 11520yj−5 + 10952yj−4 + 476928yj−3

−3047987yj−2 + 4677632yj−1 − 4677632yj+1 + 3047987yj+2

−476928yj+3 − 10952yj+4 + 11520yj+5 − 225yj+6}+O(h8).

(2.25)

Similarly, we have

Z ′′′(xj) =23

15039360h3 {225zj−6 − 11520zj−5 + 10952zj−4 + 476928zj−3

−3047987zj−2 + 4677632zj−1 − 4677632zj+1 + 3047987zj+2

−476928zj+3 − 10952zj+4 + 11520zj+5 − 225zj+6} .

(2.26)

35

If we define error function e(x) = Z(x)− y(x) and error vectors at the nodes by

e(xj) = Z(xj)− y(xj + jh), −6 ≤ j ≤ N + 6,

or equivalently ej = Zj − y(xj + jh), −6 ≤ j ≤ N + 6, then this implies

e′j = Z ′j − y′(x+ jh),

e′′j = Z ′′j − y′′(x+ jh),

e′′′j = Z ′′′j − y′′′(x+ jh).

By subtracting (2.25) from (2.26), we get

Z ′′′(xj)− y′′′(xj) =23

15039360h3 {225(zj−6 − yj−6)− 11520(zj−5 − yj−5) + 10952(zj−4

−yj−4) + 476928(zj−3 − yj−3)− 3047987(zj−2 − yj−2) + 4677632(zj−1 − yj−1)

−4677632(zj+1 − yj+1) + 3047987(zj+2 − yj+2)− 476928(zj+3 − yj+3)

−10952(zj+4 − yj+4) + 11520(zj+5 − yj+5)− 225(zj+6 − yj+6)} .

This implies

e′′′(xj) =1

15039360h3 {225ej−6 − 11520ej−5 + 10952ej−4 + 476928ej−3 − 3047987ej−2

+4677632ej−1 − 4677632ej+1 + 3047987ej+2 − 476928ej+3 − 10952ej+4

+11520ej+5 − 225ej+6} .

By Lemma 2.1.4, we get the following expression

e′′′j =1

h3{ϕ′′′

6 ej−6 + ϕ′′′5 ej−5 + ϕ′′′

4 ej−4 + ϕ′′′3 ej−3 + ϕ′′′

2 ej−2 + ϕ′′′1 ej−1 + ϕ′′′

0 ej

+ϕ′′′−1ej+1 + ϕ′′′

−2ej+2 + ϕ′′′−3ej+3 + ϕ′′′

−4ej+4 + ϕ′′′−5ej+5 + ϕ′′′

−6ej+6

}+O(h8),

(2.27)

where j = 0, 1, 2, · · · , N . By subtracting (2.1) from( 2.11), we get

Z ′′′j − y′′′j = aj (Zi − Yj) .

This implies

e′′′j = ajej, 0 ≤ i ≤ N.

36

Using (2.27), we get

ϕ′′′6 ej−6 + ϕ′′′

5 ej−5 + ϕ′′′4 ej−4 + ϕ′′′

3 ej−3 + ϕ′′′2 ej−2 + ϕ′′′

1 ej−1 + qjej + ϕ′′′−1ej+1

+ϕ′′′−2ej+2 + ϕ′′′

−3ej+3 + ϕ′′′−4ej+4 + ϕ′′′

−5ej+5 + ϕ′′′−6ej+6 = 0, (2.28)

where qj = ϕ′′′0 − h3aj and j = 0, 1, 2, · · · , N .

As 0 6 x 6 1 and xj = jh, j = 0, 1, 2, · · · , N so e0, e1, · · · , eN are non zero while

e−6, e−5, · · · , e−1 and eN+1, eN+2, · · · , eN+6 are zero because they lie outside the

interval [0, 1]. Let us define these (the left and right end) error values as

ej =

max0≤k≤7

{|ek|}O(h8), −6 ≤ j ≤ 0,

maxN−7≤k≤N

{|ek|}O(h8), N ≤ j ≤ N + 6.(2.29)

Thus system (2.28) is equivalent to(B +O(h6)

)E = 0,

where B + O(h6) is the matrix obtained by deleting the first and last six rows

and columns of the matrix B, where

E = (e−6, e−5, e−4, · · · , eN+4, eN+5, eN+6)T .

By using (2.6) and (2.7)(B +O(h6)

)E = O(h8)||Z(xj)− y(xj)|| = O(h8)||E|| = O(h8).

Hence, for small h, the coefficients matrix B +O(h6) will be invertible and thus

using the standard result from linear algebra and effect of ∥B−1∥, we have

||E|| 6(

||B−1||1−O(h6)

O(h8))= O(h2). (2.30)

This completes the proof.

The above discussion suggest that the approximations of the solution com-

puted by the method developed in pervious section are second order accurate

approximations. This suggestion is supported by the numerical experiments

given in the next section.

37

2.4 Numerical examples and discussions

In this section, the numerical collocation algorithm based on 8-point interpolat-

ing subdivision scheme described in Section 2.2, with the 8th order boundary

conditions at the end points, is tested on the two point third order boundary

value problems. Absolute errors in the analytical solutions are also calculated.

For the sake of comparisons, we also tabulated the results in this section.

Example 2.4.1. Consider the boundary value problem

y′′′(x) = y(x)− 3ex, 0 < x < 1,

with boundary conditions y′(0) = 0, y(1) = 0, y(0) = 1. The analytical solution

of this problem is

y(x) = (1− x)ex.

By using the collocation algorithm for N = 10, we get following solution of the

above problem: Zj =16∑

i=−6

ziϕ(j− i), where the values of {z−6, z−5, . . . , z5, z16} are

by using (2.21) are

{24.5275284967525, 0.690493373832105, 0.811531107670859, 0.901997043612006,

0.962835576958380, 0.995101658947808, 1, 0.978925359857389, 0.93350395624740,

0.865636023693632, 0.797539554103408, 0.771795250759200, 0.651392727341382,

0.519777983609805, 0.380902186129123, 0.199271785305208, 0,−0.1311402990079,

−0.247662275449558,−0.342309352611581,−0.406997945734322,

−0.432827137019005,−0.413625213339484},

38

and by using (2.23) are

{0.8188703235427, 0.8701325409827, 0.9139995186898, 0.9498836119009,

0.9768946175522, 0.9940000846665, 1, 0.9934991816499, 0.9728768806892,

0.9362530908960, 0.8814510666459, 0.8059555490326, 0.7068662002854,

0.5808456281783, 0.4240613657868, 0.2321210447258, 0,−0.2780394553965,

−0.6085378480019,−0.9989382153934,−1.457696393548, 1.993136716176,

−2.54764506053343}.

By using two different boundary treatments presented in Section 2.2, we ob-

tained two different solutions which are presented in Table 2.2 along with their

absolute errors. The graphical representation of the analytic and approximate

solutions of above problem is shown in Figure 2.2. Figure 2.2(a) represents the

the comparison of analytic and approximate solutions obtained by (2.21) while

analytic and approximate solutions by (2.23) are shown in Figure 2.2(b). From

this table and figure, we observe that the solution obtained by (2.23) is signif-

icantly better than the solution obtained by (2.21). So our claim, that is, the

approximate solution can be improved by adjusting boundary treatment, is jus-

tified. The maximum absolute errors in the solutions obtained by (2.21) and

(2.23) at step size 10 are 9.755× 10−2 and 2.328× 10−2 respectively.

Example 2.4.2. Consider the following third order boundary values problem

y′′′(x) = xy(x)− (x3 − 2x2 − 5x− 3)ex, 0 < x < 1,

with boundary conditions y(0) = 0 = y(1), y′(0) = 1. Its exact solution is

y(x) = x(1− x)ex.

By the homogeneous process of the boundary condition, let u(x) = y(x)−x(1−x)

then above problem can be transformed into its equivalent form

u′′′(x) = xy(x)+x2(1−x)+(x3−2x2−5x−3)ex, 0 < x < 1, u(0) = u(1) = u′(0) = 0.

39

Tabl

e2.

2:So

luti

ons

and

erro

res

tim

atio

nof

Exam

ple

2.4.

1:

xj

Ana

lyti

cso

luti

onY

App

roxi

mat

eso

luti

onA

ppro

xim

ate

solu

tion

Abs

olut

eer

ror

Abs

olut

eer

ror

Z1

by(2

.21)

Z2

by(2

.23)

by(2

.21)

by(2

.23)

0.0

11

0.00

000

0

0.1

0.99

4653

8262

0.97

8925

3598

5738

90.

9934

9918

780.

0157

2846

630.

0011

5463

837

0.2

0.97

7122

2064

0.93

3503

9562

4740

00.

9728

7689

360.

0436

1825

020.

0042

4531

280

0.3

0.94

4901

1656

0.86

5636

0236

9363

20.

9362

5311

050.

0792

6514

190.

0086

4805

513

0.4

0.89

5094

8188

0.79

7539

5541

0340

80.

8814

5109

030.

0975

5526

470.

0136

4372

850

0.5

0.82

4360

6355

0.77

1795

2507

5920

00.

8059

5557

840.

0525

6538

470.

0184

0505

71

0.6

0.72

8847

5200

0.65

1392

7273

4138

20.

7068

6623

360.

0774

5479

270.

0219

8128

64

0.7

0.60

4125

8121

0.51

9777

9836

0980

50.

5808

4566

790.

0843

4782

850.

0232

8014

41

0.8

0.44

5108

1856

0.38

0902

1861

2912

30.

4240

6141

196

0.06

4205

9995

90.

0210

4677

36

0.9

0.24

5960

3111

0.19

9271

7853

0520

80.

2321

2109

670

0.04

6688

5258

0.01

3839

2144

1.0

00

0.00

000.

0000

.000

00.

000

40

The solutions of this problem and their absolute errors obtained by two differ-

ent boundary treatments are shown in Table 2.3. The graphical representations

of the analytic and approximate solutions are shown in Figure 2.3. Figure 2.3(a)

represents the the comparison of analytic and approximate solutions obtained

by (2.21) and Figure 2.3(b) represents the the comparison of analytic and ap-

proximate solutions obtained by (2.23). We observe that the solution obtained

by (2.23) has less absolute error than that of the solution obtained by (2.21). A-

gain this supports our claim.

Comparison:- The maximum absolute errors in the solutions of Example 2.4.2

obtained by (2.21) and (2.23) at step size 10 are 2.143× 10−2 and 1.437× 10−2 re-

spectively. Caglar et al. (1999) obtained the same maximumabsolute errors but

at the step size 32 and 50 respectively. Therefore we conclude that our method

is more efficient than that of Caglar et al. (1999).

2.5 Conclusion and future work

In this work, we present an interpolatory symmetric subdivision algorithm for

the numerical solution of third order linear problems. Septic polynomials were

used for the adjustment of boundary conditions at the end points. We estab-

lished collocation algorithm and obtained stable system of linear equations which

can be solved by any well-known numerical method. The numerical result

showed that the adjustment of boundary conditions at the end points influence

the accuracy of approximate solution. That is, the accuracy of the solution can

be improved by the proper adjustment of boundary conditions. So our algorith-

m has flexibility to improve the results by adjusting boundary conditions. The

automatic selection and adjustment of the boundary conditions which improve

the approximation order of the solution is possible future research direction.

41

Tabl

e2.

3:So

luti

ons

and

erro

res

tim

atio

nof

Exam

ple

2.4.

2:

xj

Ana

lyti

cso

luti

onY

App

roxi

mat

eso

luti

onA

ppro

xim

ate

solu

tion

Abs

olut

eer

ror

Abs

olut

eer

ror

Z1

by(2

.21)

Z2

by(2

.23)

by(2

.21)

by(2

.23)

0.0

00

0.00

000.

000

0.00

0

0.1

0.09

9465

3826

20.

1142

7450

060.

0971

0744

670

0.01

4809

1179

80.

0023

5793

592

0.2

0.19

5424

4413

0.21

6856

1360

0.18

7602

4920

0.02

1431

6947

0.00

7821

9493

0.3

0.28

3470

3497

0.30

4520

3260

0.27

6779

2681

0.02

1049

9763

0.00

6691

0816

0.4

0.35

8037

9275

0.37

3200

2884

0.34

3710

7723

0.01

5162

3609

0.01

4327

1552

0.5

0.41

2180

3178

0.41

7799

6505

0.40

1831

6295

0.00

5619

3327

0.01

0348

6883

0.6

0.43

7308

5120

0.43

1980

8639

0.42

2936

8864

0.00

5327

6481

0.01

4371

6256

0.7

0.42

2888

0685

0.40

7925

8312

0.41

9957

5084

0.01

4962

2373

0.00

2930

5601

0.8

0.35

6086

5485

0.33

6064

7276

0.34

7344

2367

0.02

0021

8209

0.00

8742

3118

0.9

0.22

1364

2800

0.20

4768

4968

0.20

6999

8108

0.01

6595

7832

0.01

4364

4692

1.0

00

0.00

000.

000

0.00

0

42

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X

Sol

utio

n

Analytic solution YApproximate solution Z

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X

Sol

utio

n


(b)

Figure 2.2: Comparison between analytic and approximating solutions Example 2.4.1.

43

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

X

Sol

utio

n


(a)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

X

Sol

utio

n


(b)

Figure 2.3: Comparison between analytic and approximating solutions of Example

2.4.2.

44

Chapter 3

Subdivision Schemes Based

Collocation Algorithms for the

Solution of Fourth Order Boundary

Value Problems

In this chapter, we present two collocation algorithms based on interpolating

and approximating subdivision schemes for the solution of fourth order bound-

ary value problems. Main purpose of this chapter is to explore and seek the ap-

plications of interpolating and approximating subdivision schemes in the field

of boundary value problems along with intrinsic comparison of the results ob-

tain by algorithms based on these schemes. We consider following interpolat-

ing Deng and Ma (2013) and approximating Mustafa et al. (2014a) subdivision

45

schemes

pk+12i = pki ,

pk+12i+1 =

3565536

(pki−4 + pki+5

)− 405

65536

(pki−3 + pki+4

)+ 567

16384

(pki−2 + pki+3

)− 2205

16384

(pki−3 + pki+4

)+ 19845

32768

(pki + pki+1

),

(3.1)

and

pk+12i = 23

16384(pki−3 + pki+3)− 285

8192(pki−2 + pki+2) +

207316384

(pki−1 + pki+1) +33334096

pki ,

pk+12i+1 = − 3

32768(pki−3 + pki+4)− 33

32768(pki−2 + pki+3)− 1931

32768(pki−1 + pki+2)

+1835132768

(pki + pki+1)

(3.2)

with order of continuity C4. The schemes (3.1) and (3.2) reproduce polynomial

curves of degree nine and three by Conti and Hormann (2011) and Mustafa et al.

(2014a) respectively. Cardinal supports of these schemes are [−8, 8] and [−6, 6]

respectively.

We construct collocation algorithms by using the basis functions of above inter-

polating and approximating subdivision schemes for the numerical solution of

linear fourth order boundary value problems. The mathematical form of fourth

order boundary value problems is given by

y(iv)(x) = a(x)y(x) + b(x), 0 6 x 6 1, (3.3)

subject to the boundary condition

y(0) = α, y′(0) = β, y(1) = γ, y

′(1) = ω (3.4)

where a(x) and b(x) are continuous and a(x) > 0 on [0, 1]. Analytic solution of

such type of boundary value problem is possible only in very rare cases.

46

The outline of this chapter is as follows. In Section 3.1, we construct subdi-

vision matrices of subdivision schemes (3.1) and (3.2) for the computation of

eigenvalues and their corresponding (right and left) eigenvectors. Basis func-

tions and their derivatives have been also discussed in this section. In Sec-

tion 3.2, subdivision based collocation algorithms for solution of fourth order

boundary value problems are formulated. Approximation properties of these

algorithms are also given in Section 3.2. In Section 3.3, numerical examples are

presented. Comparison of approximate solutions by interpolating and approxi-

mating schemes based collocation algorithms is also given. Conclusion is given

in Section 3.4.

3.1 Basic properties of the schemes

In this section, we construct subdivision matrices of the schemes defined in (3.1)

and (3.2) for the computation of eigenvalues and their corresponding eigenvec-

tors. Basis functions of these schemes and their derivatives have also been dis-

cussed in this section.

3.1.1 Subdivision matrices

If S1 and S2 are subdivision matrices of the schemes (3.1) and (3.2) then these

matrices are defined as

47

S1 =

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

,

where L1 =1984532768

, L2 = − 220532768

, L3 =567

16384, L4 = − 405

65536, L5 =

3565536

and

48

S2 =

s1 s2 s3 s4 s3 s2 s1 0 0 0 0 0 0

s′1 s

′2 s

′3 s

′4 s

′4 s

′3 s

′2 s

′1 0 0 0 0 0

0 s1 s2 s3 s4 s3 s2 s1 0 0 0 0 0

0 s′1 s

′2 s

′3 s

′4 s

′4 s

′3 s

′2 s

′1 0 0 0 0

0 0 s1 s2 s3 s4 s3 s2 s1 0 0 0 0

0 0 s′1 s

′2 s

′3 s

′4 s

′4 s

′3 s

′2 s

′1 0 0 0

0 0 0 s1 s2 s3 s4 s3 s2 s1 0 0 0

0 0 0 s′1 s

′2 s

′3 s

′4 s

′4 s

′3 s

′2 s

′1 0 0

0 0 0 0 s1 s2 s3 s4 s3 s2 s1 0 0

0 0 0 0 s′1 s

′2 s

′3 s

′4 s

′4 s

′3 s

′2 s

′1 0

0 0 0 0 0 s1 s2 s3 s4 s3 s2 s1 0

0 0 0 0 0 s′1 s

′2 s

′3 s

′4 s

′4 s

′3 s

′2 s

′1

0 0 0 0 0 0 s1 s2 s3 s4 s3 s2 s1

,

where

s1 =23

16384, s2 = − 285

8192, s3 =

2073

16384, s4 =

3333

4096,

s′

1 = − 3

32768, s

′

2 = − 33

32768, s

′

3 = − 1931

32768, s

′

4 =18351

32768.

The first ten real eigenvalues of matrices S1 and S2 are same which are given

below

λi = 1, 12, 1

4, 1

8, 1

16, 1

32, 1

64, 1

128, 1

256, 1

512, i = 0, 1, 2, · · · , 9.

The remaining eigenvalues are complex which are not required. For above

eigenvalues λi, the eigenvectors υRiand υLi

that satisfies S1υRi= λiυRi

and

υLiST1 = υLi

λi are called right and left eigenvectors of the matrix S1 respectively.

We can also define the right νRiand left νLi

eigenvectors of S2 in a similar way.

The normalized left and right eigenvectors corresponding to first five eigenval-

ues of S1 and S2 are given in Table 3.1 and 3.2 respectively.

49

Tabl

e3.

1:Ei

genv

alue

san

dei

genv

ecto

rsof

the

mat

rixS1

Eige

nval

uesλi

Cor

resp

ondi

ngri

ghta

ndle

ftei

genv

ecto

rs

1υR

0=

(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)T,

υL0=

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)T

1 2υR

1=

(−8,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,8)

T

υL1=

(−1575,−

1474560,−315738080,−1397587968,43588613880,−

4311679549440,1336741045920,

−4824847319040,0,4824847319040,−1336741045920,4311679549440,−43588613880,

1397587968,315738080,1474560,1575)

T/5

8418

8424

5680

1 4υR

2=

(64,49,36,25,16,9,4,1,0,1,4,9,16,25,36,49,64)T

υL2=

(459375,215040000,47660030080,103151616000,−

3882427261296,23490017902592,−

84313449846912,

313469774708736,−497829885297150,313469774708736,−84313449846912,23490017902592,

−3882427261296,103151616000,47660030080,215040000,459375)

T/2

5962

4836

2131

201 8

υR

3=

(−512,−343,−216,−125,−64,−

27,−

8,−1,0,1,8,27,64,125,216,343,512)

T

υL3=

(−104125,−

24371200,1177382520,−5986263040,−

10571778214,207884427264,

−972244098856,

−1386160480256,0,1386160480256,972244098856,−

207884427264,10571778214,5986263040,

−1177382520,24371200,104125)T

/946

0416

8599

041 16

υR

4=

(4096,2401,1296,625,256,81,16,1,0,1,16,81,256,625,1296,2401,4096)T

υL4=

(392875,45977600,−

1296269280,5912719360,1180083476,−

86261280768,332951715808,−

67767008256,

850467338370,−

67767008256,332951715808,−

86261280768,1180083476,5912719360,−1296269280,

45977600,392875)

T/1

8376

8238

080

50

Tabl

e3.

2:Ei

genv

alue

san

dei

genv

ecto

rsof

the

mat

rixS2

Eige

nval

uesλi

Cor

resp

ondi

ngri

ghta

ndle

ftei

genv

ecto

rs

1ν R

0=

(1,1,1,1,1,1,1,1,1,1,1,1,1)T

,ν L

0=

(0,0,0,0,0,0,1,0,0,0,0,0,0)T

1 2ν R

1=

(−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6)

T

ν L1=

(−271,1475866,286711948,−3680077858,73546185237,

−507541369116,0,507541369116,−73546185237,3680077858,

−286711948,−1475866,271)

T/7

4067

0013

440

1 4ν R

2=

(36,25,16,9,4,1,0,1,4,9,16,25,36)T

ν L2=

(−3927,10663114,−582891346,−38170879262,32176934487,

2719571096148,−5426009838428,2719571096148,32176934487,

−38170879262,−582891346,10663114,−

3927)T

/499

1362

1913

601 8

ν R3=

(−216,−125,−64,−

27,−

8,−1,0,1,8,27,−

64,−

125,−216)

T

ν L3=

(23,−31050,−1351836,21998258,−

507201588,193168476,0,

−193168476,507201588,−21998258,1351836,31050,−

23)T

/635

3510

401 16

ν R4=

1 27(51867,25027,10267,3267,667,67,27,67,667,3267,10267,25027,51867)

T

ν L4=

(−6693,4466462,117537274,−

1293454266,6372836901,

−15842276196,21281793036,−15842276196,6372836901,−

1293454266,

117537274,4466462,−6693)T

/421

5029

7600

51

3.1.2 Basis functions

The basis functions for the convergent subdivision schemes (3.1) and (3.2) are

the limit curves ϕ(x) and Φ(x) generated from the cardinal data {pi = (i, δ0)T}.

The ϕ(x) and Φ(x) are also known as fundamental solutions of the subdivision

schemes, so

ϕ(i) = Φ(i) =

1, i = 0,

0, i = 0.(3.5)

Furthermore, ϕ(x) = Φ(x) satisfies the following two-scale equation

ϕ(x) =

p∑j=−p

aiϕ(2x− i), Sup(ϕ) = (−p− 1, p+ 1). (3.6)

The lth derivative of basis function at integers satisfy the relation

ϕ(l)(x) =

p∑j=−p

aiϕ(l)(2x− i), Sup(ϕ) = [−p, p]. (3.7)

Since υTRiυLj

= 1 and νTRiνLj

= 1 for i = j and 0 otherwise then by Qu (1996), we

get following result.

Lemma 3.1.1. The fundamental solution (Cardinal basis) ϕ(x) of the subdivision scheme

(3.1) is fourth times continuously differentiable, supported on [−8, 8] and its derivatives

at integers are given by

ϕ′(t) = 2sgn(t)eT|t|υL1 , ϕ′′(t) = 22eT|t|υL2 ,

ϕ′′′(t) = 23sgn(t)eT|t|υL3 , ϕiv(t) = 24eT|t|υL4 , −8 6 t 6 8, (3.8)

where υLi, 0 6 i 6 4 are defined in Table 3.1, the sgn function of a real number t is

defined as

sgn(t) =

−1, t < 0,

0, t = 0,

1, t > 0,

(3.9)

52

et’s are the column matrices defined as

et = (a8t, a7t, a6t, a5t, a4t, a3t, a2t, a1t, a0t, a−1t, a−2t, a−3t, a−4t, a−5t, a−6t, a−7t, a−8t)T ,

where 0 ≤ t ≤ 8 and

ait =

1, i = t,

0, i = t.(3.10)

Lemma 3.1.2. The fundamental solution Φ(x) of subdivision scheme (3.2) defined in

(3.6) is fourth times continuously differentiable, supported on [−6, 6] and its derivatives

at integers are defined as

Φ′(t) = 2sgn(t)eT|t|νL1 , Φ′′(t) = 22eT|t|νL2 ,

Φ′′′(t) = 23sgn(t)eT|t|νL3 , Φiv(t) = 24eT|t|νL4 − 6 6 t 6 6, (3.11)

where νLifor 0 6 i 6 4 are defined in Table 3.2, the sgn function of a real number t is

defined by (3.9) et’s are the column matrices defined as

et = (a6t, a5t, a4t, a3t, a2t, a1t, a0t, a−1t, a−2t, a−3t, a−4t, a−5t, a−6t)T , 0 6 t 6 6,

where ait are defined by (3.10).

From (3.8) and (3.11), we get values of derivatives at the integers given in

Table 3.3 and Table 3.4 respectively.

3.2 Description of numerical algorithms

In this section, first we formulate two collocation algorithms which are based on

interpolating (3.1) and approximating (3.2) subdivision schemes for the solution

of (3.3). Then we settle down the boundary conditions to get unique solution.

3.2.1 Collocation algorithms

Here we formulate two collocation algorithms based on two subdivision schemes.

These collocation algorithms are defined in coming subsections.

53

Table 3.3: Derivatives of ϕ at cardinal data by (3.8)

i ϕ′(i) ϕ′′(i) ϕ′′′(i) ϕiv(i)

0 0 −2370618501415309077185968

0 33869667457408

±1 ∓19146219521159104017

3265310153216676106344305

±4331751500815295995855

−529505475289730585

±2 ± 5304527961159104017

−878265102572676106344305

∓12153051235761183983420

10404741119358922340

±3 ∓ 147046413780629

7340630594562028319032915

±240606976566518365

−748795849970065

±4 ± 172970691159104017

− 808839012771352212688610

∓ 5285889107244735933680

2950208692871378720

±5 ∓ 27729925795520085

214899200135221268861

∓ 374141443059199171

923862417946117

±6 ∓ 112763610431936153

297875188405663806583

± 1090169453214692

− 9001877976052

±7 ∓ 40968113728119

6400019317324123

∓ 21760437028453

7184017946117

±8 ∓ 59272832136

4375618154371936

∓ 297513984910496

11225328157568

54

Table 3.4: Derivatives of Φ at cardinal data by (3.11)

i Φ′(i) Φ′′(i) Φ′′′(i) Φiv(i)

0 0 −1356502459607311960136960

0 1773482753219532800

±1 ∓4229511409330861250560

226630924679103986712320

±596199245120

−1320189683219532800

±2 ± 350219929717635000320

3575214943138648949760

∓1565437980480

708092989292710400

±3 ∓ 1840038929185167503360

− 19085439631623920273920

±1099912939709440

−215575711439065600

±4 ± 7167798792583751680

− 291445673623920273920

∓ 12517735360

587686371317196800

±5 ± 10541926452500480

5331557623920273920

∓ 115294144

22332311317196800

±6 ∓ 271370335006720

− 1309415946849280

± 2379418880

− 2231878131200

55

3.2.2 Interpolating collocation algorithm

The collocation algorithm based on interpolating scheme (3.1) say interpolating

collocating algorithm is given below. In this algorithm, we assume approximate

solution Z1(x) of (3.3) as

Z1(x) =N+8∑i=−8

ziϕ(x−xi

h

), 0 6 x 6 1, (3.12)

where N is the positive integer N > 8, h = 1/N and xi = i/N = ih, and {zi}

are the unknown to be determined for the solution of (3.3). The collocation

algorithm together with the boundary conditions to be discussed, is given by

Ziv1 (xj) = a(xj)Z1(xj) + b(xj), j = 0, 1, 2, · · · , N, (3.13)

with the following type of boundary conditions

Z1(0) = α, Z′

1(0) = β, Z1(N) = γ, Z′

1(N) = ω, (3.14)

where α, β, γ and ω are constants. Let aj = a(xj), bj = b(xj), then equation (3.13)

can be written as

Ziv1 (xj) = ajZ1(xj) + bj, j = 0, 1, 2, · · · , N, (3.15)

where

Ziv1 (xj) =

1

h4

N+8∑i=−8

ziϕiv

(xj − xi

h

). (3.16)

Using (3.12) and (3.16) in (3.15), we get following N + 1 system of equations

N+8∑i=−8

ziϕiv

(xj − xi

h

)− h4aj

N+8∑i=−8

ziϕ

(xj − xi

h

)= h4bj, j = 0, 1, 2, · · · , N.

(3.17)

56

3.2.3 Approximating collocation algorithm

In approximating collocating algorithm (i.e. algorithm based on approximating

scheme (3.2)), we assume approximate solution Z2(x) of (3.3) as

Z2(x) =N+6∑i=−6

ziΦ(x−xi

h

), 0 6 x 6 1, (3.18)

where N is the positive integer N > 6, h = 1/N and xi = i/N = ih and {zi}

are the unknown to be determined for the solution of (3.3). The collocation

algorithm together with the boundary conditions to be discussed, is given by

Ziv2 (xj) = a(xj)Z2(xj) + b(xj), j = 0, 1, 2, · · · , N, (3.19)


Z2(0) = α, Z′

2(0) = β, Z2(N) = γ, Z′

2(N) = ω. (3.20)

The equation (3.19) can be written as

Ziv2 (xj) = ajZ2(xj) + bj, j = 0, 1, 2, · · · , N, (3.21)

where

Ziv2 (xj) =

1

h4

N+6∑i=−6

ziΦiv

(xj − xi

h

). (3.22)

Using (3.18) and (3.22) in (3.21), we get following N + 1 system of equationsN+6∑i=−6

ziΦiv

(xj − xi

h

)− h4aj

N+6∑i=−6

ziΦ

(xj − xi

h

)= h4bj, j = 0, 1, 2, · · · , N.

(3.23)

Now we simplify the above systems (3.17) and (3.23) in following theorems.

Theorem 3.2.1. Interpolating collocation algorithm: For j = 0 by (3.17), we get

z−8ϕiv−8 + z−7ϕ

iv−7 + z−6ϕ

iv−6 + z−5ϕ

iv−5 + z−4ϕ

iv−4 + z−3ϕ

iv−3 + z−2ϕ

iv−2 + z−1ϕ

iv−1

+z0q0 + z1ϕiv1 + z2ϕ

iv2 + z3ϕ

iv3 + z4ϕ

iv4 + z5ϕ

iv5 + z6ϕ

iv6 + z7ϕ

iv7 + z8ϕ

iv8 = h4b0,

(3.24)

where ϕivj = ϕiv(j) and q0 = ϕiv

0 − a0h4.

57

Proof. Substituting j = 0 in (3.17), we get{z−8ϕ

iv(x0−x−8

h) + z−7ϕ

iv(x0−x−7

h) + · · ·+ zN+7ϕ

iv(x0−xN+7

h) + zN+8ϕ

iv(x0−xN+8

h)}

−a0h4{z−8ϕ(

x0−x−8

h) + z−7ϕ(

x0−x−7

h) + · · ·+ zN+7ϕ(

x0−xN+7

h) + zN+8ϕ(

x0−xN+8

h)}

= h4b0.

For xi = ih, i = 0, 1, 2 · · ·N , this implies

z−8ϕiv(8) + z−7ϕ

iv(7) + · · ·+ zN+7ϕiv(−N − 7) + zN+8ϕ

iv(−N − 8)− a0h4 {z−8ϕ(8)

+z−7ϕ(7) + · · ·+ zN+7ϕ(−N − 7) + zN+8ϕ(−N − 8)} = h4b0.

Since the cardinal support of basis function ϕ(x) is [−8, 8], so ϕ′(x), ϕ′′(x), ϕ′′′(x)

and ϕiv(x) are zero out side the interval [−8, 8], also by Table 3.3, we get

z−8ϕiv(8) + z−7ϕ

iv(7) + z−6ϕiv(6) + z−5ϕ

iv(5) + z−4ϕiv(4) + z−3ϕ

iv(3) + z−2ϕiv(2)

+z−1ϕiv(1) + z0ϕ

iv(0) + z1ϕiv(−1) + z2ϕ

iv(−2) + z3ϕiv(−3) + z4ϕ

iv(−4) + z5ϕiv(−5)

+z6ϕiv(−6) + z7ϕ

iv(−7) + z8ϕiv(−8)− a0h

4z0ϕ(0) = h4b0.

If ϕivi = ϕiv(i), then

z−8ϕiv8 + z−7ϕ

iv7 + z−6ϕ

iv6 + z−5ϕ

iv5 + z−4ϕ

iv4 + z−3ϕ

iv3 + z−2ϕ

iv2 + z−1ϕ

iv1 + z0(ϕ

iv0

−a0h4) + z1ϕ

iv−1 + z2ϕ

iv−2 + z3ϕ

iv−3 + z4ϕ

iv−4 + z5ϕ

iv−5 + z6ϕ

iv−6 + z7ϕ

iv−7 + z8ϕ

iv−8 = h4b0.

As we observe from Table 3.3, ϕiv−i = ϕiv

i , we have


iv−7 + z−6ϕ

iv−6 + z−5ϕ

iv−5 + z−4ϕ

iv−4 + z−3ϕ

iv−3 + z−2ϕ

iv−2 + z−1ϕ

iv−1 + z0(ϕ

iv0

−a0h4) + z1ϕ

iv1 + z2ϕ

iv2 + z3ϕ

iv3 + z4ϕ

iv4 + z5ϕ

iv5 + z6ϕ

iv6 + z7ϕ

iv7 + z8ϕ

iv8 = h4b0.

For q0 = ϕiv0 − a0h


Theorem 3.2.2. Interpolating collocation algorithm: For j = 1, 2, 3, · · · , N the system

(3.17) is equivalent to

z−8ϕiv−j−8 + z−7ϕ

iv−j−7 + · · ·+ z0ϕ

iv−j + z1(ϕ

iv1−j − ajh

4ϕ1−j) + z2(ϕiv2−j − ajh

4ϕ2−j)

+ · · ·+ zN(ϕivN−j − ajh

4ϕN−j) + zN+1ϕivN+1−j + · · ·+ zN+8ϕ

ivN+8−j = h4bj. (3.25)

58


z−8ϕiv(

xj−x−8

h) + z−7ϕ

iv(xj−x−7

h) + · · ·+ zN+7ϕ

iv(xj−xN+7

h) + zN+8ϕ

iv(xj−xN+8

h)

−ajh4{z−8ϕ(

xj−x−8

h) + z−7ϕ(

xj−x−7

h) + · · ·+ zN+7ϕ(

xj−xN+7

h) + zN+8ϕ(

xj−xN+8

h)}

= h4bj.

For xj = jh, j = 1, 2 · · ·N , we get

z−8ϕiv(j + 8) + z−7ϕ

iv(j + 7) + · · ·+ zN+7ϕiv(j −N − 7) + zN+8ϕ

iv(j −N − 8)

−ajh4 {z−8ϕ(j + 8) + z−7ϕ(j + 7) + · · ·+ zN+7ϕ(j −N − 7) + zN+8ϕ(j −N − 8)}

= h4bj.

This implies

z−8(ϕiv(j + 8)− ajh

4ϕ(j + 8)) + z−7(ϕiv(j + 7)− ajh

4ϕ(j + 7)) + · · ·+ zN+7(ϕiv(j

−N − 7)− ajh4ϕ(j −N − 7)) + zN+8(ϕ

iv(j −N − 8)− ajh4ϕ(j −N − 8)) = h4bj.

If ϕivj = ϕiv(j), for j = 1, 2 · · ·N then

z−8(ϕivj+8 − ajh

4ϕj+8) + z−7(ϕivj+7 − ajh

4ϕj+7) + · · ·+ zN+7(ϕivj−N−7 − ajh

4ϕj−N−7)

+zN+8(ϕivj−N−8 − ajh

4ϕj−N−8) = h4bj.

As we observe from Table 3.3, ϕiv−j = ϕiv

j j = 1, 2, · · · , N , then above equation

can be written as

z−8(ϕiv−j−8 − ajh

4ϕ−j−8) + z−7(ϕiv−j−7 − ajh

4ϕ−j−7) + · · ·+ zN+7(ϕivN+7−j

−ajh4ϕN+7−j) + zN+8(ϕ

ivN+8−j − ajh

4ϕN+8−j) = h4bj.

Since ϕ′(x), ϕ′′(x), ϕ′′′(x) and ϕiv(x) are zero out side the interval [−8, 8] then by

Table 3.3, we get (3.25).

Theorem 3.2.3. Approximating collocation algorithm: For j = 0 by (3.23), we get

z−6Φiv−6 + z−5Φ

iv−5 + z−4Φ

iv−4 + z−5Φ

iv−5 + z−4Φ

iv−4 + z−3Φ

iv−3 + z−2Φ

iv−2

+z−1Φiv−1 + z0q0 + z1Φ

iv1 + z2Φ

iv2 + z3Φ

iv3 + z4Φ

iv4 + z5Φ

iv5 + z6Φ

iv6 = h4b0

, (3.26)

where Φivj = Φiv(j) and Υ0 = Φiv

0 − a0h4.

59

Proof. Substituting j = 0 in (3.23), we get{z−6Φ

iv(x0−x−6

h) + z−5Φ

iv(x0−x−5

h) + · · ·+ zN+5Φ

iv(x0−xN+5

h) + zN+6Φ

iv(x0−xN+6

h)}

−a0h4{z−6Φ(

x0−x−6

h) + z−5Φ(

x0−x−5

h) + · · ·+ zN+5Φ(

x0−xN+5

h) + zN+6Φ(

x0−xN+6

h)}

= h4b0.

For xi = ih, i = 0, 1, 2 · · ·N , this implies

z−6Φiv(6) + z−5Φ

iv(5) + · · ·+ zN+5Φiv(−N − 5) + zN+6Φ

iv(−N − 6)− a0h4 {z−6Φ(6)

+z−5Φ(5) + · · ·+ zN+5Φ(−N − 5) + zN+6Φ(−N − 6)} = h4b0.

Since the cardinal support of basis function Φ(x) is [−6, 6], so Φ′(x), Φ′′(x), Φ′′′(x)

and Φiv(x) are zero out side the interval [−6, 6], also by Table 3.4, we get

z−6Φiv(6) + z−5Φ

iv(5) + z−4Φiv(4) + z−3Φ

iv(3) + z−2Φiv(2) + z−1Φ

iv(1) + z0Φiv(0)

+z1Φiv(−1) + z2Φ

iv(−2) + z3Φiv(−3) + z4Φ

iv(−4) + z5Φiv(−5) + z6Φ

iv(−6)

−a0h4z0Φ(0) = h4b0.

If Φivi = Φiv(i), then

z−6Φiv6 + z−5Φ

iv5 + z−4Φ

iv4 + z−3Φ

iv3 + z−2Φ

iv2 + z−1Φ

iv1 + z0(Φ

iv0 − a0h

4) + z1Φiv−1

+z2Φiv−2 + z3Φ

iv−3 + z4Φ

iv−4 + z5Φ

iv−5 + z6Φ

iv−6 = h4b0.

As we observe from Table 3.4, Φiv−i = Φiv

i , we have

z−6Φiv−6 + z−5Φ

iv−5 + z−4Φ

iv−4 + z−3Φ

iv−3 + z−2Φ

iv−2 + z−1Φ

iv−1 + z0(Φ

iv0 − a0h

4) + z1Φiv1

+z2Φiv2 + z3Φ

iv3 + z4Φ

iv4 + z5Φ

iv5 + z6Φ

iv6 + z7Φ

iv7 + z8Φ

iv8 = h4b0.

For Υ0 = Φiv0 − a0h


Theorem 3.2.4. Approximating collocation algorithm: For j = 1, 2, 3, · · · , N the sys-

tem (3.23) is equivalent to

z−6Φiv−j−6 + z−5Φ

iv−j−5 + · · ·+ z0Φ

iv−j + z1(Φ

iv1−j − ajh

4Φ1−j) + z2(Φiv2−j − ajh

4Φ2−j)

+ · · ·+ zN(ΦivN−j − ajh

4ΦN−j) + zN+1ΦivN+1−j + · · ·+ zN+6Φ

ivN+6−j = h4bj. (3.27)

60


z−6Φiv(

xj−x−6

h) + z−5Φ

iv(xj−x−5

h) + · · ·+ zN+5Φ

iv(xj−xN+5

h) + zN+6Φ

iv(xj−xN+6

h)

−ajh4{z−8Φ(

xj−x−8

h) + z−7Φ(

xj−x−7

h) + · · ·+ zN+5Φ(

xj−xN+5

h) + zN+6Φ(

xj−xN+6

h)}

= h4bj.

For xj = jh, j = 1, 2 · · ·N , we get

z−6Φiv(j + 6) + z−5Φ

iv(j + 5) + · · ·+ zN+5Φiv(j −N − 5) + zN+6Φ

iv(j −N − 6)

−ajh4 {z−6Φ(j + 6) + z−5Φ(j + 5) + · · ·+ zN+5Φ(j −N − 5) + zN+6Φ(j −N − 6)}

= h4bj.

This implies

z−6(Φiv(j + 6)− ajh

4Φ(j + 6)) + z−5(Φiv(j + 5)− ajh

4Φ(j + 5)) + · · ·+ zN+5(Φiv(j

−N − 5)− ajh4Φ(j −N − 5)) + zN+6(Φ

iv(j −N − 6)− ajh4Φ(j −N − 6)) = h4bj.

If Φivj = Φiv(j), for j = 1, 2 · · ·N then

z−6(Φivj+6 − ajh

4Φj+6) + z−5(Φivj+5 − ajh

4Φj+5) + · · ·+ zN+5(Φivj−N−5 − ajh

4Φj−N−5)

+zN+6(Φivj−N−6 − ajh

4Φj−N−6) = h4bj.

As we observe from Table 3.4, Φiv−j = Φiv

j j = 1, 2, · · · , N , then above equation

can be written as

z−6(Φiv−j−6 − ajh

4Φ−j−6) + z−5(Φiv−j−5 − ajh

4Φ−j−5) + · · ·+ zN+5(ΦivN+5−j

−ajh4ΦN+5−j) + zN+6(Φ

ivN+6−j − ajh

4ΦN+6−j) = h4bj.

Since Φ′(x), Φ′′(x), Φ′′′(x) and Φiv(x) are zero out side the interval [−6, 6] then by

Table 3.4, we get (3.27).

3.2.4 Boundary conditions at end points

We have two different systems of (N +1) equations defined by (3.17) and (3.23).

In order to get unique solution of these systems, we need sixteen more con-

ditions for the system (3.17) and twelve more conditions for the system (3.23).

61

Four conditions can be achieved from boundary conditions given in (3.4) for

both systems of linear equations in which first order derivatives are involved

and remaining conditions are achieved by some extrapolation method at the

end points. First we find the approximation of the first derivative by difference

operators and after that we define the extrapolation method at end points for

both system of linear equations.

3.2.5 Approximation of derivative boundary conditions

In this section, we approximate the derivative boundary conditions by differ-

ence operators. Since approximation order of interpolating scheme (3.1) and

approximating scheme (3.2) is ten and four respectively. So we approximate

derivative boundary conditions at end points with approximation orders ten

and four for interpolating and approximating collocation algorithms.

If we use interpolating collocation algorithm for the solution of (3.3) then ap-

proximation of derivative conditions at ends point is defined as

Z ′1(0) =

(N

2520

){−7381z0 + 25200z1 − 56700z2 + 100800z3 − 132300z4

+127008z5 − 88200z6 + 43200z7 − 14175z8 + 28800z9 − 252z10}

+O(h10), (3.28)

Z ′1(N) =

(N

2520

){7381zN − 25200zN−1 + 56700zN−2 − 100800zN−3 + 132300zN−4

−127008zN−5 + 88200zN−6 − 43200zN−7 + 14175zN−8 − 28800zN−9

+252zN−10}+O(h10), (3.29)

and if we use approximating collocation algorithm for the solution of (3.3) then

approximation of derivative conditions at end points is defined as

Z ′2(0) =

(N

12

){−25z0 + 48z1 − 36z2 + 16z3 − 3z4}+O(h4), (3.30)

Z ′2(N) =

(N

12

){25zN − 48zN−1 + 36zN−2 − 16zN−3 + 3zN−4}+O(h4). (3.31)

62

3.2.6 Adjustment of boundary conditions

Still we need twelve and eight more conditions for the systems (3.17) and (3.23)

respectively to get stable systems for the solution of (3.3). For this we made

some adjustment of boundary conditions for the system (3.17) and (3.23), which

are defined below.

Case 1:- If we use interpolating collocation algorithm for the approximate solu-

tion of (3.3) then we define six conditions at left end points and six conditions at

the right end points. Since subdivision scheme (3.1) reproduces nine degree (i.e.

tenth order) polynomials, so we define boundary conditions of order ten for so-

lution of (3.17). For simplicity only left end points z−7, z−6, z−5, z−4, z−3, z−2 are

discussed and the values of right end points zN+2, zN+3, zN+4, zN+5, zN+6, zN+7

can be treated similarly.

The values z−7, z−6, z−5, z−4, z−3, z−2 can be determined by the nonic polynomial

q(x) interpolating (xi, zi), 2 ≤ i ≤ 7. Precisely, we have

z−i = q(−xi), i = 2, 3, 4, 5, 6, 7,

where

q(xi) =10∑j=1

10

j

(−1)j+1Z1(xi−j).

Since by (3.22) Z1(xi) = zi for i = 2, 3, 4, 5, 6, 7 then by replacing xi by −xi, we

have

q(−xi) =10∑j=1

10

j

(−1)j+1z−i+j.


10∑j=0

10

j

(−1)jz−i+j = 0, i = 7, 6, 5, 4, 3, 2. (3.32)

63


N + 5, N + 6, N + 7 and

q(xi) =10∑j=1

10

j

(−1)j+1zi−j.


10∑j=0

10

j

(−1)jzi−j = 0, i = N + 2, N + 3, N + 4, N + 5, N + 6, N + 7.

(3.33)


unknowns {zi}, in which N + 1 equations are obtained from (3.24) and (3.25),

four equations from boundary conditions (3.14) and twelve from boundary con-

ditions (3.32) and (3.33).

Case 2:- If we use approximating collocation algorithm (3.23) for the solution of

(3.3) then we need eight more conditions. So in this case, we define four extra

conditions at the left end points and four conditions at the right end points by

some extrapolation method. Since the subdivision scheme reproduce cubic (i.e.

fourth order) polynomial, so we define quartic polynomial for the adjustment of

boundary treatment. The values z−4, z−3, z−2, z−1 are determined by the quartic

polynomial p(x) interpolating (xi, zi). This polynomial is defined as

z−i+1 = p(−xi+1), i = 1, 2, 3, 4,

where

p(xi+1) =4∑

j=1

4

j

(−1)j+1Z2(xi−j+1).

Since by (3.28) Z2(xi) = zi for i = 1, 2, 3, 4 then by replacing xi by −xi, we have

p(−xi+1) =4∑

j=1

4

j

(−1)j+1z−i+j+1.

64


4∑j=0

4

j

(−1)jz−i+j+1 = 0, i = 1, 2, 3, 4. (3.34)

Similarly for the right end, we can define zi+1 = p(xi+1), i = N + 1 N + 2, N + 3,

N + 4 and

p(xi+1) =4∑

j=1

4

j

(−1)j+1zi−j+1.


4∑j=0

4

j

(−1)jzi−j+1 = 0, i = N + 1, N + 2, N + 3, N + 4.

(3.35)


unknowns {zi}, in which N + 1 equations are obtained from (3.26) and (3.27),

four equations from boundary conditions (3.20) and eight from boundary con-

ditions (3.34) and (3.35).

3.2.7 Stable systems of linear equations

In this section, we present stable systems of linear equations for both interpolat-

ing and approximating collocation algorithms.

3.2.8 Stable system for interpolating collocation algorithm

From (3.24) and (3.25), we get following un-determine system of (N + 1) equa-

tions with (N + 17) unknowns {zi}

A1Z1 = G1, (3.36)

65

where the matrices A1, Z1, and G1 of orders (N +1)× (N +17), (N +17)× 1 and


A1 =

ϕiv−8 ϕiv

−7 ϕiv−6 ϕiv

−5 ϕiv−4 ϕiv

−3 ϕiv−2 ϕiv

−1 q0 ϕiv1 ϕiv

2 ϕiv3 ϕiv

4

0 ϕiv−8 ϕiv

−7 ϕiv−6 ϕiv

−5 ϕiv−4 ϕiv

−3 ϕiv−2 ϕiv

−1 q1 ϕiv1 ϕiv

2 ϕiv3

0 0 ϕiv−8 ϕiv

−7 ϕiv−6 ϕiv

−5 ϕiv−4 ϕiv

−3 ϕiv−2 ϕiv

−1 q2 ϕiv1 ϕiv

2

0 0 0 ϕiv−8 ϕiv

−7 ϕiv−6 ϕiv

−5 ϕiv−4 ϕiv

−3 ϕiv−2 ϕiv

−1 q3 ϕiv1

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

ϕiv5 ϕiv

6 ϕiv7 ϕiv

8 · · · 0 0 0

ϕiv4 ϕiv

5 ϕiv6 ϕiv

7 · · · 0 0 0

ϕiv3 ϕiv

4 ϕiv5 ϕiv

6 · · · 0 0 0

ϕiv2 ϕiv

3 ϕiv4 ϕiv

5 · · · 0 0 0

· · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · · · · · · · · · · ϕ′′′7 ϕ′′′

8 0

· · · · · · · · · · · · · · · ϕ′′′6 ϕ′′′

7 ϕ′′′8

, (3.37)

Z1 = (z−8, z−7, z−6, z−5, z−4, · · · , zN+8)T , (3.38)

G1 = (b0h4, b1h

4, b2h4, b4h

3, · · · , bNh4)T , (3.39)

where ϕivj = ϕiv(j) and qj = ϕiv

0 − ajh4.

For obtaining the unique solution of (3.36), we made some adjustment of bound-

ary conditions in previous section which is defined in (3.28), (3.29), (3.32) and

(3.33). By using this adjustment, we get a following system of (N + 17) linear

equations with (N + 17) unknowns {zi}, defined as

D1Z1 = R1, (3.40)

66

where the coefficient matrix

D1 = (BT0 , A

T1 , B

T1 )

T , (3.41)

A1 is defined by (3.37), B0 and B1 are defined as

B0 =

0 1 −10 45 −120 210 −252 210 −120 45 −10

0 0 1 −10 45 −120 210 −252 210 −120 45

0 0 0 1 −10 45 −120 210 −252 210 −120

0 0 0 0 1 −10 45 −120 210 −252 210

0 0 0 0 0 1 −10 45 −120 210 −252

0 0 0 0 0 0 1 −10 45 −120 210

0 0 0 0 0 0 0 0 7381N2520

25200N2520

−56700N2520

0 0 0 0 0 0 0 0 1 0 0

1 0 0 0 0 0 0 0 · · · 0

−10 1 0 0 0 0 0 0 · · · 0

45 −10 1 0 0 0 0 0 · · · 0

−120 45 −10 1 0 0 0 0 · · · 0

210 −120 45 −10 1 0 0 0 · · · 0

−252 210 −120 45 −10 1 0 0 · · · 0

100800N2520

−132300N2520

127008N2520

−88200N2520

43200N2520

−14175N2520

2800N2520

−252N2520

· · · 0

0 0 0 0 0 0 0 0 · · · 0

,

(3.42)

first six rows of B0 are obtained from (3.32), second last row is obtained from

(3.28) and last row is taken from given boundary conditions Z1(0) which is de-

67

fined in (3.14) and

B1 =

0 0 · · · 0 0 0 0 0 0 0 0 0 0

0 0 · · · N10 − 10N

945N8 − 120N

7 35N − 252N5

105N2 −40N 45N

2 −10N

0 0 · · · 0 0 1 −10 45 −120 210 −252 210 −120

0 0 · · · 0 0 0 1 −10 45 −120 210 −252 210

0 0 · · · 0 0 0 0 1 −10 45 −120 210 −252

0 0 · · · 0 0 0 0 0 1 −10 45 −120 210

0 0 · · · 0 0 0 0 0 0 1 −10 45 −120

0 0 · · · 0 0 0 0 0 0 0 1 −10 45

1 0 0 0 0 0 0 0 0

7381N2520 0 0 0 0 0 0 0 0

45 −10 1 0 0 0 0 0 0

−120 45 −10 1 0 0 0 0 0

210 −120 45 −10 1 0 0 0 0

−252 210 −120 45 −10 1 0 0 0

−120 210 −252 210 −120 45 −10 1 0

, (3.43)

first row of B1 is obtained from Z1(N) which is defined in (3.14), second row is

obtained from (3.29) and the last six rows are obtained from (3.33), Z1 which is

defined in (3.38) and R1 is defined as

R1 = (0, 0, 0, 0, 0, 0, β, α,GT1 , γ, ω, 0, 0, 0, 0, 0, 0)

T ,

where G1 is defined by (3.39).

3.2.9 Stable system for approximating collocation algorithm

From (3.26) and (3.27), we get following un-determine system of (N + 1) equa-

tions with (N + 13) unknowns {zi}

A2Z2 = G2, (3.44)

68

where the matrices A2, Z2, and G2 of orders (N +1)× (N +13), (N +13)× 1 and


A2 =

Φiv−6 Φiv

−5 Φiv−4 Φiv

−3 Φiv−2 Φiv

−1 Υ0 Φiv1 Φiv

2 Φiv3 Φiv

4

0 Φiv−6 Φiv

−5 Φiv−4 Φiv

−3 Φiv−2 Φiv

−1 Υ1 Φiv1 Φiv

2 Φiv3

0 0 0 Φiv−6 Φiv

−5 Φiv−4 Φiv

−3 Φiv−2 Φiv

−1 Υ2 Φiv1

0 0 0 0 Φiv−6 Φiv

−5 Φiv−4 Φiv

−3 Φiv−2 Φiv

−1 Υ3

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

Φiv5 Φiv

6 · · · 0 0 0

Φiv4 Φiv

5 · · · 0 0 0

Φiv3 Φiv

4 · · · 0 0 0

Φiv2 Φiv

3 · · · 0 0 0

· · · · · · · · · · · · · · · · · ·

· · · · · · · · · Φ′′′5 Φ′′′

6 0

· · · · · · · · · Φ′′′4 Φ′′′

5 Φ′′′6

, (3.45)

Z2 = (z−6, z−5, z−4, z−3, z−2, · · · , zN+6)T , (3.46)

G2 = (b0h4, b1h

4, b2h4, b4h

3, · · · , bNh4)T , (3.47)

where Φivj = Φiv(j) and Υj = Φiv

0 − ajh4.

In order to get the unique solution of system (3.44), we have defined some extra

conditions in (3.30), (3.31), (3.34) and (3.35). By using these extra conditions we

get a following system of (N + 13) linear equations with (N + 13) unknowns

{zi}

D2Z2 = R2, (3.48)

where the coefficient matrix

D2 = (BT0 , A

T2 ,BT

1 )T , (3.49)

69

A2 is defined by (3.45), B0 and B1 are defined as

B0 =

0 0 0 1 −4 6 −4 1 0 0 0 0 0

0 0 0 0 1 −4 6 −4 1 0 0 0 0

0 0 0 0 0 1 −4 6 −4 1 0 0 0

0 0 0 0 0 0 1 −4 6 4 1 0 0

0 0 0 0 0 0 −251N12

48N12

−36N12

16N12

−3N12

0 0

0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 · · · 0 0

0 0 · · · 0 0

0 0 · · · 0 0

0 0 · · · 0 0

0 0 · · · 0 0

0 0 · · · 0 0

0 0 · · · 0 0

0 0 · · · 0 0

, (3.50)

first four rows of B0 are obtained from (3.34), second last row is obtained from

(3.30) and the last row is taken from the given boundary conditions Z2(0) which

is defined in (3.20) and

B1 =

0 0 0 0 · · · 0 0 0 0 0 0 1 0 0 0

0 0 0 0 · · · 0 0 3N12

−16N12

36N12

−48N12

25N12

0 0 0

0 0 0 0 · · · 0 0 0 0 0 0 0 1 −4 6

0 0 0 0 · · · 0 0 0 0 0 0 0 0 1 −4

0 0 0 0 · · · 0 0 0 0 0 0 0 0 0 1

0 0 0 0 · · · 0 0 0 0 0 0 0 0 0 0

70

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

−4 1 0 0 0 0 0 0

6 −4 1 0 0 0 0 0

−4 6 −4 1 0 0 0 0

1 −4 6 −4 1 0 0 0

, (3.51)

first row of B1 is obtained from Z2(N) which is defined in (3.20), second row

is obtained from (3.31) and the last four rows are obtained from (3.35), Z2 is

defined in (3.46) and R2 is defined as

R2 = (0, 0, 0, 0, 0, 0, β, α,GT2 , γ, ω, 0, 0, 0, 0, 0, 0)

T ,

where G2 is defined by (3.47). Hence to obtain the approximate solution of the

fourth order boundary value problem (3.3) by interpolating and approximating

collocation algorithms we need to solve the systems (3.40) and (3.48) respective-

ly.


In this section, we discuss the non-singularity of the coefficient matrices D1 and

D2 defined in (3.41) and (3.49) respectively. We observe that the coefficient ma-

trices D1 and D2 are neither symmetric nor diagonally dominant. However it

can be shown that D1 and D2 are non-singular. Since D1 and D2 are band matri-

ces with half band width 9 and 7, numerical complexities for solving the linear

systems using Gaussian elimination are about 81(N + 17) and 49(N + 13) mul-

tiplications respectively. For large N , the matrices are almost symmetric except

the first and last eight rows and columns of D1 and first and last four rows and

columns of D2 due to the boundary conditions. Therefore we first consider their

71

symmetric part i.e. square symmetric matrices E1 and E2 of orders N+1 defined

as

E1 =

ϕiv0 ϕiv

1 ϕiv2 ϕiv

3 ϕiv4 ϕiv

5 ϕiv6 ϕiv

7 ϕiv8 · · · 0 0 0 0

ϕiv−1 ϕiv

0 ϕiv1 ϕiv

2 ϕiv3 ϕiv

4 ϕiv5 ϕiv

6 ϕiv7 · · · 0 0 0 0

ϕiv−2 ϕiv

−1 ϕiv0 ϕiv

1 ϕiv2 ϕiv

3 ϕiv4 ϕiv

5 ϕiv6 · · · 0 0 0 0

ϕiv−3 ϕiv

−2 ϕiv−1 ϕiv

0 ϕiv1 ϕiv

2 ϕiv3 ϕiv

4 ϕiv5 · · · 0 0 0 0

0 0 0 0 0 0 0 0 0 · · · ϕiv4 ϕiv

3 ϕiv2 ϕiv

1

0 0 0 0 0 0 0 0 0 · · · ϕiv3 ϕiv

2 ϕiv1 ϕiv

0

,

and

E2 =

Φiv0 Φiv

1 Φiv2 Φiv

3 Φiv4 Φiv

5 Φiv6 0 0 · · · 0 0 0 0

Φiv−1 Φiv

0 Φiv1 Φiv

2 Φiv3 Φiv

4 Φiv5 Φiv

6 0 · · · 0 0 0 0

Φiv−2 Φiv

−1 Φiv0 Φiv

1 Φiv2 Φiv

3 Φiv4 Φiv

5 Φiv6 · · · 0 0 0 0

Φiv−3 Φiv

−2 Φiv−1 Φiv

0 Φiv1 Φiv

2 Φiv3 Φiv

4 Φiv5 · · · 0 0 0 0

0 0 0 0 0 0 0 0 0 · · · Φiv4 Φiv

3 Φiv2 Φiv

1

0 0 0 0 0 0 0 0 0 · · · Φiv3 Φiv

2 Φiv1 Φiv

0

.

So E1 and E2 are symmetric matrices obtained from D1 and D2 respectively.

It can be shown that E1 and E2 are non-singular when N increase. The non-

singularity of the matrices E1 and E2 is shown in Table 3.5 by finding their

determinants. The matrices E1 and E2 remain non-singular for N 6 500 and

N 6 100 respectively. For large N the determinants of the matrices may or may

not be equals to zero. The non-singularity of the matrices D1 and D2 have been

checked by finding their eigenvalues for N 6 500 and for N 6 100 respectively.

Since the eigenvalues for both the matrices are non-zero, then by Strang (2011)

matrices D1 and D2 are non-singular. However the matrices D1 for N > 500

and D2 for N > 100 may or may not be non-singular. Therefore we claim that

systems of equations (3.40) and (3.48) are stable.

72

Table 3.5: Determinants of the matrices

N E1 E2

10 −8667/56 1.11401292× 103

50 -177183 3.2517495× 102

100 -552709050 0.753776508473953

200 -5.033491472×1036 · · ·

300 -4.477989536×1071 · · ·

400 3987757210720454 · · ·

500 3987757210720454 · · ·

3.2.11 Error estimation

In this section, we discuss the approximation properties of the interpolating and

approximating collocation algorithms. Since the scheme (3.1) and (3.2) repro-

duce polynomial curves of degree nine and three so by Dyn (2002) and Mustafa

et al. (2014a) schemes have approximation order ten and four respectively. Here

we present our main results for error estimation.


by (3.40) then absolute error by interpolating collocation algorithm is

||err1(x)||∞ = ||Z(l)1 (x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3,

where l denotes the order of derivative.


by (3.48) then absolute error by approximating collocation algorithm is

||err2(x)||∞ = ||Z(l)2 (x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3.

Proof of these results are similar to the proof of proposition by Mustafa and

Ejaz (2014) and Qu and Agarwal (1996).

73

3.3 Numerical examples and comparison

In this section, the interpolating and approximating collocation algorithms de-

scribed in Section 3.2, are tested on the problems given below. Absolute errors

between exact and approximate solutions are also calculated. For the sake of

comparisons, we also tabulated the results in this section. Graphical illustra-

tions of solutions are presented.

3.3.1 Numerical examples

Here we find the numerical solutions of some of the boundary value problems

arising in the mathematical modeling of viscoelastic and inelastic flows etc.

Example 3.3.1. Consider the fourth order linear boundary value problem

yiv(x) + xy = −(8 + 7x+ x3)ex, 0 < x < 1, (3.52)

subject to the boundary conditions

y(0) = y(1) = 0, y′(0) = 1, y′(1) = −e. (3.53)

By comparing the above problem with equation (3.3), we have a(x) = −x and

b(x) = −(8 + 7x + x3)ex. The exact solution for the above problem is y =

x(1− x)ex.

Here we present the numerical solution of above problem by interpolating and

approximating collocation algorithms.

74

Solution by interpolating collocation algorithm:

In this method, by solving the system of linear equations (3.40) at N = 10, we

obtain the approximate solution (3.12) of (3.52) where {zi}, −8 ≤ i ≤ 18 are

{−0.736798818, −0.643536669, −0.554623724, −0.467529093, −0.379799398,

−0.289680649, −0.196215973, −0.099344276, 0.0000, 0.0997855, 0.196780435,

0.286445566, 0.362823930, 0.4184258296, 0.444109480, 0.428957387, 0.360149377,

0.222833881, 0.000, −0.327646133,−0.781795936, −1.386637856, −2.168934169,

−3.158065904, −4.386034900, −5.887410047,−7.698656102}.

Solution by approximating collocation algorithm:

In this method, we solve the system of linear equations (3.48) at N = 10 and get

solution (3.18) of (3.52) where {zi}, −6 ≤ i ≤ 16 are

{240576495.97007, 346838.29496, 518.9347067, −0.27530396, −0.192116155,

−0.098802082, 0.00, 0.099651832, 0.195515154, 0.282951708, 0.357323235,

0.412290577, 0.439106327, 0.426338357, 0.359800241, 0.223138586, 0,

−0.325968911, −0.771121538, −1.351811276, 759.3687002, 508693.855464,

352843031.1188856}.

Example 3.3.2. Consider the following fourth order linear boundary value prob-

lem

yiv(x) = (x4 + 14x3 + 49x2 + 32x− 12)ex 0 ≤ x ≤ 1 (3.54)

with

y(0) = y(1) = y′(0) = y′(1) (3.55)

corresponds to the bending of a thin beam clamped at both ends. The unique

solution of (3.54) is

y(x) = x2(1− x)2ex.

75


By using this method, we solve system (3.40) at N = 10 and get solution (3.12)

of (3.54) where {zi}, −8 ≤ i ≤ 18 are

{−1.29218842, 0.538942241, 0.376822099, 0.25497974, 0.157378182, 0.084436111,

0.035355124, 0.008209932, 0, 0.006713642, 0.023450361, 0.044645810, 0.064445941,

0.077282873, 0.078714604, 0.066604028, 0.042729735, 0.014941769, 0, 0.027260994,

0.143411251, 0.418480429, 0.953407669, 1.889477424, 3.419988190, 5.826254985,

6.993262174}.


In this method, by solving system of linear equations (3.48) at N = 10, we obtain

the approximate solution (3.18) of (3.54) where {zi}, −6 ≤ i ≤ 16 are

{1.711850528× 1011,−2.4699609× 108, 3.560795701× 105,−5.33546354× 105,

0.06034270857, 0.013560666, 0, 0.01051064293, 0.03594252611, 0.067145581105,

0.09496973950, 0.11026493288, 0.10721651489, 0.08657152262, 0.05642424602,

0.02486897512, 0, 0.01008838960, 0.00269809658,−3.898592698× 103,

2.597091086× 106,−1.801597056× 109, 1.2486272867× 1012}.

Example 3.3.3. Consider the boundary value problem

y(iv) − y = −4(2x cos(x) + 3 sin(x)) (3.56)

with boundary conditions

y(0) = 0, y(1) = 0, y′(0) = −1, y′(1) = 2 sin(1). (3.57)

The exact solution of this problem is y = (x2 − 1) sin(x).

76


Here, we solve system (3.40) at N = 10 and get solution (3.12) of (3.56) where

{zi}, −8 ≤ i ≤ 18 are

{0.0600152940, 0.18160335240, 0.25833493990, 0.29216208360, 0.2870204837,

0.1824477485, 0.09700293, 0,−0.1001836945,−0.1951709144,−0.2768975252,

−0.3717065517,−0.372918743,−0.3376558661,−0.3379193763,−0.26367852,

−0.1505934691, 0, 0.1844062122, 0.3967742453, 0.6290856272, 0.87118909250,

1.11088175850, 1.33403484280, 1.52476071733, 1.6661398769}.


By solving system of linear equations (3.48) at N = 10, we obtain the approxi-

mate solution (3.18) of (3.56) where {zi}, −6 ≤ i ≤ 16 are

{115585876195.8271, 166975210.2723412, 240725.36249556, 360.797052040,

0.1912504195, 0.0989171622, 0,−0.0988737231,−0.1910766633,−0.2699814765,

−0.328960819,−0.361387346,−0.3620271786,−0.3276693903,−0.2568529513,

−0.1481168313, 0, 0.1889585727, 0.4202199172, 648.2401207, 432617.012,

300071307.86, 207719711144.92}.

77

Tabl

e3.

6:N

umer

ical

resu

lts

ofEx

ampl

e3.

3.1

xi

Ana

lyti

cA

ppro

xim

ate

solu

tion

App

roxi

mat

eso

luti

on

solu

tion

Z1

byin

terp

olat

ing

Z2

byap

prox

imat

ing

||err

1(x

i)|| ∞

||err

2(x

i)|| ∞

yco

lloca

tion

algo

rith

mco

lloca

tion

algo

rith

m

0.0

00

00

0

0.1

0.09

9465

380.

0997

8551

520.

0996

5183

170.

0003

2013

0.00

0186

45

0.2

0.19

5424

440.

1967

8040

150.

1955

1515

400.

0013

5596

0.00

0090

71

0.3

0.28

3470

350.

2864

4551

080.

2829

5170

800.

0029

7516

0.00

0518

64

0.4

0.35

8037

930.

3628

2385

780.

3573

2323

450.

0047

8593

0.00

0714

69

0.5

0.41

2180

320.

4184

2575

120.

4122

9057

650.

0062

4543

0.00

0110

25

0.6

0.43

7308

510.

4441

0940

750.

4391

0632

740.

0068

0090

0.00

1797

81

0.7

0.42

2888

070.

4289

5733

140.

4263

3835

690.

0060

6926

0.00

3450

29

0.8

0.35

6086

550.

3601

4934

300.

3598

0024

080.

0040

6279

0.00

3713

69

0.9

0.22

1364

280.

2228

3386

920.

2231

3858

610.

0014

6959

0.00

1774

31

1.0

00

00

0

78

Tabl

e3.

7:N

umer

ical

resu

lts

ofEx

ampl

e3.

3.2

xi

Ana

lyti

cA

ppro

xim

ate

solu

tion

App

roxi

mat

eso

luti

on

solu

tion

Z1

byin

terp

olat

ing

Z2

byap

prox

imat

ing

||err

1(x

i)|| ∞

||err

2(x

i)|| ∞

yco

lloca

tion

algo

rith

mco

lloca

tion

algo

rith

m

0.0

00

00

0

0.1

0.00

8951

884

0.00

6713

642

0.01

0510

643

0.00

2238

243

0.00

1558

758

0.2

0.03

1267

9106

0.02

3450

361

0.03

5942

526

0.00

7817

550

0.00

4674

615

0.3

0.05

9528

773

0.04

4645

810

0.06

7145

581

0.01

4882

9636

0.00

7616

808

0.4

0.08

5929

102

0.06

4445

941

0.09

4969

740

0.02

1483

162

0.00

9040

637

0.5

0.10

3045

079

0.07

7282

873

0.11

0264

933

0.02

5762

206

0.00

7219

853

0.6

0.10

4954

043

0.07

8714

604

0.10

7216

515

0.02

6239

439

0.00

2262

472

0.7

0.08

8806

494

0.06

6604

028

0.08

6571

523

0.02

2202

467

0.00

2234

972

0.8

0.05

6973

848

0.04

2729

735

0.05

6424

256

0.01

4244

112

0.00

0549

602

0.9

0.01

9922

785

0.01

4941

769

0.02

4868

975

0.00

4981

016

0.00

4946

1810

1.0

00

00

0

79

Tabl

e3.

8:N

umer

ical

resu

lts

ofEx

ampl

e3.

3.3

xi

Ana

lyti

cA

ppro

xim

ate

solu

tion

App

roxi

mat

eso

luti

on

solu

tion

Z1

byin

terp

olat

ing

Z2

byap

prox

imat

ing

||err

1(x

i)|| ∞

||err

2(x

i)|| ∞

yco

lloca

tion

algo

rith

mco

lloca

tion

algo

rith

m

0.0

00

00

0

0.1

-0.0

9883

508

-0.1

0018

369

-0.0

9887

3723

0.00

1348

60.

0000

3864

1

0.2

-0.1

9072

256

-0.1

9517

091

-0.1

9107

6663

30.

0044

484

0.00

0354

11

0.3

-0.2

6892

339

-0.2

7689

752

-0.2

6998

1476

50.

0079

741

0.00

1058

0.4

-0.3

2711

141

-0.3

3791

938

-0.3

2896

0819

0.01

0808

0.00

1849

4

0.5

-0.3

5956

915

-0.3

7170

655

-0.3

6138

7346

0.01

2137

0.00

1818

2

0.6

-0.3

6137

118

-0.3

7291

874

-0.3

6202

7178

60.

0115

480.

0006

5600

0.7

-0.3

2855

102

-0.3

3765

587

-0.3

2766

9390

30.

0091

048

0.00

0881

63

0.8

-0.2

5824

819

-0.2

6367

852

-0.2

5685

2951

30.

0054

303

0.00

1395

2

0.9

-0.1

4883

211

-0.1

5059

347

-0.1

4811

6831

30.

0017

614

0.00

0715

28

1.0

00

00

0

80

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

X

Sol

utio

n


1

Approximate solution Z2

Figure 3.1: Comparison between analytic and approximate solutions of Example 3.3.1

obtained by interpolating and approximating collocation algorithms.

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

X

Sol

utio

n


1




81

0 0.2 0.4 0.6 0.8 1−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

X

Sol

utio

n


1




3.3.2 Comparison:

The numerical results of Examples 3.3.1, 3.3.2 and 3.3.3 by interpolating and

approximating collocation algorithms are presented in Tables 3.6, 3.7 and 3.8

respectively. The maximum absolute errors in the solution of Examples 3.3.1,

3.3.2 and 3.3.3 obtained by interpolating and approximating collocation algo-

rithms are given in Table 3.9. Graphical representation of these results is shown

in Figures 3.1, 3.2 and 3.3. In these figures solid curve represents the exact so-

lutions, dashed lines represent approximate solutions obtained by (3.40) and

dotted lines represent the approximate solutions obtained by (3.48). Following

is the comparison of the numerical solutions obtained by proposed algorithms

and other approaches of this type of boundary value problems:

• From above results we see that approximating schemes based collocation

algorithms give better results then interpolating schemes based colloca-

tion algorithms.

82

• Example 3.3.1 is also solved by Usmani (1983). He solved this problem by

second order finite difference method and obtained the maximum abso-

lute errors at different step sizes h. We observe that the maximum absolute

error at the step size h = 110

by the proposed approximating collocation al-

gorithm is better than the maximum absolute error obtained by Usmani

(1983) at step size h = 116

. The comparison of proposed methods with

second order finite difference method of Usmani (1983) at difference step

sizes is shown in Table 3.10.

• Example 3.3.2 is also solved by Russell and Shampine (1972) by quintic

spline based collocation methods. We observe that order of error approx-

imation obtained by Russell and Shampine (1972) and proposed approxi-

mating collocation algorithm are same.

Table 3.9: Maximum absolute errors of Examples 3.3.1, 3.3.2 and 3.3.3

Example Max. absolute errors Max. absolute errors

by interpolating by approximating

collocation algorithm collocation algorithm

3.3.1 6.8010× 10−3 3.7137× 10−3

3.3.2 2.6239× 10−2 0.9041× 10−2

3.3.3 1.2137× 10−2 0.18494× 10−2

3.4 Conclusion

In this chapter, we have presented interpolating and approximating collocation

algorithms based on interpolating and approximating subdivision schemes for

the solution of linear fourth order boundary value problems. The proposed

83

Table 3.10: Comparison of Example 3.3.1 with different methods

Max. absolute errors Max. absolute errors Second order finite

h by interpolating by approximating difference method

collocation algorithm collocation algorithm by Usmani (1983)

14

· · · · · · 8.50× 10−2

18

· · · 1.4098× 10−2 2.09× 10−2

110

6.8010× 10−3 3.7137× 10−3 · · ·116

6.7469× 103 4.4572× 10−3 5.27× 10−3

algorithms have been applied on different linear fourth order boundary val-

ue problems. Results show that the approximating collocation algorithm gives

better results comparative to interpolating collocation algorithm. We have also

observed that the accuracy of the solution can be improved by choosing differ-

ent subdivision schemes with the proper adjustment of boundary conditions.

Approximating subdivision scheme based collocation algorithm gives better

results comparative to second order finite difference method. However, ap-

proximating subdivision scheme based collocation algorithm and quintic spline

based collocation algorithm have same order of approximation.

84

Chapter 4

A Subdivision Based Iterative

Collocations Algorithm for

Nonlinear Third Order Boundary

Value Problems

Many problems in physics, chemistry and engineering science are demonstrated

mathematically by third order boundary value problems. These boundary value

problems can be found in different areas of applied mathematics and physics as,

in the deflection of a curved beam having a constant or varying cross section, a

three layer beam, electromagnetic waves, or gravity driven flows. In this chap-

ter, we develop subdivision schemes based collocation iterative algorithm for

the solution of nonlinear third order boundary value problems.

An outline of this chapter is as follows: In Section 4.1, some results about ex-

istence and uniqueness of the solution of third order boundary value problem

are given. In Section 4.2, subdivision algorithm, basis function and their deriva-

tives are briefed. In Section 4.3, subdivision based iterative algorithm for the

solution of nonlinear third order BVPs using the derivatives of basis functions

85

is formulated. Convergence of the proposed algorithm is also discussed in this

section. Error analysis is given in Section 4.4. Numerical examples illustrating

the usefulness of our proposed algorithm are given in Section 4.5.

4.1 Existence and uniqueness of the solution

In this section, we present some results about the existence and uniqueness of

the solution of third order nonlinear boundary value problems. The detail of

these results can be found in Agarwal (1973).

The general third order nonlinear boundary value problem can be prescribed as

y′′′= f(x, y, y

′, y

′′) (4.1)

with boundary conditions define as

y(0) = α1, y′(0) = α2, y(1) = α3. (4.2)

where αi, i = 1, 2, 3 are constants.

Proposition 4.1.1. If the function f(x, y, y′, y′′) is continuous and satisfy the following

uniform Lipschitz condition∣∣f(x, y, y′, y′′)− f(x, y, y′, y′′)∣∣ ≤ M0 |y − y|+M1

∣∣y′ − y′∣∣+M2 |y′′ − y′′| ,

(4.3)

∀ (x, y, y′, y′′), (x, y, y′, y′′) ∈ [a, b]×R3

where the constants M0, M1 and M2 satisfy

2

81M0(b− a)3 +

1

6M1(b− a)2 +

2

3M2(b− a) < 1, (4.4)

or

3

160M0(b− a)3 +

17

150M1(b− a)2 +

2

3M2(b− a) < 1, (4.5)

86

then the boundary value problem has one and only one solution

y′′′= f(x, y, y

′, y

′′), a ≤ x ≤ b, y(a) = y1, y′(a) = y2, y(b) = y3. (4.6)

The existence and uniqueness of the differential equation (4.1) with boundary

conditions at two or three points are presented in Agarwal (1973).

Remark 4.1.1. Throughout this paper the function f(x, y, y′, y′′) satisfies Lipschitz

conditions (4.3) along with condition (4.4)-(4.6). So the existence and uniqueness

of the solutions of (4.1) is guaranteed.

4.2 Subdivision scheme and basis function

In this section, we define binary subdivision algorithm and their basis function

that are used to construct the approximate solutions of (4.1).

4.2.1 Interpolating subdivision scheme

We consider the following 8-point binary interpolating subdivision scheme in-

troduced in Deng and Ma (2013) and Deslauriers and Dubuc (1989)

pk+12i = pki

pk+12i+1 =

12252048

(pki + pki+1

)− 245

2048

(pki−1 + pki+2

)+ 49

2048

(pki−2 + pki+3

)− 5

2048

(pki−3 + pki+4

)(4.7)

where pki and pk+1i are points at kth and (k+1)th iterative level. The scheme (4.7)

is C3-continuous and support length (−7, 7) with 8th order approximation.

4.2.2 Basis function and their derivatives

The basis function is the limit function resulting from cardinal data, where all

vertices of the polygon have value zero except for one. Let g(x), x ∈ R be the

87

fundamental solution of (4.7) satisfies the two scale equation

g(x) = g(2x) +1

2048[1225{g(2x− 1) + g(2x+ 1)} − 245{g(2x− 3)

+g(2x+ 3)}+ 49{g(2x− 5) + g(2x+ 5)} − 5{g(2x− 7)

+g(2x+ 7)}] , x ∈ R

and

g(x) ∈ C3, g(x) = 0, if x /∈ [-6, 6], g(i) = δ0, i ∈ Z.

Proposition 4.2.1. The fundamental solution g(x) is three time continuously differen-

tiable over the interval [−6, 6]. Its derivatives at integers are given by

g′(i) = 2sign(i)ET|i|η1, g′′(i) = 22ET

|i|η2, g′′′(i) = 23sign(i)ET|i|η3,

where

sgn(t) =

−1, t < 0,

0, t = 0,

1, t > 0,

and Et’s for 0 ≤ t ≤ 6, are defined below

Et = (e6t, e5t, e4t, e3t, e2t, e1t, e0t, e−1t, e−2t, e−3t, e−4t, e−5t, e−6t)T ,

where

eit =

1, i = t,

0, i = t,

and ηj , 1 ≤ j ≤ 3 are defined in Mustafa and Ejaz (2014).

Furthermore, the numeric values of first, second and third derivatives of g(i)

at i ∈ [−6, 6] are

g′(0) = 0, g′(±1) = ∓7859249553

,

g′(±2) = ± 76113198212

, g′(±3) = ∓ 332849553

,

g′(±4) = ± 2645594636

, g′(±5) = ± 256743295

,

g′(±6) = ∓ 1594636

,

(4.8)

88

g′′(0) = −34264341124

, g′′(±1) = 57042561079505

,

g′′(±2) = −120536518636040

, g′′(±3) = 3259521079505

,

g′′(±4) = − 608712878680

, g′′(±5) = − 704215901

,

g′′(±6) = 551727208

,

(4.9)

g′′′(0) = 0, g′′′(±1) = ∓292352117495

,

g′′′(±2) = ±30479871879920

, g′′′(±3) = ∓ 331213055

,

g′′′(±4) = ∓ 1369234990

, g′′′(±5) = ± 162611

,

g′′′(±6) = ∓ 541776

.

(4.10)

The above derivative values are found by using the left eigenvectors of the sub-

division process (4.7). The detailed description about these left eigenvectors ηj ,

1 ≤ j ≤ 3 and derivatives can be found in Mustafa and Ejaz (2014).

4.3 Subdivision based iterative algorithm

In this section, we describe the algorithm for the numerical solution of nonlinear

boundary value problem (4.1).


In this subsection, we construct the collocation method based on the interpolat-

ing subdivision scheme (4.7). Let U(x) be the assumed solution of (4.1)

U(x) =N+6∑i=−6

uig

(x− xi

h

), 0 6 x 6 1, (4.11)

where N(> 6) ∈ Z+, h = 1/N , xi = i/N = ih and {ui} are the unknown to be

determined for the solution of (4.1). The collocation algorithm together with the

boundary conditions is defined as follows:

U′′′(xj) = f(xj, U(xj), U

′(xj)), j = 0, 1, 2, · · · , N, (4.12)

89


U(0) = α1, U′(0) = α2, U(N) = α3, (4.13)

by taking the third derivative of (4.11) we get

U′′′(x) =

1

h3

N+6∑i=−6

uig′′′(x− xi

h

), 0 6 x 6 1, (4.14)

using (4.14) into (4.12), we get

N+6∑i=−6

uig′′′(xj − xi

h

)= h3f(xj, U(xj), U

′(xj)), j = 0, 1, 2, · · · , N.

This can be written asN+6∑i=−6

uig′′′

j−i = h3f(xj, U(xj), U′(xj)), j = 0, 1, 2, · · · , N.

As we know that g′′′i = −g

′′′−i, then above system of equations become

N+6∑i=−6

(−1)uig′′′

i−j = h3f(xj, U(xj), U′(xj)), j = 0, 1, 2, · · · , N. (4.15)

Now we simplify the nonlinear system of equations (4.15) in following theorem-

s.

Theorem 4.3.1. The nonlinear system of equations (4.15) for j = 0 becomes

6∑i=−6

(−1)uig′′′

i = h3f(x0, U(x0), U′(x0)). (4.16)

Proof. By expanding (4.15) for j = 0, we get

(−1){u−6g′′′

−6 + u−5g′′′

−5 + · · ·+ uN+5g′′′

N+5 + uN+6g′′′

N+6} = h3f(x0, U(x0), U′(x0)).

As we know that g′′′(i) exist only for the interval for i ∈ [−6, 6] and outside the

interval it will be zero. Then above equation can be written as

(−1){u−6g′′′

−6 + u−6g′′′

−6 + · · ·+ u5g′′′

5 + u6g′′′

6 } = h3f(x0, U(x0), U′(x0)).

This implies (4.16).

90

Theorem 4.3.2. For j = 1, 2, · · · , N , the nonlinear system of equations (4.15) becomes

6+j∑i=−6+j

(−1)uig′′′

i−j = h3f(xj, U(xj), U′(xj)). (4.17)

Proof. Substituting j = 1 in (4.15), it becomes

(−1){u−6g′′′

−7 + u−5g′′′

−6 · · ·+ u7g′′′

6 · · ·+ uN+6g′′′

N+5} = h3f(x1, U(x1), U′(x1)).

Since g′′′(i) is non-zero only for the interval for i ∈ [−6, 6] and outside the inter-

val it will be zero. Then above equation becomes

(−1){u−5g′′′

−6 + u−4g′′′

−5 + · · ·+ u4g′′′

5 + u5h′′′

6 } = h3f(x1, U(x1), U′(x1)). (4.18)

For j=2, (4.15) becomes

(−1){u−6g′′′

−8 + u−5g′′′

−7 + u−4g′′′

−6 + · · ·+ u4g′′′

2 + u5g′′′

3 + u6g′′′

4 + u7g′′′

5 u8g′′′

6

+ · · ·+ uN+5g′′′

N+3 + uN+6g′′′

N+4} = h3f(x2, U(x2), U′(x2)).

This implies

(−1){u−4g′′′

−6 + u−3g′′′

−5 + · · ·+ u7g′′′

5 + u8g′′′

6 } = h4f(x2, U(x2), U′(x2)). (4.19)

Similarly, we can find the expression for j = 3, 4, · · · , N , i-e.

(−1){u−3g′′′

−6 + u−4g′′′

−5 + · · ·+ u8g′′′

5 + u9g′′′

6 } = h3f(x3, U(x3), U′(x3)). (4.20)

(−1){u−2g′′′

−6 + u−1g′′′

−5 + · · ·+ u9g′′′

5 + u10g′′′

6 } = h3f(x4, U(x4), U′(x4)). (4.21)

...

(−1){uN−6g′′′

−6 + uN−5g′′′

−5 + · · ·+ uN+5g′′′

5 + uN+6g′′′

6 } = h3f(xN , U(xN), U′(xN)).

(4.22)

Hence by combining the equations (4.18)-(4.22) we get (4.17).

91

4.3.2 Unstable nonlinear system

The nonlinear system of (4.15) is equivalent to the following nonlinear system

of N + 1 equations with (N+13) unknowns {ui}:

A1U = F (u), (4.23)

where A1 is banded matrix of order (N +1)× (N +13), U is the unknown vector

of order N + 13 and F (u) is the vector of order N + 1 depends on u. The matrix

A, vectors U and F (u) are given explicitly by

A1 = (−1)(g′′′

pq(q − p− 6))(N+1)×(N+13), (4.24)

where p and q represent the row and column respectively i.e. p = 1, 2, 3 · · · , N+1

and q = 1, 2, 3, · · · , N + 13,

F (u) =(h3f(x0, U(x0), U

′(x0)), · · · , h3f(xN , U(xN), U

′(xN))

)T

, (4.25)

U = (u−6, u−5, z−4, · · · , uN+4, uN+5, uN+6)T , (4.26)

U′(xj) =

N+6∑i=−6

ujg′(xj − xi

h

),

g′(i) is defined in (4.8) and g(i) = gi. The system (4.23) is unstable and we

need to make it stable to get unique solution. The detail for the stable system of

nonlinear equations is given in next section.

4.3.3 Stable nonlinear system

For unique solution of nonlinear systems (4.23), we need twelve more condi-

tions. Three conditions can be attained from given boundary conditions for

nonlinear systems of equations and remaining nine conditions are attained by

setting some extrapolation method. The detail of the given boundary conditions

and extrapolation method are given below:

92

4.3.4 Approximated boundary condition

The given boundary conditions are :

U(0) = α1, U′(0) = α2, U(N) = α3. (4.27)

We see that first derivative involve in the given boundary conditions, since ap-

proximation order of interpolating scheme (4.7) is eight, so we approximate

derivative boundary conditions at end point with approximation order eight.

The approximation of derivative conditions at end point is defined as

U ′(0) =

(N

840

){−2283u0 + 6720u1 − 11760u2 + 15680u3 − 14700u4 + 9408u5

−3920u6 + 960u7 − 105u8}+O(h8). (4.28)

4.3.5 Imposed boundary conditions

Remaining nine conditions for the nonlinear systems (4.23) to get stable sys-

tems for the solution of (4.1) are obtained by setting the following extrapolation

method.

We define five conditions at left end points and four conditions at the right

end points. Since subdivision scheme reproduces seven degree polynomials, so

we define boundary conditions of order eight for solution of (4.23). For simplic-

ity only the left end points u−5, u−4, u−3, u−2, u−1are discussed and the values of

right end points uN+2, uN+3, uN+4, uN+5 can be treated similarly.

The values u−5, u−4, u−3, u−2, u−1 can be determined by the septic polynomial

R(x) interpolating (xi, ui), 0 ≤ i ≤ 5. Precisely, we have

u−i = R(−xi), i = 1, 2, 3, 4, 5,

where

R(xi) =8∑

r=1

8

r

(−1)r+1U(xi−r).

93

Since by (4.11), U(xi) = ui for i = 1, 2, 3, 4, 5 then by replacing xi by −xi, we

have

R(−xi) =8∑

r=1

8

r

(−1)r+1u−i+r.


8∑r=0

8

r

(−1)ru−i+r = 0, i = 5, 4, 3, 2, 1. (4.29)

Similarly for the right end, we can define ui = R(−xi), i = N + 2, N + 3, N + 4,

N + 5 and

R(xi) =8∑

r=0

8

r

(−1)r+1ui−r.


8∑r=0

8

r

(−1)rui−r = 0, i = N + 2, N + 3, N + 4, N + 5. (4.30)

Finally, we get a following new system of (N + 13) linear equations with (N +

13) unknowns {ui}, in which N + 1 equations are obtained from (4.15), three

equations from boundary conditions (4.27) and nine from boundary conditions

(4.29) and (4.30).

Hence the stable nonlinear system of equations define as:

AU = G(u), (4.31)

where the matrix A is given by

A = (AT0 , A

T1 , A

T2 )

T , (4.32)

the matrix A1 is defined in (4.24). The matrix (A0)7×(N+13) is constructed as by

taking first five rows from (4.29), sixth row taking from (4.28) and the last row

U(0) from (4.27). Hence

94

A0 =

0 1 −8 28 −56 70 −56 28 −8

0 0 1 −8 28 −56 70 −56 28

0 0 0 1 −8 28 −56 70 −56

0 0 0 0 1 −8 28 −56 70

0 0 0 0 0 1 −8 28 −56

0 0 0 0 0 0 −2283N840

6720N840 − 11760N

840

0 0 0 0 0 0 1 0 0

1 0 0 0 0 0 0 · · · 0 0

−8 1 0 0 0 0 0 · · · 0 0

28 −8 1 0 0 0 0 · · · 0 0

−56 28 −8 1 0 0 0 · · · 0 0

70 −56 28 −8 1 0 0 · · · 0 0

15680N840 − 14700N

8409408N840 − 3920N

840960N840 −105N

840 0 · · · 0 0

0 0 0 0 0 0 0 · · · 0 0

.

The matrix (A2)6×(N+13) is constructed as by taking first row U(N) from (4.27)

and last five rows obtained from (4.30). Hence

A2 =

0 0 · · · 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 · · · 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0 0

0 0 · · · 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0

0 0 · · · 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0

0 0 · · · 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0

.

The column vectors U is defined in (4.26) and G is defined as

G(u) = (0, 0, 0, 0, 0, U ′(0), U(0), F T (u), U(1), 0, 0, 0, 0)T , (4.33)

where F (u) is defined by (4.25).

4.3.6 Non-singularity of a matrix

The non-singularity of coefficient matrix A, which is defined in (4.32), can be

checked by different methods:

95

Since the determinant of matrix A is non-zero for N ≤ 2000 then the non linear

system of equations give solution for N ≤ 2000.

It is also observed that all the eigenvalues are non-zero for N ≤ 2000. Hence by

Strang (2011) matrix A is non-singular. For large N > 2000 the matrix may or

may not be non-singular.

The coefficient matrix A is neither symmetric nor diagonally dominant. Though

it can be prove that A is non-singular/invertible. The matrix is almost symmet-

ric except the first and last few rows and columns due to its boundary treatment

for large value of N . Therefore, we first consider the square band matrix B of

order N + 1 defined as

B = (−1)×

g′′′1 g′′′2 g′′′3 g′′′4 · · · · · · 0 0 0

g′′′0 g′′′1 g′′′2 g′′′3 · · · · · · 0 0 0

g′′′−1 g′′′0 g′′′1 g′′′2 · · · · · · 0 0 0

g′′′−2 g′′′−1 g′′′0 g′′′1 · · · · · · 0 0 0

g′′′−3 g′′′−2 g′′′−1 g′′′0 · · · · · · 0 0 0

· · · · · · · · · · · · · · · · · · · · · · · · · · ·

0 0 0 0 · · · · · · g′′′2 g′′′2 g′′′3

0 0 0 0 · · · · · · g′′′0 g′′′1 g′′′2

0 0 0 0 · · · · · · g′′′−1 g′′′0 g′′′1

.

Kilic and Stanica (2013) presented a method to find the inverse of banded matrix

of order n by using the LU factorization of the banded matrix. As B is a band

matrix of order (N + 1) so by LU factorization method its inverse exist.

96

4.3.7 Iterative algorithm and its convergence

In this section, we propose an iterative algorithm and discuss its convergence.

4.3.8 Iterative algorithm based on basis function

The iterative algorithm is based on basis function of the subdivision scheme

(4.7) as defined in the following three steps.

First step: Initial approximation

Initial approximation is important because numerical solution depends on the

initial approximation. We define the process for finding the initial approxima-

tion as follows:

Let initial approximate solution U0 be the solution of the following linear sys-

tem

AU0 = G0, (4.34)

where

G0 = (0, 0, 0, 0, 0, y′(a), y(a), p0, p1, p2, · · · , pN , y(b), 0, 0, 0, 0)T ,

pi = h3p(xi,mi, S), i = 0, 1, 2, · · ·N, a ≤ x ≤ b,

mi = y(a) + ih(

y(b)−y(a)b−a

),

S = y(b)− y(a).

G0 is the initial linear approximation of the nonlinear vector G(u). By solving

the linear system of equations (4.34) we get initial approximate solution.

97

Second step: Numerical solution

The numerical solutions U∗ of the nonlinear system are obtained by using sim-

ple iterative scheme

AU (m+1) = G(Um), m = 0, 1, 2, 3, · · · (4.35)

Third step: Stopping condition

The above iterative processes will terminate when the following condition is

satisfied

||Um − Um−1|| ≤ tol (4.36)

where tol is supposed value i.e. tol = 10−6. The convergence of the above it-

erative algorithm is guaranteed by the following propositions. The solutions of

linear system of equations (4.34) and (4.35) are obtained by Gaussian elimina-

tions method.

Proposition 4.3.3. The successive solutions {Um} for the nonlinear system (4.31) gen-

erated by the iterative algorithm (4.35) linearly converge to the solution U∗ provided

that the M0 and M1 are Lipschitz constants and step size h is small.

i.e.

∥∥A−1∥∥(M0h

3 +867307

212370M1h

2

)≤ 1. (4.37)

Proof. Let U∗ and U (m) be the solutions of the nonlinear system (4.31) then by

definition, for small h we have

AU∗ = G(U∗),

AU (m+1) = G(U (m)).

98

Let the error vector be defined as e(m) = Um−U∗ at mth iteration which satisfies

AU (m+1) − AU∗ = G(U (m))−G(U∗),

G(U (m+1) − U∗) = G(U (m))−G(U∗),

Ae(m+1) = G(Um)−G(U∗).

For i = 0, 1, 2, · · · , N ,

D3e(m+1)i = (F (U (m))− F (U∗))i,

by using the mean value theorem, which is stated as “If a function f(x, y, z) is

continuously differentiable in an open set of R3 containing points (x1, y1, z1) and

(x2, y2, z2) and the line segment connecting them, then an equation

f(x2, y2, z2)− f(x1, y1, z1) = f′

x(r, s, t)(x2 − x1) + f′

y(r, s, t)(y2 − y1)

+f′

z(r, s, t)(z2 − z1),

is valid for the interior point (a, b, c) of the segment.”, we have

D3e(m+1)i = f(xi, U

(m)i , U ′(m))− f(xi, U

(∗)i , U ′(∗)).

The above equation can be written as (by using mean value theorem)

D3e(m+1)i = f ∗

x(xi − xi) + f ∗y (U

(m)i − U

(∗)i ) + f ∗

y′(U′(m) − U ′(∗)),

by using the definition of error vector, we have

D3e(m+1)i = f ∗

y e(m)i + f ∗

y′e′(m)i ,

D3e(m+1)i = f ∗

y e(m)i + f ∗

y′D1e(m)i ,

where D1 and D3 are the derivative difference operators defined as

D1fi =1

2973180h[−5(fi+6 − fi−6) + 1024(fi+5 − fi−5) + 13225(fi+4 − fi−4)

−199680(fi+3 − fi−3) + 1141695(fi+2 − fi−2)− 4715520(fi+1 − fi−1)] ,

99

D3fi =1

1879920h3[−225(fi+6 − fi−6) + 11520(fi+5 − fi−5)− 10952(fi+4 − fi−4)

−476928(fi+3 − fi−3) + 3047987(fi+2 − fi−2)− 4677632(fi+1 − fi−1)] .

This implies

D3e(m+1)i = h3f ∗

y e(m)i + h2f ∗

y′D1e(m)i .

Since ei = eN−i = 0, i = 0,−1,−2, · · · ,−6, therefore we have

Ae(m+1)i = h3f ∗

y e(m)i + h2f ∗

y′D1e(m)i .


e(m+1)i = A−1(h3f ∗

y e(m)i + h2f ∗

y′D1e(m)i ).

By taking norm on both sides, we get

∥e(m+1)i ∥ = ∥A−1∥∥h3f ∗

y e(m)i + h2f ∗

y′D1e(m)i ∥.

By using the definition of Lipschitz condition, we get

∥e(m+1)i ∥ ≤ ∥A−1∥(h3M0∥e(m)

i ∥+ h2M1∥D1∥∥e(m)i ∥). (4.38)

This implies

∥e(m+1)i ∥

∥e(m)i ∥

≤ ∥A−1∥(h3M0 + h2M1∥D1∥

),

which is equivalent to

∥e(m+1)i ∥

∥e(m)i ∥

≈ h2M1∥A−1∥∥D1∥,

The results follows immediately from this inequality and the following fact

∥D1∥ =867307

212370.

A simple approximation of condition by omitting the cubic term is

h ≤(212370

867307M−1

1

∥∥A−1∥∥−1

) 12

.

This complete the proof.

100

Proposition 4.3.4. If f satisfies the Lipschitz condition (4.3) and Lipschitz constants

M0, M1 and M2 and mesh size h are small enough then nonlinear system (4.23) has

a unique solution U∗. A sufficient condition for the existence of a solution is given by

(4.37).

Proof. From the proof of previous proposition, we observe that if (4.37) holds,

than an inequality similar to that of (4.38) holds, which implies that the sequence

{Um} is contracting and hence converges. The limit U∗ of (4.35) also satisfies

(4.31) due to the continuity of the right hand side function f(x, y, y′, y′′).

Remark 4.3.1. The numerical complexity for the solution of the linear system

(4.35), where the matrix A is almost band matrix with half band 7, using Gaus-

sian elimination method is about 49(N + 13) multiplications. The number of

complexity depends upon the efficient boundary treatment. If more efficient

boundary treatment is constructed than number of complexity will be reduced.

4.4 Error analysis

From the approximation properties of the basis function g(x), it is shown that

the collocation method (4.11) with nonic precision treatments at end points has

at least the power of approximation O(h3). Here we present our main results for

error estimation. Proof of these results are similar to the proof of proposition in

Mustafa and Ejaz (2014).

Proposition 4.4.1. Suppose the exact solution y(x) ∈ C3[0, 1] and {ui} are obtained


||err(x)||∞ = ||U (l)(x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3.


101

Proof. Since the order of approximation of subdivision scheme (4.2) is ten so

by direct calculation (third left eigenvector), we can find derivative of smooth

function y(x) as

y′′′(xj) =1

1879920h3 [−225{y(xj + 6h)− y(xj − 6h)}+ 11520{y(xj + 5h)− y(xj − 5h)}

−10952{y(xj + 4h) + y(xj − 4h)} − 476928{y(xj + 3h) + y(xj − 3h)}

+3047987{y(xj + 2h)− y(xj − 2h)} − 4677632{y(xj + h)− y(xj − h)}]

+O(h8).


y′′′(xj) =1

1879920h3[−225(yj+6 − yj−6) + 11520(yj+5 − yj−5)− 10952(yj+4 − yj−4)

−476928(yj+3 − yj−3) + 3047987(yj+2 − yj−2)− 4677632(yj+1 − yj−1)]

+O(h8). (4.39)

Similarly, we have

U ′′′(xj) =1

1879920h3[−225(uj+6 − uj−6) + 11520(uj+5 − uj−5)− 10952(uj+4 − uj−4)

−476928(uj+3 − uj−3) + 3047987(uj+2 − uj−2)− 4677632(uj+1 − uj−1)]

+O(h8). (4.40)

If we define error function e(x) = U(x)− y(x) and error vectors at the nodes by

e(xj) = U(xj)− y(xj + jh), −6 ≤ j ≤ N + 6,

or equivalently ej = Zj − yj, −6 ≤ j ≤ N + 6, then this impliese′j = U ′

j − y′j,

e′′j = U ′′j − y′′j ,

e′′′j = U ′′′j − y′′′j .


102

y′′′

j − U′′′

j =1

1879920h3[−225{(yj+6 − yj−6)− (uj+6 − uj−6)}+ 11520{(yj+5

−yj−5)− (uj+5 − uj−5)} − 10952{(yj+4 − yj−4)− (uj+4 − uj−4)}

−476928{(yj+3 − yj−3)− (uj+3 − uj−3)}+ 3047987{(yj+2 − yj−2)

−(uj+2 − uj−2)} − 4677632{(yj+1 − yj−1)− (uj+1 − uj−1)}] +O(h8).

This implies

e′′′

j =1

1879920h3[−225(ej+6 − ej−6) + 11520(ej+5 − ej−5)− 10952(ej+4

−ej−4)− 476928(ej+3 − ej−3) + 3047987(ej+2 − ej−2)− 4677632(ej+1

−ej−1)] +O(h8). (4.41)

From (4.1), (4.11), (4.41) and by assuming the eighth order boundary treatments

at the end points, we have

e′′′

j = ajej + bje′

j, 0 ≤ i ≤ N (4.42)

and

ej =

max0≤k≤6

{|ek|}O(h8), −6 ≤ j ≤ 0

maxN−6≤k≤N

{|ek|}O(h8), N ≤ j ≤ N + 6(4.43)

where j = 0, 1, · · · , N ,

aj = fy(tj, y∗j , y

′∗j ), bj = fy′(tj, y

∗j , y

′∗j ),

and

y∗j = yj + θjej, y′∗j = y′j + θje

′

j, 0 ≤ θj ≤ 1.

Using the results (4.41) and

D1Ui =1

2973180h[−5(ui+6 − ui−6) + 1024(ui+5 − ui−5) + 13225(ui+4 − ui−4)

−199680(ui+3 − ui−3) + 1141695(ui+2 − ui−2)− 4715520(ui+1 − ui−1)] .

103

It can be conclude that relation (4.42) and (4.43) is equivalent to

(A+O(h6)−O(h3)−D1O(h))E = O(h8)∥E∥,

where E = (e−6, e−5, · · · , e5, e6).

Hence for small h, the coefficient matrix A+O(h2), will be invertible, thus us-

ing the standard result from algebra and effect of ∥A−1∥ , we have the following

estimate

∥E∥ ≤ ∥A−1∥1−O(h2)

O(h8) = O(h3).

This completes the result.

4.5 Examples, comparison and conclusion

In this section, we use subdivision based collocation algorithm to find the solu-

tion of some nonlinear third order boundary value problems. We present nu-

merical results in table format along with their graphical representations. We

also give comparison of the results obtained by our algorithm and the results

computed by existing algorithms. We end this section with precise conclusion.

4.5.1 Numerical examples

We find the approximate solutions of the following nonlinear problems to check

the accuracy and convergence of subdivision based iterative collocation algo-

rithm.

Example 4.5.1. The nonlinear boundary value problem

y′′′= −2e−3y + 4(1 + x)−3, (4.44)


y(0) = 0, y′(0) = 1, y(1) = ln(2).

104


y′′′(x) = e−xy2(x) (4.45)


y(0) = 1, y′(0) = 1, y(1) = e.


y′′′(x) = −exy2(x) (4.46)


y(0) = 1, y′(0) = −1, y(1) =1

e.

The exact solutions of the problems (4.44), (4.45) and (4.46) are y = ln(1 + x),

y = ex and y = e−x respectively.

Example 4.5.4. We consider the third order ordinary differential equation

y′′′ = y−k, k > 0, (4.47)

where k is constant. The initial conditions imposed by Tanner (1979) are

y(0) = 1, y′(0) = 0. (4.48)

The problem is closed by the boundary condition

y(r) = 0, (4.49)

and the problem become singular at y = 0. The boundary condition is imposed

Momonait (2011) as

y(r) = ϵ, (4.50)

105

where r is a constant satisfying r > 0. The analytic solution of (4.47) by Momon-

ait (2011)

y(x) =1

6(6− 7x2 + 6ϵx2 + x3). (4.51)

Case-1( Momonait (2011)): For k = 0 the problem (4.47) takes the form

y′′′ = y0 = 1, (4.52)

We solve (4.52) along with the conditions (4.48) and (4.50) by letting r = 1

then ϵ = 1112

and obtain the results which also support our algorithm. That is the

numerical results have the order of approximation O(h3). The numerical results

are tabulated in Table 4.4 and graphical representation of these results is shown

in Figure 4.4. These results are obtained after first iteration level. The maximum

absolute error is 6.1250× 10−3.

Case-2 (Momonait (2011)): For k = 12

the problem (4.47) takes the form

y′′′ = y−12 , (4.53)

with the boundary conditions y(0) = 1, y′(0) = 0 and y(r) = ϵ. The numerical

solution of (4.53) is obtained by using the proposed numerical algorithm. The

solution after third iteration level is presented in Table 4.5 and their graphical

representation is presented in Figure 4.5. The maximum absolute error for this

problem is 6.4004× 10−3.

Case-3 (Duffy and Wilson (1979) and Momonait (2011)): In this case, we nu-

merically solve the problem (4.47) for k = 2 i. e.

y′′′ = y−2, (4.54)

together with the boundary conditions y(0) = 1, y′(0) = 0 and y(r) = ϵ. Its

exact solution is given in (4.52). The numerical solution of (4.54) with given and

106

imposed condition at node points is tabulated in Table 4.6 and graphical repre-

sentation is given in Figure 4.6. The maximum absolute error for this problem

is 4.93276× 10−3.

4.5.2 Comparison and discussion

For Examples 4.5.1, 4.5.2 and 4.5.3, we use the iterative collocation algorithm

described in Section 4.3, for h = 10−1, 20−1, 50−1 (i-e. for N=10, 20, 50) and tol =

10−6 along with eighth order boundary treatment at end points, to get solutions

of nonlinear boundary value problems. The numerical results are obtained after

third iteration with condition (4.36).

• The numerical solutions of the problems (4.44), (4.45) and (4.46) are pre-

sented in Tables 4.1, 4.2 and 4.3 respectively.

• Caglar et al. (1999) solved the problem (4.44) by fourth degree B-spline

algorithm. The maximum absolute error obtained by the proposed algo-

rithm and by Caglar et al. (1999) are 4.77×10−3 and 5.80×10−2 respective-

ly. The graphical comparison between exact and approximate solutions is

shown in Figure 4.1. We observe that numerical results obtained by pro-

posed algorithm are better than the results of caglar et al. (1999).

• Hasan (2012) solved the problem (4.45) by modified Adomian decompo-

sition method (MADM). We also solve this problem by subdivision based

collocation algorithm. Here we observe that the order of approximation

by proposed and MADM algorithms is same (i.e.O(h3)). The graphical

comparison between exact and approximate solutions is shown in Figure

4.2.

• The comparison between exact and approximate solutions of problem (4.46)

is given in Figure 4.3.

107

In Example 4.5.4, we consider the problem related to thin film flows. We solve

(4.47) by assuming different values of k and we observer from the numerical

results tabulated in Table 4.4, 4.5 and 4.6 accuracy of the approximate solution

is O(h3). The numerical results are obtained after first and third iteration for

k = 0 and k = 12, 1 with condition (4.36) respectively.

The numerical solutions of Examples 1, 2 and 3 at different step sizes are shown

in Figure 4.7. From this figure, we see that step sizes have small effect on the

numerical solutions of BVPs.

4.5.3 Conclusion

In this chapter, we have presented subdivision based iterative collocation algo-

rithm for the solution of nonlinear third order boundary value problems. The

proposed algorithm has been applied on different nonlinear third order bound-

ary value problems. Numerical results show that the accuracy of approximate

solution is O(h3). We have also observed that the accuracy of the solution can

be improved by choosing different subdivision schemes with the proper adjust-

ment of boundary conditions. Our proposed algorithm gives better results com-

parative to the solution obtained by fourth degree B-spline Caglar et al. (1999).

The order of approximation by the proposed algorithm and modified Adomian

decomposition method Hasan (2012) is the same.

108

Table 4.1: Numerical results of Example 4.5.1: h = 10−1

xi Analytic Approximate Error by prop- Error by Cag-

solution yi solution Zi osed method lar et al. (1999)

0.0 0 0 0 0

0.1 0.0953101798 0.0957100706 0.0003998908 0.009

0.2 0.1823215568 0.1836519852 0.0013304284 0.013

0.3 0.2623642645 0.2648256813 0.0024614168 0.031

0.4 0.3364722366 0.3400108156 0.0035385790 0.045

0.5 0.4054651081 0.4098275640 0.0043624559 0.054

0.6 0.4700036292 0.4747771017 0.0047734725 0.058

0.7 0.5306282511 0.5352702927 0.0046420416 0.058

0.8 0.5877866649 0.5916474874 0.0038608225 0.053

0.9 0.6418538862 0.6441934825 0.0023395963 0.044

1.0 0.6931471806 0.6931471806 0 0.000

109

Table 4.2: Numerical results of Example 4.5.2: h = 10−1

xi Analytic Approximate Error by prop-

solution yi solution Zi osed method

0.0 1 1 0

0.1 1.105170918 1.1056664532 0.000495535

0.2 1.221402758 1.2232054699 0.001802712

0.3 1.349858808 1.3534912400 0.003632432

0.4 1.491824698 1.4974920669 0.005667369

0.5 1.648721271 1.6562805667 0.007559296

0.6 1.822118800 1.8310443828 0.008925583

0.7 2.013752707 2.0230974700 0.009344763

0.8 2.225540928 2.2338920118 0.008351084

0.9 2.459603111 2.4650310392 0.005427928

1.0 2.718281828 2.7182818285 0

110

Table 4.3: Numerical results of Example 4.5.3 : h = 10−1



0.0 1 1 0

0.1 0.9048374180 0.9045439989 0.0002934191

0.2 0.8187307531 0.8177120092 0.0010187439

0.3 0.7408182207 0.7388586762 0.0019595445

0.4 0.6703200460 0.6674013997 0.0029186463

0.5 0.6065306597 0.6028146540 0.0037160057

0.6 0.5488116361 0.5446245817 0.0041870544

0.7 0.4965853038 0.4924038346 0.0041814692

0.8 0.4493289641 0.4457666384 0.0035623257

0.9 0.4065696597 0.4043640577 0.0022056020

1.0 0.3678794412 0.3678794412 0

111

Table 4.4: Numerical results of Example 4.5.4 : k = 0 and h = 10−1



0.0 1 1 0

0.1 0.9976666667 0.9980416667 0.0003750000

0.2 0.9913333333 0.9926666667 0.0013333334

0.3 0.9820000000 0.9846250000 0.0026250000

0.4 0.9706666667 0.9746666667 0.0040000000

0.5 0.9583333333 0.9635416667 0.0052083334

0.6 0.9460000000 0.9520000000 0.0060000000

0.7 0.9346666667 0.9407916667 0.0061250000

0.8 0.9253333333 0.9306666667 0.0053333334

0.9 0.9190000000 0.9223750000 0.0033750000

1.0 0.9166666667 0.9166666667 0

112

Table 4.5: Numerical results of Example 4.5.4 : k = 12

and h = 10−1



0.0 1 1 0

0.1 0.9976666667 0.9980508018 0.0003841351

0.2 0.9913333333 0.9927028673 0.0013695340

0.3 0.9820000000 0.9847044600 0.0013695340

0.4 0.9706666667 0.9748013302 0.0041346635

0.5 0.9583333333 0.9637358441 0.0054025108

0.6 0.9460000000 0.9522463844 0.0062463844

0.7 0.9346666667 0.9410670255 0.0064003588

0.8 0.9253333333 0.9309274865 0.0055941532

0.9 0.9190000000 0.9225533666 0.0035533666

1.0 0.9166666667 0.9166666692 0

113

Table 4.6: Numerical results of Example 4.5.4 : k = 2 and h = 10−1



0.0 1 1 0

0.1 0.9976666667 0.9980024045 0.0003357378

0.2 0.9913333333 0.9925110306 0.0011776973

0.3 0.9820000000 0.9842830603 0.0022830603

0.4 0.9706666667 0.9740862436 0.0034195769

0.5 0.9583333333 0.9627028681 0.0043695348

0.6 0.9460000000 0.9509327630 0.0049327630

0.7 0.9346666667 0.9395952385 0.0049285718

0.8 0.9253333333 0.9295297470 0.0041964137

0.9 0.9190000000 0.9215950588 0.0025950588

1.0 0.9166666667 0.9166666667 0

114

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X

Sol

utio

n

Figure 4.1: Comparison of the analytic and approximate solution of Example 4.5.1

by proposed algorithm and Caglar et al. (1999). In this figure solid line shows exact

solution, dotted lines show approximate solution by proposed algorithm and dashed

lines show the solution obtained by Caglar et al. (1999).

115

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

X

Sol

utio

n

Figure 4.2: Comparison of the analytic and approximate solution of Example 4.5.2 by

proposed algorithm and Hasan (2012). In this figure solid line shows exact solution,

dotted lines show approximate solution by proposed algorithm and dashed lines show

the solution obtained by Hasan (2012).

116

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Sol

utio

n

Figure 4.3: Comparison of the analytic and approximate solution of Example 4.5.3 by

proposed algorithm. In this figure solid line shows exact solution and dashed lines show

approximate solution by proposed algorithm.

117

0 0.2 0.4 0.6 0.8 10.9

0.92

0.94

0.96

0.98

1

X

Sol

utio

n

Figure 4.4: Comparison between exact and approximate solutions of Example 4.5.4

for k = 0. Solid line represents exact solution and dash line represents approximate

solution.

118

0 0.2 0.4 0.6 0.8 10.9

0.92

0.94

0.96

0.98

1

X

Sol

utio

n



solution.

119

0 0.2 0.4 0.6 0.8 10.9

0.92

0.94

0.96

0.98

1

X

Sol

utio

n



solution.

120

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X

Sol

utio

n

h=1/10h=1/20h=1/50

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

X

Sol

utio

n

h=1/10h=1/20h=1/50

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Sol

utio

n

h=1/10h=1/20h=1/50

Figure 4.7: Approximate solutions of Examples 4.5.1, 4.5.2 and 4.5.3 at different step

sizes. 121

Chapter 5

A Numerical Approach Based on

Subdivision Schemes for Solving

Nonlinear Fourth Order Boundary

Value Problems

In this chapter, we present an iterative collocation algorithm based on interpo-

lating subdivision schemes for the solution of nonlinear fourth order boundary

value problems. Main purpose of this chapter is to explore and seek the ap-

plications of subdivision schemes in the field of physics and engineering. We

consider the following type of nonlinear boundary value problem

y(iv) = f(x, y, y′), (5.1)

with the boundary condition y(a) = α1, y′(a) = α2,

y(b) = α3, y′(b) = α4,(5.2)

where αi, i = 1, 2, 3, 4 are constants. We assume that the problem is well-posed.

In Section 5.1, some results about subdivision algorithms and basis function are

122

given. In Section 5.2, a numerical method to solve (5.1) using the refinable ba-

sis functions is formulated and its convergence properties are discussed. Error

properties are given in Section 5.3. Numerical examples illustrating the feasibil-

ity of our proposed algorithm are given in Section 5.4.

5.1 Basis functions and their derivatives

Some useful results for the solution of nonlinear boundary value problem are

discussed in this section. Introduction to the basis functions of subdivision

schemes that are used to construct the approximate solutions of (5.1) is also

part of this section.

5.1.1 Interpolating subdivision scheme

A mathematical formulation of binary subdivision scheme is defined asP k+12i =

∑j∈Z

a−2jPki+j

P k+12i+1 =

∑j∈Z

ai−2jPki+j

(5.3)

thus the scheme is a stepwise interpolatory scheme iff the coefficient ai satisfy

a2i = δi ∀ i ∈ Z. We consider the following binary interpolating subdivision

scheme by Deslauriers and Dubuc (1989), Qu (1994) and Deng and Ma (2013)

pk+12i = pki ,

pk+12i+1 =

3565536

(pki−4 + pki+5

)− 405

65536

(pki−3 + pki+4

)+ 567

16384

(pki−2 + pki+3

)− 2205

16384

(pki−3 + pki+4

)+ 19845

32768

(pki + pki+1

).

(5.4)

The scheme (5.4) is C4-continuous, having support length (−9, 9) and approxi-

mation order is ten.

123

5.1.2 Basis functions

The basis functions is the limit function resulting from cardinal data, where all

vertices of the polygon have value zero except for one. Let ϕ(x), x ∈ R be the

fundamental solution of (5.4) satisfies the two scale equation

ϕ(x) = ϕ(2x) +1

65536[39690{ϕ(2x− 1) + ϕ(2x+ 1)} − 8820{ϕ(2x− 3)

+ϕ(2x+ 3)}+ 2268{ϕ(2x− 5) + ϕ(2x+ 5)} − 405{ϕ(2x− 7)

+ϕ(2x+ 7)}+ 35{ϕ(2x− 9) + ϕ(2x+ 9)}] , x ∈ R (5.5)

and

ϕ(x) ∈ C4, ϕ(x) = 0, x ∈]− 8, 8[, ϕ(i) = δ0, i ∈ Z. (5.6)

Furthermore, it has the following derivatives, first derivatives of ϕ(i) at i ∈

[−8, 8] are

ϕ(i)(0) = 0, ϕ(i)(±1) = ∓19146219521159104017

,

ϕ(i)(±2) = ± 5304527961159104017

, ϕ(i)(±3) = ∓ 147046413780629

,

ϕ(i)(±4) = ± 172970691159104017

, ϕ(i)(±5) = ∓ 27729925795520085

,

ϕ(i)(±6) = ∓ 112763610431936153

, ϕ(i)(±7) = ∓ 40968113728119

,

ϕ(i)(±8) = ∓ 59272832136

,

(5.7)

second derivatives of ϕ(i) at i ∈ [−8, 8] are

ϕ(ii)(0) = −2370618501415309077185968

, ϕ(ii)(±1) = 3265310153216676106344305

,

ϕ(ii)(±2) = −878265102572676106344305

, ϕ(ii)(±3) = 7340630594562028319032915

,

ϕ(ii)(±4) = − 808839012771352212688610

, ϕ(ii)(±5) = 214899200135221268861

,

ϕ(ii)(±6) = 297875188405663806583

, ϕ(ii)(±7) = 6400019317324123

,

ϕ(ii)(±8) = 4375618154371936

,

(5.8)

124

third derivatives of ϕ(i) at i ∈ [−8, 8] are

ϕ(iii)(0) = 0, ϕ(iii)(±1) = ±4331751500815295995855

,

ϕ(iii)(±2) = ∓12153051235761183983420

, ϕ(iii)(±3) = ±240606976566518365

,

ϕ(iii)(±4) = ∓ 5285889107244735933680

, ϕ(iii)(±5) = ∓ 374141443059199171

,

ϕ(iii)(±6) = ± 1090169453214692

, ϕ(iii)(±7) = ∓ 21760437028453

,

ϕ(iii)(±8) = ∓ 297513984910496

,

(5.9)

fourth derivatives of ϕ(i) at i ∈ [−8, 8] are

ϕ(iv)(0) = 33869667457408

, ϕ(iv)(±1) = −529505475289730585

,

ϕ(iv)(±2) = 10404741119358922340

, ϕ(iv)(±3) = −748795849970065

,

ϕ(iv)(±4) = 2950208692871378720

, ϕ(iv)(±5) = 923862417946117

,

ϕ(iv)(±6) = − 9001877976052

, ϕ(iv)(±7) = 7184017946117

,

ϕ(iv)(±8) = 11225328157568

.

(5.10)

The above derivative values are found by using the left eigenvectors of the sub-

division process (5.4). The detailed description about these left eigenvectors

and derivatives can be found in Qu (1996) and Mustafa and Ejaz (2014). The

graphical representations of above derivatives are given in Figure 5.1

5.2 Description of iterative numerical algorithm

This section describes the method for the numerical solution of nonlinear bound-

ary value problem (5.1). The detail of the method is given below:


In this subsection, the collocation method is constructed based on the interpo-

lating subdivision scheme (5.4). Our numerical approach for nonlinear fourth

order boundary value problem using collocation method based on subdivision

125

Figure 5.1: Graphical representation of basis functions is shown in figure (a), and first,

second, third and fourth derivatives of basis function are shown in figure (b), (c), (d)

and (e). 126

scheme is to seek an approximate solution as

Z(x) =N+8∑i=−8

ziϕ

(x− xi

h

), 0 6 x 6 1, (5.11)

where N is the positive integer N > 8, h = 1/N and xi = i/N = ih, and {zi}

are the unknown to be determined for the solution of (5.1). In order to solve

the problem, a collocation method Z(x) is considered to be the solution of the

above differential equation at x = xj and we substitute equation (5.11) into the

equation (5.1). This leads to

Z(iv)(xj) = f(xj, Z(xj), Z′(xj)), j = 0, 1, 2, · · · , N, (5.12)

and boundary conditions

Z(0) = α1, Z′(0) = α2, Z(N) = α3, Z

′(N) = α4, (5.13)

From (5.11), we get

Z(iv)(x) =1

h4

N+8∑i=−8

ziϕ(iv)

(x− xi

h

), 0 6 x 6 1, (5.14)

substituting (5.14) into (5.12), we obtain

N+8∑i=−8

ziϕ(iv)

(xj − xi

h

)= h4f(xj, Z(xj), Z

′(xj)), j = 0, 1, 2, · · · , N.

This can be written asN+8∑i=−8

ziϕ(iv)j−i = h4f(xj, Z(xj), Z

′(xj)), j = 0, 1, 2, · · · , N.

Since ϕ(iv)i = ϕ

(iv)−i , the above system of equations become

N+8∑i=−8

ziϕ(iv)i−j = h4f(xj, Z(xj), Z

′(xj)), j = 0, 1, 2, · · · , N. (5.15)

The nonlinear system of equations (5.15) can be simply in following Theorems

5.2.1 and 5.2.2.

127

Theorem 5.2.1. The nonlinear system of equations (5.15) for j = 0 becomes

8∑i=−8

ziϕ(iv)i = h4f(x0, Z(x0), Z

′(x0)). (5.16)

Proof. By expanding (5.15) for j = 0, we get

N+8∑i=−8

ziϕ(iv)i = h4f(x0, Z(x0), Z

′(x0)),


iv−7 + z−6ϕ

iv−6 + · · ·+ z7ϕ

iv7 + z8ϕ

iv8 + z9ϕ

iv9 + · · ·+ zN+7ϕ

ivN+7

+zN+8ϕivN+8 = h4f(x0, Z(x0), Z

′(x0)).

Since ϕiv(i) exist only for the interval for i ∈ [−8, 8] and outside the interval it

will be zero. Then above equation can be written as


iv−7 + z−6ϕ

iv−6 + · · ·+ z7ϕ

iv7 + z8ϕ

iv8 = h4f(x0, Z(x0), Z

′(x0)).

Theorem 5.2.2. For j = 1, 2, · · · , N , the nonlinear system of equations (5.15) becomes

8+j∑i=−8+j

ziϕ(iv)i−j = h4f(xj, Z(xj), Z

′(xj)). (5.17)

Proof. By expanding (5.15), for j = 1, 2, 3, · · · , N, we get

z−8ϕiv−8−j + z−7ϕ

iv−7−j + z−6ϕ

iv−6−j + · · ·+ z7ϕ

iv7−j + z8ϕ

iv8−j + z9ϕ

iv9−j + z10ϕ

iv10−j

+ · · ·+ zN+6ϕivN+6−j + zN+7ϕ

ivN+7−j + zN+8ϕ

ivN+8−j = h4f(xj, Z(xj), Z

′(xj)).

(5.18)

Substituting j = 1 in (5.18), it becomes

z−8ϕiv−8−1 + z−7ϕ

iv−7−1 + z−6ϕ

iv−6−1 + · · ·+ z7ϕ

iv7−1 + z8ϕ

iv8−1 + z9ϕ

iv9−1 + z10ϕ

iv10−1

+ · · ·+ zN+6ϕivN+6−1 + zN+7ϕ

ivN+7−1 + zN+8ϕ

ivN+8−1 = h4f(x1, Z(x1), Z

′(x1)).

128

This implies


iv−8 + z−6ϕ

iv−7 + · · ·+ z7ϕ

iv6 + z8ϕ

iv7 + z9ϕ

iv8 + z10ϕ

iv9 + · · ·

+zN+6ϕivN+5 + zN+7ϕ

ivN+6 + zN+8ϕ

ivN+7 = h4f(x1, Z(x1), Z

′(x1)).

Since ϕiv(i) is non-zero only for the interval for i ∈ [−8, 8] and outside the inter-

val it will be zero. Then above equation becomes


iv−7 + · · ·+ z7ϕ

iv6 + z8ϕ

iv7 + z9ϕ

iv8 = h4f(x1, Z(x1), Z

′(x1)). (5.19)

For j=2, (5.18) becomes

z−8ϕiv−8−2 + z−7ϕ

iv−7−2 + z−6ϕ

iv−6−2 + · · ·+ z7ϕ

iv7−2 + z8ϕ

iv8−2 + z9ϕ

iv9−2 + z10ϕ

iv10−2

+ · · ·+ zN+6ϕivN+6−2 + zN+7ϕ

ivN+7−2 + zN+8ϕ

ivN+8−2 = h4f(x2, Z(x2), Z

′(x2)).

This implies

z−8ϕiv−10 + z−7ϕ

iv−9 + z−6ϕ

iv−8 + · · ·+ z7ϕ

iv5 + z8ϕ

iv6 + z9ϕ

iv7 + z10ϕ

iv8

+ · · ·+ zN+6ϕivN+4 + zN+7ϕ

ivN+5 + zN+8ϕ

ivN+6 = h4f(x2, Z(x2), Z

′(x2)).

By using the definition of ϕivi given in (5.10), above equation yields


iv−7 + · · ·+ z7ϕ

iv5 + z8ϕ

iv6 + z9ϕ

iv7 + z10ϕ

iv8 = h4f(x2, Z(x2), Z

′(x2)).

(5.20)

By using the similar pattern for j = 1, 2, we can find the expression for j =

3, 4, · · ·N ,

z−8+jϕiv−8−j + z−7+jϕ

iv−7−j + z−6+jϕ

iv−6−j + · · ·+ z7+jϕ

iv7−j + z8+jϕ

iv8−j + z9+jϕ

iv9−j

+ · · ·+ zN+6+jϕivN+6−j + zN+7+jϕ

ivN+7−j + zN+8+jϕ

ivN+8−j = h4f(xj, Z(xj), Z

′(xj)).

(5.21)

129

The nonlinear system of equation (5.15) is equivalent to the following non-

linear system of N + 1 equations with (N+17) unknowns {zi}

AZ = F (z) (5.22)

where A is banded matrix of order (N + 1)× (N + 17), Z is the unknown vector

of order N + 17 and F (z) is the vector of order N + 1 depends on z. The matrix

A, vectors Z and F (z) are given explicitly by

A = [ϕivpq(q − p− 8)](N+1)×(N+17), (5.23)

where p = 1, 2, 3 · · · , N + 1 and q = 1, 2, 3, · · · , N + 17 represent the row and

column respectively.

F (z) =(h4f(x0, Z(x0), Z

′(x0)), · · · , h4f(xN , Z(xN), Z

′(xN))

)T

, (5.24)

Z = (z−8, z−7, z−6, · · · , zN+6, zN+7, zN+8)T , (5.25)

Z′(xj) =

N+8∑i=−8

zjϕ′(xj − xi

h

),

where ϕ′(i) is already defined in (5.7) with ϕ(i) = ϕi.

5.2.2 Boundary conditions at end points

For unique solution of nonlinear systems (5.15), we need sixteen more condi-

tions. Four conditions can be attained from given boundary conditions for non-

linear systems of equations and remaining conditions are attained by setting

some extrapolation method. The detail of the given boundary conditions and

extrapolation method are given below:

5.2.3 Boundary conditions

The given boundary conditions are :

Z(0) = α1, Z′(0) = α2, Z(N) = α3, Z

′(N) = α4.

130

The approximation of derivative conditions at ends point is defined as

Z ′(0) =

(N

2520

){−7381z0 + 25200z1 − 56700z2 + 100800z3 − 132300z4 + 127008z5

−88200z6 + 43200z7 − 14175z8 + 28800z9 − 252z10}+O(h10), (5.26)

Z ′(N) =

(N

2520

){7381zN − 25200zN−1 + 56700zN−2 − 100800zN−3 + 132300zN−4

−127008zN−5 + 88200zN−6 − 43200zN−7 + 14175zN−8 − 28800zN−9

+252zN−10}+O(h10). (5.27)

5.2.4 Extrapolation method

The remaining twelve conditions for the nonlinear systems (5.15) to obtain sta-

ble systems for the solution of (5.1) are obtained by setting the following extrap-

olation method.

We define six conditions at left end points and six conditions at the right end

points. Since subdivision scheme (5.4) reproduces nine degree (i.e. tenth or-

der) polynomials, so we define boundary conditions of order ten for solution of

(5.15). For simplicity only left end points z−7, z−6, z−5, z−4, z−3, z−2 are discussed

and the values of right end points zN+2, zN+3, zN+4, zN+5, zN+6, zN+7 can be treat-

ed similarly.

The values z−7, z−6, z−5, z−4, z−3, z−2 can be determined by the polynomial q(x)

interpolating (xi, zi), 2 ≤ i ≤ 7. Precisely, we have

z−i = q(−xi), i = 2, 3, 4, 5, 6, 7,

where

q(xi) =10∑j=1

10

j

(−1)j+1Z(xi−j).

From (5.13), Z1(xi) = zi for i = 2, 3, 4, 5, 6, 7 and replacing xi by −xi, we have

q(−xi) =10∑j=1

10

j

(−1)j+1z−i+j.

131


10∑j=0

10

j

(−1)jz−i+j = 0, i = 7, 6, 5, 4, 3, 2. (5.28)


N + 5, N + 6, N + 7 and

q(xi) =10∑j=1

10

j

(−1)j+1zi−j.


10∑j=0

10

j

(−1)jzi−j = 0, i = N + 2, N + 3, N + 4, N + 5, N + 6, N + 7.

(5.29)

Finally, we obtain a new system of (N + 17) linear equations with (N + 17)

unknowns {zi}. The N + 1 equations are obtained from (5.15), four equations

from boundary conditions (5.13) and twelve from boundary conditions (5.28)

and (5.29).

Hence the stable nonlinear system of equations is define as

BZ = R(z), (5.30)

where the matrix B is given by

B = (CT0 , A

T , CT1 )

T , (5.31)

A is defined in (5.23), C0, C1 and the vector R(z) is defined as

132

C0 =

0 1 −10 45 −120 210 −252 210 −120 45 −10 1

0 0 1 −10 45 −120 210 −252 210 −120 45 −10

0 0 0 1 −10 45 −120 210 −252 210 −120 45

0 0 0 0 1 −10 45 −120 210 −252 210 −120

0 0 0 0 0 1 −10 45 −120 210 −252 210

0 0 0 0 0 0 1 −10 45 −120 210 −252

0 0 0 0 0 0 0 0 7381N2520

25200N2520 −56700N

2520100800N

2520

0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 · · · 0 0

1 0 0 0 0 0 0 · · · 0 0

−10 1 0 0 0 0 0 · · · 0 0

45 −10 1 0 0 0 0 · · · 0 0

−120 45 −10 1 0 0 0 · · · 0 0

210 −120 45 −10 1 0 0 · · · 0 0

− 132300N2520

127008N2520 − 88200N

252043200N2520 −14175N

25202800N2520 −252N

2520 · · · 0 0

0 0 0 0 0 0 0 · · · 0 0

,(5.32)

the first six rows of C0 are obtained from (5.28), second last row is obtained

from (5.26) and last row is taken from given boundary conditions Z1(0) which

is defined in (5.13) and

C1 =

0 0 · · · 0 0 0 0 0 0 0 0 0 0

0 0 · · · N10 −10N

945N8 − 120N

7 35N − 252N5

105N2 −40N 45N

2 −10N

0 0 · · · 0 0 1 −10 45 −120 210 −252 210 −120

0 0 · · · 0 0 0 1 −10 45 −120 210 −252 210

0 0 · · · 0 0 0 0 1 −10 45 −120 210 −252

0 0 · · · 0 0 0 0 0 1 −10 45 −120 210

0 0 · · · 0 0 0 0 0 0 1 −10 45 −120

0 0 · · · 0 0 0 0 0 0 0 1 −10 45

133

1 0 0 0 0 0 0 0 0

7381N2520 0 0 0 0 0 0 0 0

45 −10 1 0 0 0 0 0 0

−120 45 −10 1 0 0 0 0 0

210 −120 45 −10 1 0 0 0 0

−252 210 −120 45 −10 1 0 0 0

−120 210 −252 210 −120 45 −10 1 0

, (5.33)

first row of C1 is obtained from Z1(N) which is defined in (5.13), second row is

obtained from (5.27) and the last six rows are obtained from (5.29), Z which is

defined in (5.25) and R(z) is defined as

R(z) = (0, 0, 0, 0, 0, 0, Z ′(0), Z(1), F T (z), Z(1), Z ′(1), 0, 0, 0, 0, 0, 0)T , (5.34)

where F (z) is defined by (5.24).

5.2.5 Non-singularity of a matrix

We can check the non-singularity of coefficient matrix B defined in (5.31) by

different methods. We observe that the determinant of matrix B is non-zero for

N ≤ 500. Hence the nonlinear system of equations have a solution for N ≤

500. We also check the non singularity of matrix by finding eigenvalues up to

N ≤ 500 and we observe that all the eigenvalues are non-zero. Hence by Strang

(2011) we conclude that the B is non-singular. For large N > 500 the matrix may

or may not be singular.

5.2.6 Iterative algorithm and its convergence

An iterative algorithm and its convergence are described in this section.

134

5.2.7 Iterative algorithm based on basis function

The iterative algorithm based on basis function of the subdivision scheme (5.4)

are as defined in the following three steps.

First step: Initial approximation

The initial approximation is important because numerical solution depends on

the initial approximation. We define the process for finding the initial approxi-

mation as follows:

Let initial approximate solution Z0 be the solution of the following linear system

BZ0 = F 0, (5.35)

where

F 0 = (0, 0, 0, 0, 0, 0, y′(a), y(a), f0, f1, f2, · · · , fN , y(b), y′(b), 0, 0, 0, 0, 0, 0)T ,

fi = h4f(xi, Li, D), i = 0, 1, 2, · · ·N

Li = y(0) + ih(

y(b)−y(a)b−a

),

D = y(b)− y(a).

(5.36)

F 0 is the initial linear approximation of the nonlinear vector R(z).

Second step: Numerical solution

The numerical solutions Z∗ of the nonlinear system are obtained by using sim-

ple iterative scheme

BZ(m+1) = R(Zm), m = 0, 1, 2, 3, · · · (5.37)

135

Third step: Stopping condition

The above iterative processes will terminate when the following condition sat-

isfied

||z(m) − z(m−1)|| ≤ tol. (5.38)

where tolerance is supposed value i.e. tol = 10−6. The convergence of the above

iterative algorithm is guaranteed by the following proposition.

Proposition 5.2.3. The successive solutions {Z(m)} generated by the iterative algorith-

m (5.37) linearly converges to the solution Z∗ of the nonlinear solution of the system

(5.30) provided that the M0 and M1 are Lipschitz constants and step size h is small.

i.e.

∥∥B−1∥∥ ≤

(M0h

4 +4994220330463

1460471061420M1h

3

). (5.39)

Proof. Let Z∗ and Z(m) be the solutions of the nonlinear system (5.30). Then by

definition, for small h we have

BZ∗ = R(Z∗), (5.40)

BZm+1 = R(Zm). (5.41)

Let the error vector is defined as e(m) = Z(m)−Z∗ at mth iteration which satisfies

BZ(m+1) −BZ∗ = R(Z(m))−R(Z∗),

B(Z(m+1) − Z∗) = R(Z(m))−R(Z∗),

Be(m+1) = R(Z(m))−R(Z∗). (5.42)

For i = 0, 1, 2, · · · , N ,

D4e(m+1)i = (F (Z(m))− F (Z∗))i.

136

By mean value theorem, which is stated as “If a function f(x, y, z) is continuous-

ly differentiable in an open set of R3 containing points (x1, y1, z1) and (x2, y2, z2)

and the line segment connecting them, then an equation

f(x2, y2, z2)− f(x1, y1, z1) =

f′

x(r, s, t)(x2 − x1) + f′

y(r, s, t)(y2 − y1) + f′

z(r, s, t)(z2 − z1),

is valid for the interior point (a, b, c) of the segment.”, we have

D4e(m+1)i = f(xi, Z

(m)i , Z ′(m))− f(xi, Z

(∗)i , Z ′(∗)).

The above equation can be written as (by using mean value theorem)

D4e(m+1)i = f ∗

x(xi − xi) + f ∗y (Z

(m)i − Z

(∗)i ) + f ∗

y′(Z′(m) − Z ′(∗)),

by using the definition of error vector, we have

D4e(m+1)i = f ∗

y e(m) + f ∗

y′e′(m),

D4e(m+1)i = f ∗

y e(m) + f ∗

y′D1e(m),

where D1 and D4 are the derivative difference operators defined as

D1fi =1

2920942122840h[1575(fi−8 − fi+8) + 1474560(fi−7 − fi+7)

+315738080(fi−6 − fi+6) + 1397587968(fi−5 − fi+5)

−43588613880(fi−4 − fi+4) + 311679549440(fi−3 − fi+3)

−1336741045920(fi−2 − fi+2) + 4824847319040(fi−1 − fi+1)] ,

D4fi =1

183768238080h4[392875(fi+8 − fi−8) + 45977600(fi+7 − fi−7)

−1296269280(fi+6 − fi−6) + 5912719360(fi+5 − fi−5)

+1180083476(fi+4 − fi−4)− 86261280768(fi+3 − fi−3)

+332951715808(fi+2 − fi−2)− 677767008256(fi+1 − fi−1)

+850467338370fi] .

137

This implies

D4e(m+1)i = h4f ∗

y e(m) + h3f ∗

y′D1e(m).

Since ei = eN−i = 0, i = 0,−1,−2, · · · ,−8, we have

Be(m+1)i = h4f ∗

y e(m) + h3f ∗

y′D1e(m).


e(m+1)i = B−1(h4f ∗

y e(m) + h3f ∗

y′D1e(m)).

By taking norm on both sides, we get

∥e(m+1)i ∥ = ∥B−1(h4f ∗

y e(m) + h3f ∗

y′D1e(m))∥.

This implies

∥e(m+1)i ∥ = ∥B−1∥∥(h4f ∗

y e(m) + h3f ∗

y′D1e(m))∥.

By using the definition of Lipschitz condition, we get

∥e(m+1)∥ ≤ h4M0(b− a)∥B−1∥∥e(m)∥+ h3M1∥D1∥∥e(m)∥.

This implies

∥e(m+1)i ∥

∥e(m)∥≤ ∥B−1∥

(h4M0(b− a) + h3M1∥D1∥

),

which is equivalent to

∥e(m+1)i ∥

∥e(m)∥≈ h3M1∥B−1∥∥D1∥ ≤ hM1∥B−1∥∥D1∥,

i-e

∥e(m+1)i ∥

∥e(m)∥≈ hM1∥B−1∥∥D1∥.

The results follows immediately from this inequality and the following fact

∥D1∥ =4994220330463

1460471061420.

A simple approximation of condition by omitting the quatric term is

h ≤ 1460471061420

4994220330463M−1

1

∥∥B−1∥∥−1

.

This complete the proof.

138

5.3 Error estimation

From the approximation properties of the basis function ϕ(x), it is shown that

the collocation method (5.11) with nonic precision treatments at end points has

at least the power of approximation O(h3). Here we present our main results for

error estimation. Proof of these results are similar to the proof of proposition by

Qu and Agarwal (1996) and Mustafa and Ejaz (2014).



||err(x)||∞ = ||Z(l)(x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3.


Proof. Since the order of approximation of subdivision scheme (5.4) is ten so

by direct calculation (fourth left eigenvector), we can find derivative of smooth

function y(x) as

yiv(xj) =24

183768238080h4{392875y(xj − 8h) + 45977600y(xj − 7h)

−1296269280y(xj − 6h) + 5912719360y(xj − 5h) + 1180083476y(xj − 4h)

−86261280786y(xj − 3h) + 332951715808y(xj − 2h)− 677767008256y(xj − h)

+850467338370y(xj)− 677767008256y(xj + h) + 332951715808y(xj + 2h)

−86261280786y(xj + 3h) + 1180083476y(xj + 4h) + 5912719360y(xj + 5h)

−1296269280y(xj + 6h) + 45977600y(xj + 7h) + 392875y(xj + 8h)}+O(h10).


yivj =24

183768238080h4{392875yj−8 + 45977600yj−7 − 1296269280yj−6

+5912719360yj−5 + 1180083476yj−4 − 86261280786yj−3 + 332951715808yj−2

−677767008256yj−1 + 850467338370yj − 677767008256yj+1 + 332951715808yj+2

−86261280786yj+3 + 1180083476yj+4 + 5912719360yj+5 − 1296269280yj+6

+45977600yj+7 + 392875yj+8}+O(h10). (5.43)

139

Similarly, we have

Zivj =

24

183768238080h4{392875zj−8 + 45977600zj−7 − 1296269280zj−6

+5912719360zj−5 + 1180083476zj−4 − 86261280786zj−3 + 332951715808zj−2

−677767008256zj−1 + 850467338370zj − 677767008256zj+1 + 332951715808zj+2

−86261280786zj+3 + 1180083476zj+4 + 5912719360zj+5 − 1296269280zj+6

+45977600zj+7 + 392875zj+8}+O(h10). (5.44)

If we define error function e(x) = Z(x)− y(x) and error vectors at the nodes by

e(xj) = Z(xj)− y(xj + jh), −8 ≤ j ≤ N + 8,

or equivalently ej = Zj − yj, −8 ≤ j ≤ N + 8, then this implies

e′j = Z ′j − y′j,

e′′j = Z ′′j − y′′j ,

e′′′j = Z ′′′j − y′′′j

eivj = Zivj − yivj .

(5.45)


yivj − Zivj = 24

183768238080h4 {392875(yj−8 − zj−8) + 45977600(yj−7 − zj−7)

−1296269280(yj−6 − zj−6) + 5912719360(yj−5 − zj−5) + 1180083476(yj−4 − zj−4)

−86261280786(yj−3 − zj−3) + 332951715808(yj−2 − zj−2)− 677767008256(yj−1 − zj−1)

+850467338370(yj − zj)− 677767008256(yj+1 − zj+1) + 332951715808(yj+2 − zj+2)

−86261280786(yj+3 − zj+3) + 1180083476(yj+4 − zj+4) + 5912719360(yj+5 − zj+5)

−1296269280(yj+6 − zj+6) + 45977600(yj+7 − zj+7) + 392875(yj+8 − zj+8)}+O(h10).

This implies

eivj =24

183768238080h4{392875ej−8 + 45977600ej−7 − 1296269280ej−6

+5912719360ej−5 + 1180083476ej−4 − 86261280786ej−3 + 332951715808ej−2

−677767008256ej−1 + 850467338370ej − 677767008256ej+1 + 332951715808ej+2

−86261280786ej+3 + 1180083476ej+4 + 5912719360ej+5 − 1296269280ej+6

+45977600ej+7 + 392875ej+8}+O(h10). (5.46)

140

From (5.1), (5.11), (5.45) and by assuming the tenth order boundary treatments

at the end points, we have

eivj = ajej + bje′

j, 0 ≤ i ≤ N (5.47)

and

ej =

max0≤k≤7

{|ek|}O(h10), −8 ≤ i ≤ 0

maxN−3≤k≤N

{|ek|}O(h10), N ≤ i ≤ N + 8(5.48)

where j = 0, 1, · · ·N

aj = fy(tj, y∗j , y

′∗j ), bj = fy′(tj, y

∗j , y

′∗j ),

and

y∗j = yj + θjej, y′∗j = y′j + θje

′

j, 0 ≤ θj ≤ 1.

Using the results (5.46) and

[1575(zi−8 − zi+8) + 1474560(zi−7 − zi+7) + 315738080(zi−6 − zi+6) + 1397587968

(zi−5 − zi+5)− 43588613880(zi−4 − zi+4) + 311679549440(zi−3 − zi+3)− 1336741045920

(zi−2 − zi+2) + 4824847319040(zi−1 − zi+1)] = 2920942122840hZ′+O(h10),

it can be conclude that relation (5.47) and (5.48) is equivalent to

(B +O(h8)−O(h4)−D1O(h3))E = O(h10)∥E∥,

where E = (e−8, e−7, · · · , e7, e8).

Hence for small h, the coefficient matrix B +O(h), will be invertible, thus us-

ing the standard result from algebra and effect of ∥B−1∥ , we have the following

estimate

∥E∥ ≤ ∥B−1∥1−O(h)

O(h10) = O(h3).

This completes the result.

141

5.4 Results and discussions

In this section, we test the proposed method on some nonlinear problems. Nu-

merical results for each problems are presented in the tables. These values are

very close to the corresponding true solutions and the values of the correspond-

ing errors are also given in the table.

Example 5.4.1. ( Agarwal (1986)) Consider the following nonlinear BVP

yiv − 6e(−4y) = −12(1 + x)−4, (5.49)


y(0) = 0, y′(0) = 1, y(1) = ln(2) = y′(1) = 0.5.

The exact solution of the problem (5.49) is y = ln(1 + x). Using the collocation

method described in Section 5.2 for N = 10, h = 10−1 and tol = 10−6 with tenth

order boundary treatment at end points. The numerical results are obtained

after third iteration with the condition (5.38). The obtained numerical results for

this problem are presented in Table 5.1. The maximum absolute error obtained

by the proposed method is 1.78×10−3. The graphical comparison between exact

and approximate solutions is shown in Figure 5.2.

Example 5.4.2. (Agarwal (1986)) Consider the nonlinear BVP

y(iv) = y2 − x10 + 4x9 − 4x8 − 4x7 + 8x6 − 4x4 + 120x− 48 (5.50)


y(0) = y′(0) = 0, y(1) = y′(1) = 1.

Using the collocation method described in Section 5.2 for N = 10, h = 10−1 and

tol = 10−6 with tenth order boundary treatment at end points. The numerical

results are obtained after third iteration with the condition (5.38). The obtained

142

Table 5.1: Numerical results of Example 5.4.1

xi Analytic Approximate Error

solution Yi solution Zi = ||Yi − Zi||∞

0.0 0 0 0

0.1 0.0953101798 0.0950147533 0.0002954265

0.2 0.1823215568 0.1814496227 0.0008719341

0.3 0.2623642645 0.2609546573 0.0014096072

0.4 0.3364722366 0.3347370220 0.0017352146

0.5 0.4054651081 0.4036840381 0.0017810699

0.6 0.4700036292 0.4684459279 0.0015577013

0.7 0.5306282511 0.5294932609 0.0011349902

0.8 0.5877866649 0.5871580370 0.0006286279

0.9 0.6418538862 0.6416636708 0.0001902154

1.0 0.6931471806 0.6931471806 0

143

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X

Sol

utio

n


Figure 5.2: Comparison of the analytic and approximate solution of Example 5.4.1.

numerical results for this problem are presented in Table 5.2. The maximum

absolute error obtained by the proposed method is 1.73 × 10−2. The graphical

comparison between exact and approximate solutions is shown in Figure 5.3.

144

Table 5.2: Numerical results of Example 5.4.2

xi Analytic Approximate Error

solution Yi solution Zi = ||Yi − Zi||∞

0.0 0 0 0

0.1 0.01981 0.0202195 0.0004095

0.2 0.07712 0.0796952 0.0025752

0.3 0.16623 0.1728732 0.0066432

0.4 0.27904 0.2905995 0.0115595

0.5 0.40625 0.4219208 0.0156708

0.6 0.53856 0.5558846 0.0173246

0.7 0.66787 0.6833406 0.0154706

0.8 0.78848 0.7987412 0.0102612

0.9 0.89829 0.9019417 0.0036517

1.0 1.00000 1.0000000 0

145

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X

Sol

utio

n


Figure 5.3: Comparison of the analytic and approximate solution of Example 5.4.2.

5.5 Conclusion

This study has presented a numerical approach based on subdivision based col-

location algorithm for solving the numerical solution of nonlinear fourth order

boundary value problems. The proposed iterative algorithm has been applied

on different nonlinear fourth order boundary value problems. Numerical re-

sults show that the accuracy of approximate solution is O(h3). We have also

observed that the accuracy of the solution can be improved by choosing differ-

ent subdivision schemes with the proper adjustment of boundary conditions.

146

Chapter 6

Subdivision Based Collocation

Algorithm for Singularly Perturbed

Boundary Value Problems

This chapter investigates the approximate solutions of singularly perturbed sec-

ond order boundary value problems using binary interpolating subdivision sch-

eme based collocation algorithm. These type of problems frequently occur in

various field of science and engineering. We consider a second order singularly

perturbed boundary value problem

εy′′(x) = A(x)y

′(x) +B(x)y(x) + F (x), (6.1)

y(a) = α0, y(b) = α1 , a 6 x 6 b (6.2)

where A(x), B(x), F (x) are smooth bounded real functions and ε is a parameter

such that 0 < ε ≪ 1. Generally, the inhomogeneities F and α0, α1 may depends

on ε as well. It is well known that boundary value problem (6.1) with condition

(6.2) shows boundary layers at one or both ends of the interval depending on

the choice of the function A(x) detail is given in Ascher et al. (1988).

147

In this chapter, we are using subdivision based technique for the solution of

second order singularly perturbed boundary value problems. We develop sub-

division collocation methods based on following 6-point binary interpolating

C2 scheme by Lee et al. (2006)pk+12i = pki ,

pk+12i+1 = ω

(pki−2 + pki+3

)−

(3ω + 1

16

)(pki−1 + pki+2) +

(2ω + 9

16

)(pki + pki+1),

(6.3)

where 0 < ω < 0.042, for the solution of (6.1). The scheme (6.3) has support

length (−5, 5), approximation order four and satisfies following two scale rela-

tion

Ψ(x) = Ψ(2x) +

[ω {Ψ(2x− 1) + Ψ(2x+ 1)} −

(3ω +

1

16

){Ψ(2x− 3) + Ψ(2x

+3)}+(2ω +

9

16

){Ψ(2x− 5) + Ψ(2x+ 5)}

], x ∈ R (6.4)

where

Ψ(x) =

1 for x = 0,

0 for x = 0.(6.5)

This chapter is arranged in following way: In Section 6.1, we discuss derivatives

of the subdivision scheme and construct the subdivision based numerical algo-

rithm for the approximate solution of the singularly perturbed boundary value

problems. In Section 6.2, convergence of the method has been discussed. Sec-

tion 6.3, contains three numerical examples for the illustration of the algorithm

and their results are compared with other methods to show the efficiency of the

method. Conclusion about the numerical results is presented in Section 6.4.

6.1 Derivation of numerical algorithm

In this section, first we discuss the derivatives of (6.4), then we formulate the

subdivision based collocation algorithm for the solution of second order singu-

148

larly perturbed boundary value problems.

6.1.1 First and second derivatives of Ψ(x)

Since the function Ψ(x) ∈ C2, then the first and second derivatives can be

obtained by using the left and right eigenvectors of the subdivision matrix of

scheme (6.3). The first and second derivatives of (6.4) for the parametric value

ω = 0.04 are given in Table 6.1. Similar approach of Mustafa and Ejaz (2014) and

Ejaz et al. (2015) has been used to find derivatives.

Table 6.1: First and second derivatives of Ψ

i 0 ±1 ±2 ±3 ±4

Ψ(i) 1 0 0 0 0

Ψ′(i) 0 ±91004313

∓1967325878

± 160012939

± 12812939

Ψ′′(i) −144314224

43252112

−15752816

25132

133

6.1.2 The subdivision based collocation algorithm

Let N be a positive integer (N ≥ 4), h = 1/N and xi = i/N = ih, i = 0, 1, 2, · · ·N ,

and

U(x) =N+4∑i=−4

uiΨ

(x− xi

h

), 0 ≤ x ≤ 1, (6.6)

be the approximate solution of (6.1) where {ui} are the unknowns to be deter-

mined then

εU ′′(xj) = A(xj)U′(xj) +B(xj)U(xj) + F (xj), j = 0, 1, 2, · · · , N, (6.7)

with given boundary conditions

U(0) = α0, U(1) = α1.

149

From (6.6), we have

U ′(xj) =1h

N+4∑i=−4

uiΨ′ (xj−xi

h

),

U ′′(xj) =1h2

N+4∑i=−4

uiΨ′′ (xj−xi

h

).

(6.8)


εN+4∑i=−4

uiΨ′′(xj − xi

h

)− hAj

N+4∑i=−4

uiΨ′(xj − xi

h

)− h2Bj

N+4∑i=−4

uiΨ

(xj − xi

h

)= h2Fj,

where Aj = A(xj), Bj = B(xj) and Fj = F (xj). This implies

N+4∑i=−4

ui

{εΨ′′

(xj − xi

h

)− hAjΨ

′(xj − xi

h

)− h2BjΨ

(xj − xi

h

)}= h2Fj.

Further implies

N+4∑i=−4

ui

{εΨ′′(j − i)− hAjΨ

′(j − i)− h2BjΨ(j − i)}= h2Fj, (6.9)

where j = 0, 1, 2, · · · , N and xi = ih or xj = jh . By using Ψ(i) = Ψi, (6.9) can be

written as

N+4∑i=−4

ui

{εΨ′′

j−i − hAjΨ′j−i − h2BjΨj−i

}= h2Fj, j = 0, 1, 2, · · · , N. (6.10)

As we observe from Table 6.1, Ψ′−i = −Ψ′

i and Ψ′′−i = Ψ′′

i so (6.10) becomes

N+4∑i=−4

ui

{εΨ′′

i−j + hAjΨ′i−j − h2BjΨi−j

}= h2Fj, j = 0, 1, 2, · · · , N. (6.11)

The following proposition is the simplified form of the above system of equa-

tions.

Proposition 6.1.1. The system (6.11) is equivalent to

4∑i=−4

uj+iPji = h2Fj, j = 0, 1, 2, · · · , N, (6.12)

150

where

P ji =

εΨ′′0 − h2Bj, for i = 0,

εΨ′′i − hAjΨ

′i, for i = 0.

(6.13)

Proof. Substituting j = 0 in (6.11), we get

N+4∑i=−4

ui

{εΨ′′

i + hA0Ψ′i − h2B0Ψi

}= h2F0, j = 0, 1, 2, · · · , N.

By expanding above equation, we get

u−4{εΨ′′−4 + hA0Ψ

′−4 − h2B0Ψ−4}+ u−3{εΨ′′

−3 + hA0Ψ′−3 − h2B0Ψ−3}+ · · ·

+u0{εΨ′′0 + hA0Ψ

′0 − h2B0Ψ0}+ · · ·+ uN+3{εΨ′′

N+3 + hA0Ψ′N+3 − h2B0ΨN+3}

+uN+4{εΨ′′N+4 + hA0Ψ

′N+4 − h2B0ΨN+4} = h2F0.

Since Ψ′i and Ψ′′

i are non-zero in the interval [−4, 4], outside the interval these

are zero, then above equation becomes

u−4{εΨ′′−4 + hA0Ψ

′−4 − h2B0Ψ−4}+ u−3{εΨ′′

−3 + hA0Ψ′−3 − h2B0Ψ−3}+ u−2{εΨ′′

−2

+hA0Ψ′−2 − h2B0Ψ−2}+ u−1{εΨ′′

−1 + hA0Ψ′−1 − h2B0Ψ−1}+ u0{εΨ′′

0 + hA0Ψ′0

−h2B0Ψ0}+ u1{εΨ′′1 + hA0Ψ

′1 − h2B0Ψ1}+ u2{εΨ′′

2 + hA0Ψ′2 − h2B0Ψ2}+ u3{εΨ′′

3

+hA0Ψ′3 − h2B0Ψ3}+ u4{εΨ′′

4 + hA0Ψ′4 − h2B0Ψ4} = h2F0.

By using the definition of Ψi given in (6.5) and Ψ′0 = 0, given in Table 6.1, above

expression becomes

u−4{εΨ′′−4 + hA0Ψ

′−4}+ u−3{εΨ′′

−3 + hA0Ψ′−3}+ u−2{εΨ′′

−2 + hA0Ψ′−2}

+u−1{εΨ′′−1 + hA0Ψ

′−1}+ u0{εΨ′′

0 − h2B0Ψ0}+ u1{εΨ′′1 + hA0Ψ

′1}

+u2{εΨ′′2 + hA0Ψ

′2}+ u3{εΨ′′

3 + hA0Ψ′3}+ u4{εΨ′′

4 + hA0Ψ′4} = h2F0.

If

P 0±4 = εΨ′′

±4 + hA0Ψ′±4, P 0

±3 = εΨ′′±3 + hA0Ψ

′±3, P 0

±2 = εΨ′′±2 + hA0Ψ

′±2

P 0±1 = εΨ′′

±1 + hA0Ψ′±1, P 0

0 = εΨ′′0 − h2B0,

151

so above equation becomes

4∑i=−4

uiP0i = h2F0.

Similarly for j = 1, 2, 3, · · · , N , we get

4∑i=−4

ui+jPji = h2Fj.

where for i = −4,−3, · · · , 3, 4 and j = 1, 2, 3, · · ·N, we have

P ji =

εΨ′′0 − h2Bj for i = 0,

εΨ′′i − hAjΨ

′i for i = 0.

This completes the proof.

6.1.3 Singularly perturbed linear system of equations

The system of equations (6.12) are the singularly perturbed linear equations.

These equations can be written in matrix form as

AU = F1, (6.14)

In (6.14) matrix A is (N + 1)× (N + 9) defined as

A = (pr−1s )(N+1)×(N+9), (6.15)

where “r” represents rows i.e. r = 1, 2, · · ·N + 2 , “s” represents columns i.e.

s = −4,−3, · · · , N + 3, N + 4 and

pr−1s =

P r−1i , for − 4 6 i 6 4,

0, for otherwise,

where P r−1i defined in (6.13). The vectors U and F1 are defined as

U = (u−4, u−3, · · · , uN+3, uN+4)T , (6.16)

152

and

F1 = h2 × (F0, F1, · · · , FN−1, FN)T . (6.17)

To find the unique solution of the system (6.14), we need eight more conditions.

Two conditions are given in (6.2) i.e. U(0) and U(1), we construct remaining

conditions as given in next section.

6.1.4 Compelled conditions

By adding two given conditions in the system (6.14), the system remains un-

stable. We need six more conditions to get stable system. Since approximation

order of subdivision scheme (6.3) is four, so we define compelled conditions of

order four for solution of (6.14). In this section, we discuss two methods for

the construction of compelled condition. For simplicity only the left end points

u−1, u−2, u−3 are discussed and right end points uN+1, uN+2, uN+3 can be treated

similarly.

153

C-1: Conditions by using cubic polynomial:

The values u−3, u−2, u−1 can be determined by the fourth order polynomial S1(x)

interpolating (xi, ui), 0 ≤ i ≤ 3. Precisely, we have

u−i = S1(−xi), i = 1, 2, 3,

where

S1(xi) =4∑

j=1

4

j

(−1)j+1U(xi−j).

Since by (6.6), U(xi) = ui for i = 1, 2, 3 then by replacing xi by −xi, we have

S1(−xi) =4∑

j=1

4

j

(−1)j+1u−i+j.

Hence the following compelled conditions can be employed at the left end

4∑j=0

4

j

(−1)ju−i+j = 0, i = 1, 2, 3. (6.18)

Similarly for the right end, we can define ui = S1(xi), i = N + 1, N + 2, N + 3

and

S1(xi) =4∑

j=1

4

j

(−1)j+1ui−j.


4∑j=0

4

j

(−1)jui−j = 0, i = N + 1, N + 2, N + 3. (6.19)

C-2: Conditions by using cardinal basis functions:

The values u−3, u−2, u−1 are determined by fourth order polynomial S2(x), i.e.

u−i = S2(−xi), i = 1, 2, 3.

154

The cubic polynomial S2(x) is defined as

S2(x) = u0L0

(x− x0

h

)+ u1L1

(x− x0

h

)+ u′′

0L∗0

(x− x0

h

)+u′′

1L∗1

(x− x0

h

), (6.20)

where the basis functions are defined as

L0

(x− x0

h

)= 1−

(x− x0

h

),

L1

(x− x0

h

)=

(x− x0

h

),

L∗0

(x− x0

h

)= −1

6

(x− x0

h

)(x− x0

h− 1

)(x− x0

h− 2

),

L∗1

(x− x0

h

)=

1

6

(x− x0

h

)(x− x0

h− 1

)(x− x0

h+ 1

),

and for t = 0, 1

u′′t = A

(xt − x0

h

)ut +B

(xt − x0

h

)ut + F

(xt − x0

h

).

Similarly for the right end, we can define ui = S2(−xi), i = N + 1, N + 2, N + 3.

The cubic polynomial in this case is

S2(x) = uNLN

(x− xN

h

)+ uN+1LN+1

(x− x0

h

)+ u′′

NL∗N

(x− x0

h

)+u′′

N+1L∗N+1

(x− x0

h

), (6.21)

where the basis functions are given below

LN

(x− xN

h

)= 1−

(x− xN

h

),

LN+1

(x− xN

h

)=

(x− xN

h

),

L∗N

(x− xN

h

)= −1

6

(x− xN

h

)(x− xN

h− 1

)(x− xN

h− 2

),

L∗N+1

(x− xN

h

)=

1

6

(x− xN

h

)(x− xN

h− 1

)(x− xN

h+ 1

),

155

and for t = N,N + 1

u′′t = A

(xt − xN

h

)ut +B

(xt − xN

h

)ut + F

(xt − xN

h

).

6.1.5 Stable singularly perturbed linear system of equations

By using any of above compelled conditions, we get a following new system

of (N + 9) singularly perturbed linear equations with (N + 9) unknowns {ui},

in which N + 1 equations are obtained from (6.12) and two equations obtained

from given boundary conditions (6.2). Further six equations are obtained from

compelled conditions either from (6.18) and (6.19) or (6.20) and (6.21).

By using C-1 compelled conditions:

If we use compelled conditions defined in (6.18) and (6.19) then stable singularly

perturbed linear system of equations is define as

B1U = F, (6.22)

where the coefficients matrix B1 = (LT1 ,AT ,RT

1 )T , A is defined by (6.15). The

matrix L1 of order 4× (N + 9) for left end boundary conditions is defined as

L1 =

0 1 −4 6 −4 1 1 0 0 0 · · · 0 0

0 0 1 −4 6 −4 1 0 0 0 · · · 0 0

0 0 0 1 −4 6 −4 1 0 0 · · · 0 0

0 0 0 0 1 0 0 0 0 0 · · · 0 0

,

where the first three rows are obtained from (6.18) and last row is obtained by

using left end compelled condition (6.2) i.e. U(0).

The matrix R1 of order 4 × (N + 9) for right end boundary conditions is con-

structed as

R1 =

0 0 · · · 0 0 0 0 0 0 1 0 0 0 0

0 0 · · · 0 0 0 1 −4 6 −4 1 0 0 0

0 0 · · · 0 0 0 0 1 −4 6 −4 1 0 0

0 0 · · · 0 0 0 0 0 1 −4 6 −4 1 0

,

156

where the first row of above matrix is obtained from given condition (6.2) i.e.

U(1) and the remaining three conditions are obtained from the right end com-

pelled conditions (6.19). The vector U is defined in (6.16) and F is define as

F = (0, 0, 0, α0,FT1 , α1, 0, 0, 0)

T , (6.23)

where F1 is defined in (6.17) and α0, α1 given in (6.2).

By using C-2 compelled conditions:

If we use compelled conditions defined in (6.20) and (6.21) then stable singularly

perturbed linear system takes the form as

B2U = F, (6.24)

where the coefficients matrix B2 = (LT2 ,AT ,RT

2 )T , A is defined by (6.15). The

matrix L2 of order 4× (N +9) for left end boundary conditions is defined as: the

first three rows are obtained from (6.20) and last row is obtained by using left

end compelled condition (6.2) i.e. U(0). The matrix R2 of order 4 × (N + 9) for

right end boundary conditions is constructed as: the first row of above matrix is

obtained from given condition (6.2) i.e. U(1) and the remaining three conditions

are obtained from the right end compelled conditions (6.21). The vectors U and

F are defined in (6.16) and (6.23).

Hence for different compelled conditions we have two different singularly

perturbed linear system of equations (6.22) and (6.24). The non-singularity of

the coefficient matrices B1 and B2 has been discussed in next section.


In this section, we discuss the non-singularity of the coefficient matrix. Non-

singularity of the matrices B1 and B2 by finding the eigenvalues of both the

matrices. We observe that all the eigenvalues up to N ≤ 500 are non-zero. Hence

157

by Strang (2011), we conclude that the B1 and B2 are non-singular. For large

N > 500, the matrices may or may not be singular.

6.2 Convergence of the method

In this section, we discuss convergence of the method described in Section 6.1.

Let Y be the analytic solution of the problem (6.1) with (6.2) then

εY′′(x) = A(x)Y

′(x) +B(x)Y (x) + F (x).

The above result can be written for node points for j = 0, 1, · · ·N, as

εY′′(xj) = A(xj)Y

′(xj) +B(xj)Y (xj) + F (xj). (6.25)

Let the vector Y (x) be defined as

Y (x) = (y(x0), y(x1), · · · , y(xN))T .

By Taylor’s series

Y′(xj) =

1

25878h[−256y(xj − 4h)− 3200y(xj − 3h) + 19673y(xj − 2h)

−54600y(xj − h) + 54600y(xj + h)− 19673y(xj + 2h)

+3200y(xj + 3h) + 256y(xj + 4h)] + o(h4),

and

Y′′(xj) =

1

8448h2[256y(xj − 4h) + 1600y(xj − 3h)− 4725y(xj − 2h)

+17300y(xj − h)− 28862y(xj) + 17300y(xj + h)− 4725y(xj + 2h)

+1600y(xj + 3h) + 256y(xj + 4h)] + o(h4).

The system of equations (6.22) and (6.24) provide the required subdivision based

approximate solution U(x) for (6.1) then by (6.7), for j = 0, 1, · · · , N,

εU ′′(xj) = A(xj)U′(xj) +B(xj)U(xj) + F (xj), (6.26)

158

where U ′(xj) and U ′′(xj) are defined as

U ′(xj) =1

25878h[−256u(xj − 4h)− 3200u(xj − 3h) + 19673u(xj − 2h)

−54600u(xj − h) + 54600u(xj + h)− 19673u(xj + 2h)

+3200u(xj + 3h) + 256u(xj + 4h)] + o(h4),

and

U ′′(xj) =1

8448h2[256u(xj − 4h) + 1600u(xj − 3h)− 4725u(xj − 2h)

+17300u(xj − h)− 28862u(xj) + 17300u(xj + h)− 4725u(xj + 2h)

+1600u(xj + 3h) + 256u(xj + 4h)] + o(h4).

Let the error function E is defined as E(x) = Y (x)− U(x) and

E = (E−4, E−3, · · · , EN+3, EN+4).

Then error vector at the node points is

E(xj) = Y (xj)− U(xj), −4 6 j 6 N + 4.

This implies

E′(xj) = Y

′(xj)− U ′(xj), −4 6 j 6 N + 4,

E′′(xj) = Y

′′(xj)− U ′′(xj), −4 6 j 6 N + 4.


ε[Y

′′(xj)− U ′′(xj)

]= A(xj)

[Y

′(xj)− U ′(xj)

]+B(xj)

[Y (xj)− U(xj)

].

By definition of error vector

εE′′(xj) = A(xj)E

′(xj) + B(xj)E(xj), 0 6 j 6 N.

This implies

εE′′(xj)− A(xj)E

′(xj)−B(xj)E(xj) = 0, 0 6 j 6 N, (6.27)

159

where for 0 6 j 6 N

E′(xj) =

1

25878h[−256E(xj − 4h)− 3200E(xj − 3h) + 19673E(xj − 2h)

−54600E(xj − h) + 54600E(xj + h)− 19673E(xj + 2h)

+3200E(xj + 3h) + 256E(xj + 4h)] + o(h4),

and for 0 6 j 6 N

E′′(xj) =

1

8448h2[256E(xj − 4h) + 1600E(xj − 3h)− 4725E(xj − 2h)

+17300E(xj − h)− 28862E(xj) + 17300E(xj + h)− 4725E(xj + 2h)

+1600E(xj + 3h) + 256E(xj + 4h)] + o(h4).

As 0 ≤ x ≤ 1 and xj = jh, j = 0, 1, 2, · · · , N so E0, E1, · · · , EN are non zero while

E−4, · · · , E−1 and EN+1, · · · , EN+4 are zero because they lie outside the interval

[0, 1]. Let us define error values at the end points as

Ej =

max0≤k≤4

{|Ek|}O(h4), −4 ≤ j < 0,

maxN−4≤k≤N

{|Ek|}O(h4), N < j ≤ N + 4.

(6.28)

By expanding (6.27) similar to Proposition 6.1.1 and using the algorithm defined

in Section 6.1, we get

(B1 +O(h4))E = 0.

Similarly

(B2 +O(h4))E = 0.

These are equivalent to

(B1 +O(h4))E = O(h4) ∥ E ∥= O(h4),

and

(B2 +O(h4))E = O(h4) ∥ E ∥= O(h4).

160

Since for small h, the coefficient matrix Bi + O(h4), i = 1, 2 will be invertible

and thus using the standard result from linear algebra and effect of ∥B−1i ∥, we

have

||E|| ≤(

||B−1i ||

1−O(h4)O(h4)

)= O(h2), i = 1, 2.

Hence ∥ E ∥= O(h2). The result is summarized in the following theorem.

Theorem 6.2.1. Let Y be the exact solution of the system (6.1) and Uj , j = 0, 1, · · · , N

be the approximate solution of (6.7) then ∥ E ∥= O(h2).

6.3 Numerical examples and discussions

In this section, we have implemented our method on three examples which sup-

port the theoretical analysis of our findings about order of convergence. We

solve these examples by solving two singularly perturbed linear systems of e-

quations (6.22) and (6.24). The maximum absolute errors in the exact and ap-

proximate solutions are also calculated at the different step sizes. For the sake

of comparisons, we have also tabulated the numerical results. The physical be-

havior of analytic and approximate solutions is also presented in this section.

We consider the following three examples of second order singularly perturbed

boundary value problems:

Example 6.3.1.

εy′′(x) = y + cos2(πx) + 2επ2 cos(2πx), 0 < x < 1,


y(0) = 0 = y(1).

The analytic solution is given by

y(x) =

[e

(−(1−x)√

ε

)+ e

(−x√

ε

)][1 + e

(−1√ε

)] − cos2(πx).

161

Example 6.3.2.

εy′′(x)− (1 + x)y(x) = 40[x(x2 − 1)− 2ε], 0 < x < 1,


y(0) = 0 = y(1).

The analytic solution of this problem is

y(x) = 40x(1− x).

Example 6.3.3.

εy′′(x)− {1 + x(1− x)}y(x) = − [1 + x(1− x) + {2√ε− x2(1− x)}

e

{(1−x)√

ε

}+ {2

√ε− x(1− x)2}e

{− x√

ε

}],

where 0 6 x 6 1 with boundary conditions

y(0) = y(1) = 0.

The analytic solution for the above problem is

y(x) = 1 + (x− 1)e

{− x√

ε

}− xe

{− (1−x)√

ε

}.

6.3.1 Results and discussion

All above examples are solved by subdivision based algorithm at different val-

ues of N and small values of ε. We have observed the following facts:

• The numerical results of Example 6.3.1 are shown in Tables 6.2 - 6.6 and

in Figures 6.1 - 6.3. Tables 6.2 and 6.3 show the maximum absolute errors

at different values of N and small values of ε. Tables 6.4 - 6.6 provide a

comparison of maximum absolute error with the existing methods. i.e.

comparison with spline based method by Aziz and Khan (2002) and Bawa

162

and Natesan (2005), tension spline method by Khan and Aziz (2005), fit-

ted mesh B-spline method by Kadalbajoo and Aggarwal (2005) and Haar

wavelet based method by Pandit and Kumar (2014). It is concluded that

our method gives better results than the methods of Aziz and Khan (2002),

Bawa and Natesan (2005), Khan and Aziz (2005), Kadalbajoo and Aggar-

wal (2005) and Pandit and Kumar (2014). Figure 6.1 compares the analytic

and numerical solutions while Figures 6.2 and 6.3 delineate the physical

behavior of the Example 6.3.1 at different values of N and ε.

• We present the numerical results of Example 6.3.2 in Tables 6.7 - 6.10 and

graphical representation in Figures 6.4 - 6.6. Tables 6.7 and 6.10 show the

maximum absolute errors at different value of ε and N while Tables 6.9

and 6.10 present the comparison of maximum absolute errors with exist-

ing methods Miller (1979) and Niijima (1980a, 1980b). From these com-

parison, it is concluded that our method gives better results than that of

existing methods. Figure 6.4 compare the analytic and numerical solution-

s while Figures 6.5 and 6.6 depicts the physical behavior of the Example

6.3.2 at different values of N and ε.

• The numerical results of Example 6.3.3 are reported in Tables 6.11 - 6.13

and in Figures 6.7 - 6.9. Tables 6.11 and 6.12 shows the maximum absolute

errors at different values of N and ε. We compare numerical results of Ex-

ample 6.3.3 with wavelet and finite difference methods Kumar and Mehra

(2009) and Lubuma and Patidar (2006) and it is concluded that our method

gives better approximation comparative to other methods. The graphical

representation of the solution of Example 6.3.3 for different values of N

and ε is given in Figures 6.8 and 6.9.

• From the tabulated results of these examples, we observe that the condi-

tion C-2 gives less maximum absolute errors comparative to the condition

163

C-2.

• It is also observed that for fixed N maximum absolute errors increase with

the decrease of ε while for fixed value of ε maximum absolute errors de-

crease with the increase of N .

Table 6.2: Maximum absolute errors of Example 6.3.1

N = 10 Our method Our method

for ε with C-1 with C-2

0.1× 10−3 2.3445E-02 1.8459E-02

0.1× 10−4 2.4427E-03 1.9197E-03

0.1× 10−5 2.4525E-04 1.9270E-04

0.1× 10−6 2.4535E-05 1.9278E-05

0.1× 10−7 2.4535E-06 1.9279E-06

0.1× 10−8 2.4536E-07 1.9279E-07

0.1× 10−9 2.4536E-08 1.9279E-08

164

Tabl

e6.

3:M

axim

umab

solu

teer

rors

ofEx

ampl

e6.

3.1

Nε=

10−5

ε=

10−8

ε=

10−10

ε=

10−5

ε=

10−8

ε=

10−10

wit

hC

-1w

ith

C-2

102.

4427

E-03

2.45

35E-

062.

4536

E-08

1.91

97E-

031.

9279

E-06

1.92

78E-

08

100

1.20

05E-

012.

3115

E-04

2.31

26E-

069.

3578

E-02

1.88

87E-

041.

8895

E-06

150

1.51

52E-

015.

1955

E-04

5.20

08E-

061.

1209

E-01

4.24

58E-

044.

2498

E-06

200

1.59

50E-

019.

2276

E-04

9.24

42E-

061.

1327

E-01

7.54

13E-

047.

5541

E-06

250

1.56

92E-

011.

4402

E-03

1.44

43E-

051.

0808

E-01

1.17

71E-

031.

1802

E-05

165

Tabl

e6.

4:M

axim

umab

solu

teer

rors

ofEx

ampl

e6.

3.1

λ1

andλ2

ε=

10−5

ε=

10−5

ε=

10−8

ε=

10−10

Azi

zan

dK

han

(200

2)N

=10

N=

100

N=

200

N=

250

&K

han

and

Azi

z(2

005)

1/18

,4/9

···

1.44

463E

-03

6.22

342E

-02

6.27

380E

-02

1/14

,3/7

···

1.52

823E

-02

8.33

647E

-02

8.39

115E

-02

1/24

,11/24

···

1.00

616E

-02

4.50

702E

-02

4.55

413E

-02

1/30

,14/30

···

1.67

078E

-02

3.52

995E

-02

3.57

527E

-02

1/6,

1/3

···

1.19

71E-

012.

6683

E-01

2.67

93E-

01

Our

met

hod

2.44

27E-

031.

2005

E-01

9.22

76E-

041.

4443

E-05

wit

hC

-1

Our

met

hod

1.91

97E-

039.

3579

E-02

7.54

13E-

041.

1802

E-05

wit

hC

-2

166


for N = 32 Bawa and Pandit and Our method Our method

and ε = (2−r)2 Natesan (2005) Kumar (2014) with C-1 with C-2

r=10 5.022E-02 1.23E-02 2.2646E-03 1.8478E-03

r=20 3.125E-02 1.23E-08 2.1692E-09 1.7695E-09

r=25 3.125E-02 1.20E-11 2.1200E-12 1.7283E-12


for N = 32 Kadalbajoo and Pandit and Our method Our method

and ε = 2−r Aggarwal (2005) Kumar (2014) with C-1 with C-2

r=10 5.022E-02 1.80E-03 1.5551E-01 1.0545E-01

r=20 3.125E-02 1.23E-03 2.2645E-03 1.8478E-03

r=25 3.125E-02 4.04E-4 7.1071E-05 5.7977E-05

167



for ε with C-1 with C-2

0.1× 10−3 1.4049E-02 2.4080E-02

0.1× 10−4 1.4331E-03 2.4866E-03

0.1× 10−5 1.4360E-04 2.4948E-04

0.1× 10−6 1.4363E-05 2.4957E-05

0.1× 10−7 1.4364E-06 2.4958E-06

0.1× 10−8 1.4364E-07 2.4958E-07

0.1× 10−9 1.4364E-08 2.4958E-08

6.4 Conclusions

The subdivision techniques appear to be an ideal tool to attain the numerical so-

lution of singularly perturbed boundary value problem. The subdivision based

method is developed for the approximate solution of the singularly perturbed

boundary value problems. The proposed method is computationally efficient

and the algorithm can be easily implemented on computer. In addition, we in-

troduced some compelled conditions for the existence of approximate solution.

The method has been proved to be second order convergent. It is concluded

that our method for the solutions of singularly perturbed boundary value prob-

lems is batter than spline based methods by Aziz and Khan (2002), Bawa and

Natesan (2005), Khan and Aziz (2005), Kadalbajoo and Aggarwal (2005) , finite

difference methods by Lubuma (2006), Miller (1979), Niijima (1980a), Niijima

(1980b) and Haar wavelet methods by Kumar and Mehra (2002), Pandit and

Kumar (2014).

168

Tabl

e6.

8:M

axim

umab

solu

teer

rors

ofEx

ampl

e6.

3.2

Nε=

10−5

ε=

10−8

ε=

10−10

ε=

10−5

ε=

10−8

ε=

10−10

wit

hC

-1w

ith

C-2

101.

4331

E-03

1.43

64E-

061.

4364

E-08

2.48

66E-

032.

4958

E-06

2.49

58E-

08

100

1.50

99E-

031.

5640

E-06

1.56

44E-

081.

1720

E-02

1.72

89E-

051.

7298

E-07

150

1.52

17E-

031.

5687

E-06

1.56

95E-

081.

3907

E-02

2.68

51E-

052.

6881

E-07

200

1.52

62E-

031.

5706

E-06

1.57

21E-

081.

4855

E-02

3.63

98E-

053.

6471

E-07

250

1.52

8E-0

31.

5714

E-06

1.57

36E-

081.

5259

E-02

4.59

19E-

054.

6062

E-07

169


for N = 16 Miller Niijima Niijima Our Our

and ε (1979) (1980a) (1980b) method method

with C-1 with C-2

0.1× 10−3 0.25E-01 0.26E-01 0.65E-04 0.1408E-01 0.3079E-01

0.1× 10−4 0.21E-01 0.24E-01 0.36E-04 0.1478E-02 0.3374E-02

0.1× 10−5 0.70E-02 0.17E-01 0.33E-04 0.14862E-03 0.3407E-03

0.1× 10−6 0.75E-03 0.69E-02 0.26E-04 0.14870E-04 0.3410E-04

0.1× 10−7 0.74E-04 0.23E-02 0.20E-04 0.14871E-05 0.3411E-05

0.1× 10−8 0.67E-05 0.76E-03 0.11E-04 0.14872E-06 0.3411E-06


for N = 32 Miller Niijima Niijima Our Our

and ε (1979) (1980a) (1980b) method method

with C-1 with C-2

0.1× 10−3 0.64E-02 0.65E-02 0.59E-04 0.1414E-01 0.3861E-01

0.1× 10−4 0.61E-02 0.64E-02 0.21E-04 0.1497E-02 0.5501E-02

0.1× 10−5 0.41E-02 0.56E-02 0.35E-04 0.1529E-03 0.5756E-03

0.1× 10−6 0.77E-03 0.31E-02 0.39E-04 0.1531E-04 0.5783E-04

0.1× 10−7 0.76E-04 0.12E-02 0.21E-04 0.1532E-05 0.5786E-05

0.1× 10−8 0.67E-05 0.38E-03 0.21E-04 0.1532E-06 0.5786E-06

170



ε with C-1 with C-2

0.1× 10−3 2.0312E-02 7.4455E-03

0.1× 10−4 2.1110E-03 7.6536E-04

0.1× 10−5 2.1189E-04 7.6679E-05

0.1× 10−6 2.1197E-05 7.6682E-06

0.1× 10−7 2.1198E-06 7.6679E-07

0.1× 10−8 2.1198E-07 7.6678E-08

0.1× 10−9 2.1198E-08 7.6679E-09

171

Tabl

e6.

12:M

axim

umab

solu

teer

rors

ofEx

ampl

e6.

3.3

wit

hC

-1w

ith

C-2

Nε=

10−5

ε=

10−8

ε=

10−10

ε=

10−5

ε=

10−8

ε=

10−10

102.

1110

E-03

2.11

98E-

062.

1198

E-08

7.65

36E-

047.

6679

E-07

7.66

77E-

09

165.

5268

E-03

5.58

76E-

065.

5877

E-08

2.00

82E-

032.

0212

E-06

2.02

12E-

08

322.

1930

E-02

2.29

64E-

052.

2965

E-07

8.02

32E-

038.

3069

E-06

8.30

69E-

08

100

1.19

19E-

012.

2869

E-04

2.28

79E-

062.

3666

E-02

8.27

35E-

058.

2759

E-07

200

1.59

31E-

019.

1801

E-04

9.19

65E-

063.

7244

E-02

3.32

26E-

043.

3265

E-06

250

1.56

91E-

011.

4343

E-03

1.43

84E-

054.

6497

E-02

5.19

30E-

045.

2028

E-06

172

Tabl

e6.

13:M

axim

umab

solu

teer

rors

ofEx

ampl

e6.

3.3

forN

=16

Kum

aran

dM

ehra

Lubu

ma

and

Pati

dar

Our

met

hod

Our

met

hod

andε=

10−r

(200

9)(2

006)

wit

hC

-1w

ith

C-2

r=3

0.77

E-01

0.28

E-01

0.15

24E-

000.

2703

E-01

r=5

0.46

E-02

0.53

E-02

0.55

27E-

020.

2008

E-02

r=7

0.46

E-04

0.53

E-02

0.55

87E-

040.

2021

E-04

r=8

0.46

E-05

0.53

E-02

0.55

89E-

052.

0212

E-05

173

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

x

App

roxi

mat

e so

lutio

n

Analytic solutionApproximate solution with C−1 Approximate solution with C−2

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

x

App

roxi

mat

e so

lutio

n


(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

x

App

roxi

mat

e so

lutio

n


(c)

Figure 6.1: Physical behavior of analytic and approximate solutions of Example 6.3.1

for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c) respectively.

174

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

x

App

roxi

mat

e so

lutio

n



for N = 32 and ε = 2−25.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

x

App

roxi

mat

e so

lutio

n



for N = 32 and ε = (2−20)2.

175

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

x

App

roxi

mat

e so

lutio

n


(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

x

App

roxi

mat

e so

lutio

n


(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

x

App

roxi

mat

e so

lutio

n


(c)



176

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

x

App

roxi

mat

e so

lutio

n



for N = 16 and ε = 10−8.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

x

App

roxi

mat

e so

lutio

n



for N = 32 and ε = 10−9.

177

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

App

roxi

mat

e so

lutio

n

Analytic solutionApproximate solution with C−1Approximate solution with C−2

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

App

roxi

mat

e so

lutio

n


(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

App

roxi

mat

e so

lutio

n


(c)



178

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

App

roxi

mat

e so

lutio

n



for N = 16 and ε = 10−5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

App

roxi

mat

e so

lutio

n



for N = 16 and ε = 10−8.

179

Chapter 7

A Subdivision Collocation

Algorithm for Solving Two point

Boundary value Problems of Order

Three

In this chapter, we propose an algorithm for the numerical solution of self ad-

joint singularly perturbed third order boundary value problems in which the

highest order derivative is multiplied by a small parameter ε. This chapter is

divided into five sections. Self adjoint singularly perturbed BVP, subdivision

scheme and their derivative are presented in Section 7.1. Numerical algorithm

for the solution of third order singularly perturbed boundary value problem is

discussed in Section 7.2. Convergence analysis of the numerical algorithm is

given in Section 7.3. Numerical examples and their discussion for the illustra-

tion of algorithm is given in Section 7.4. Conclusion about the proposed algo-

rithm is given in Section 7.5.

180

7.1 Preliminaries

We present third order singularly perturbed boundary value problem, binary

subdivision scheme and their derivatives in this section.

7.1.1 Third order singularly perturbed BVP

We consider the general expression for third order singulary perturbed problem

as

−εy′′′(x) + p(x)y(x) = f(x), p(x) > 0,

y(0) = β0, y(1) = β1, y′(0) = β2,(7.1)

or

−εy′′′(x) + p(x)y(x) = f(x), p(x) > 0,

y(0) = β0, y(1) = β1, y′′(0) = β3,(7.2)

where β0, β1, β2 and β3 are constant, p(x), f(x) are smooth functions and ε is a

small positive parameter with ε ≪ 1. These type of problems usually occur in

quantum mechanics, fluid mechanics, optical control, chemical reactions etc.

7.1.2 Subdivision scheme and derivatives

The subdivision schemes have been considered esteemed in many arenas of

computational sciences. Such as computer animation, computer graphics and

computer aided geometric design due to its efficient and simple characteristics.

The subdivision scheme defines a curve out of an initial control polygon by

subdividing them according to some refining rules recursively. Consider eight

181

point binary interpolating scheme presented by Lee et al. (2006) as

pk+12i = pki ,

pk+12i+1 = −ω

(pki−3 + pki+4

)+(5ω + 3

256

)(pki−2 + pki+3)−

(9ω + 25

256

)(pki−1

+pki+2) +(5ω + 75

128

)(pki + pki+1).

(7.3)

The scheme (7.3) is C3 derivable continuous for 0.0016 < ω < 0.0084, the support

width for the mask of the scheme is [−6, 6], the approximation order is six and

satisfies following two scale relation

Ψ(x) = Ψ(2x) +[−ω {Ψ(2x− 1) + Ψ(2x+ 1)}+

(5ω + 3

256

){Ψ(2x− 3)

+Ψ(2x+ 3)} −(9ω + 25

256

){Ψ(2x− 5) + Ψ(2x+ 5)}+

(5ω + 75

128

){Ψ(2x− 7) + Ψ(2x+ 7)}] , x ∈ R

(7.4)

where

Ψ(x) =

1 for x = 0,

0 for x = 0.(7.5)

As the function Ψ(x) ∈ C3, then the first, second and third derivatives can be

obtained by using the similar approach as in Mustafa and Ejaz (2014). The third

derivatives of (7.4) for the parametric value ω = 0.0032 are given below:

Ψ′′′(0) = 0, Ψ′′′(±1) = ±1122400355000418234124847

,

Ψ′′′(±2) = ∓1502922273911836468249694

, Ψ′′′(±3) = ±1166667500038021284077

,

Ψ′′′(±4) = ± 159804307271672936499388

, Ψ′′′(±5) = ∓ 4221440000418234124847

,

Ψ′′′(±6) = ± 108068864418234124847

.

(7.6)

7.2 Subdivision collocation algorithm

This section describes an algorithm for the numerical solutions of singularly

perturbed linear third order boundary value problems with non-homogeneous

boundary conditions. An algorithm is described as follows:

182

Let N be a positive integer (N > 6), h = 1/N and xi = i/N = ih, i =

0, 1, 2, · · ·N , and

U(x) =N+6∑i=−6

uiΨ

(x− xi

h

), 0 ≤ x ≤ 1, (7.7)

be the approximate solution to (7.1) or (7.2) where {ui} are the unknown to be

determined. Then

−εU ′′′(xj) + p(xj)U(xj) = f(xj), j = 0, 1, 2, · · · , N,

with the following given boundary conditions

U(0) = β0, U(1) = β1 U ′(0) = β2,

or

U(0) = β0, U(1) = β1 U ′′(0) = β3.

(7.8)

Let we define p(xj) = pj , and f(xj) = fj , then above equation can be written as

−εU ′′′(xj) + pjU(xj) = fj, j = 0, 1, 2, · · · , N, (7.9)

From (7.7) we have

U ′′′(xj) =1

h3

N+6∑i=−6

uiΨ′′′(xj − xi

h

). (7.10)


−ε

N+6∑i=−6

uiΨ′′′(xj − xi

h

)+ h3pj

N+6∑i=−6

uiΨ

(xj − xi

h

)= h3fj.

This implies

N+6∑i=−6

ui

{−εΨ′′′

(xj − xi

h

)+ h3pjΨ

(xj − xi

h

)}= h3fj.

Further implies

N+6∑i=−6

ui

{−εΨ′′′(j − i) + h3pjΨ(j − i)

}= h3fj, (7.11)

183

where j = 0, 1, 2, · · · , N and xi = ih or xj = jh. By using Ψ(i) = Ψi, (7.11) can

be written asN+6∑i=−6

ui

{−εΨ′′′

j−i + h3pjΨj−i

}= h3fj, j = 0, 1, 2, · · · , N. (7.12)

As we observe from (7.6), Ψ′′′−i = −Ψ′′′

i , then (7.12) becomes

N+6∑i=−6

ui

{εΨ′′′

i−j + h3pjΨi−j

}= h3fj, j = 0, 1, 2, · · · , N. (7.13)

Remark 7.2.1. The system (7.13) is equivalent to

6∑i=−6

uj+iQji = h3fj, j = 0, 1, 2, · · · , N, (7.14)

where

Qji =

εΨ′′′0 + h3pj, i = 0,

εΨ′′′i , i = 0.

(7.15)

7.2.1 Singularly perturbed system

The system of equations (7.14) are the singularly perturbed linear equations.

These equations can be written in matrix form as

AU = F1, (7.16)

where

A = (qr−1s )(N+1)×(N+13), (7.17)

U = (us)(N+13)×1, (7.18)

F1 = h3 × (uℓ)(N+13)×1, (7.19)

“r” and “s” represent rows and columns respectively. Where

qr−1s =

Qr−1i , for − 6 6 i 6 6,

0, for otherwise,

184

ℓ = 0, 1, · · · , N, r = 1, 2, · · · , N + 2, s = −6,−5, · · · , N + 5, N + 6.

and Qr−1i defined in (7.15).

To find the unique solution of the system (7.16), we need twelve more condi-

tions. Three conditions are given in (7.8) i.e. U(0), U(1) and U ′(0) or U ′′(0). As

in given conditions first or second derivative is involved so first we replace first

or second derivative conditions by their approximation. The approximation of

these derivatives is given as follows:

7.2.2 Approximation of derivative conditions

We approximate first and second derivatives of the function U(x) by finite dif-

ferences algorithm. Given a non-zero value of h, the lth order derivative satisfies

the following equation where the integer order of error p > 0 may be selected

as desired

U l(x) =l!

hl

imax∑i=imin

ciU(x+ ih) +O(hp). (7.20)

A forward difference approximation occurs if we set imin = 0 and imax = l +

p − 1. The vector C = (cimin, · · · , cimax) is called the convolution mask for the

approximation. In order for equation (7.20) satisfied, it is necessary that

imax∑i=imin

inci =

0, for 0 6 n 6 l + p− 1 and n = l,

1, for n = l.(7.21)

Approximation of U ′(x) with error O(h7), we have imin = 0 and imax = 7. The

convolution matrix (c0, c1, · · · , c7) is obtained by solving the linear system

7∑i=0

inci =

0, for 0 6 n 6 7 and n = 1,

1, for n = 1.(7.22)

185

After solving (7.22) substituting the values of ci in (7.20), we obtain first deriva-

tive approximation as

U ′(0) = −(N

60

)[−147u0 + 360u1 − 450u2 + 400u3 − 225u4 + 72u5 − 10u6] .

(7.23)

Similarly approximation of U ′′(x) with error O(h7), we have imin = 0 and imax =

8. The convolution matrix (c0, c1, · · · , c8) is obtained by solving the linear system

8∑i=0

inci =

0, for 0 6 n 6 8 and n = 2,

1, for n = 2.(7.24)

After solving (7.24) substituting the values of ci in (7.20), we obtain second

derivative approximation as

U ′′(0) = −(

N

360

)[938u0 − 4014u1 + 7911u2 − 9490u3 + 7380u4 − 3618u5

+1019u6 − 126u7] . (7.25)

The remaining nine conditions are discussed in the next section.

7.2.3 Necessitated conditions

To find the unique solution of (7.16) with (7.8), we require nine more conditions.

For this purpose we define these conditions, named necessitated conditions, in

this section. These conditions can be defined as follows:

The values u−5, u−4, u−3, u−2, u−1 can be determined by the sixth order polyno-

mial S1(x) interpolating (xi, ui), 0 ≤ i ≤ 5. Precisely, we have

u−i − S1(−xi) = 0, i = 1, 2, 3, 4, 5,

where

S1(xi) =6∑

j=1

6

j

(−1)j+1U(xi−j).

186

Since by (7.7), U(xi) = ui for i = 1, 2, 3, 4, 5 then by replacing xi by −xi, we have

S1(−xi) =6∑

j=1

6

j

(−1)j+1u−i+j.

Hence the following necessitated conditions can be employed at the left end

6∑j=0

6

j

(−1)ju−i+j = 0, i = 1, 2, 3, 4, 5. (7.26)

Similarly for the right end, we can define ui−S1(xi) = 0, i =N +1, N +2, N +3,

N + 4 So we have the following necessitated boundary conditions at the right

end

6∑j=0

6

j

(−1)jui−j = 0, i = N + 1, N + 2, N + 3, N + 4. (7.27)

7.2.4 Stable linear system of equations

By using above necessitated conditions, we get a following new system of (N +

13) singularly perturbed linear equations with (N+13) unknowns {ui}, in which

N+1 equations are obtained from (7.14) and three equations obtained from giv-

en boundary conditions (7.8). Further nine equations are obtained from neces-

sitated conditions defined in (7.26) and (7.27).

If we use necessitated conditions then stable singularly perturbed linear system

of equations becomes

BU = F, (7.28)

where the coefficients matrix B = (LT ,AT ,RT )T , A is defined by (7.17).

The matrix L of order 7 × (N + 13) for left end boundary conditions is defined

as:

First five rows are obtained from (7.26), second last row is obtained from (7.23)

or (7.25) either U ′(0) or U ′′(0) and last row is also obtained from (7.8) i.e. U(0).

187

The matrix R of order 6 × (N + 13) for right end boundary conditions is con-

structed as: First row of above matrix is obtained from given condition (7.8) i.e.

U(1) and the remaining five conditions are obtained from the conditions (7.27).

The vector U is defined in (7.18) and F is define as

F = (0, 0, 0, 0, 0, β2, β0,FT1 , β1, 0, 0, 0, 0, 0)

T ,

or

F = (0, 0, 0, 0, 0, β3, β0,FT1 , β1, 0, 0, 0, 0, 0)

T ,

(7.29)

where F1 is defined in (7.19) and β0, β1, β2, β3, given in (7.8). For the existences

of the unique solution, first we check the non-singularity of the coefficient B.

We observed that B remains non-singulary for N 6 500 and for large N it may

or may not be non-singular.

7.3 Convergence of collocation algorithm

In this section, we discuss convergence of collocation algorithm described in

Section 7.2.

Let y(x) be the exact solution of the problem (7.1) or (7.2) then for j = 0, 1, · · ·N,

we have

−εy′′′(xj) + p(xj)y(xj) = f(xj). (7.30)

Let the vector y(x) be defined as

y(x) = (y(x0), y(x1), · · · , y(xN))T .

188

By Taylor’s series

y′′′(xj) = − 1

1672936499388h3

[−432275456y(j−6)h + 16885760000y(j−5)h

−15980430727y(j−4)h − 513333700000y(j−3)h + 3005844547822y(j−2)h

−4489601420000y(j−1)h + 4489601420000y(j+1)h − 3005844547822y(j+2)h

+513333700000y(j+3)h + 15980430727y(j+4)h − 16885760000y(j+5)h

+432275456y(j+6)h

]+ o(h7),

where y(xj − th) = y(j−t)h for t = −6,−5, · · · , N + 6. The system of equations

(7.28) provides the required subdivision based approximate solution U(x) for

(7.1) or (7.2) then by (7.8), for j = 0, 1, · · · , N

−εU ′′′(xj) + p(xj)U(xj) = f(xj), (7.31)

where U ′′′(xj) is defined as

U ′′′(xj) = − 1

1672936499388h3

[−432275456u(j−6)h + 16885760000u(j−5)h

−15980430727u(j−4)h − 513333700000u(j−3)h + 3005844547822u(j−2)h

−4489601420000u(j−1)h + 4489601420000u(j+1)h − 3005844547822u(j+2)h

+513333700000u(j+3)h + 15980430727u(j+4)h − 16885760000u(j+5)h

+432275456u(j+6)h

]+ o(h7),

where u(xj − th) = u(j−t)h for t = −6,−5, · · · , N + 6. Let the error function E is

defined as E(x) = y(x)− U(x) and

E = (E−6, E−5, · · · , EN+5, EN+6).

Then error vector at the node points is

E(xj) = y(xj)− U(xj), −6 6 j 6 N + 6.

This implies

E′′′(xj) = y′′′(xj)− U ′′′(xj), −6 6 j 6 N + 6.

189


−ε [y′′′(xj)− U ′′′(xj)] + p(xj) [y(xj)− U(xj)] = 0.

By definition of error vector

−εE′′′(xj) + p(xj)E(xj) = 0, 0 6 j 6 N.

This implies

−εE′′′(xj) + p(xj)E(xj) = 0, 0 6 j 6 N, (7.32)

where for 0 6 j 6 N

E′′′(xj) =1

1672936499388h3

[−432275456E(j−6)h + 16885760000E(j−5)h

−15980430727E(j−4)h − 513333700000E(j−3)h + 3005844547822E(j−2)h

−4489601420000E(j−1)h + 4489601420000E(j+1)h − 3005844547822E(j+2)h

+513333700000E(j+3)h + 15980430727E(j+4)h − 16885760000E(j+5)h

+432275456E(j+6)h

]+ o(h7).

As 0 ≤ x ≤ 1 and xj = jh, j = 0, 1, 2, · · · , N so E0, E1, · · · , EN are non zero while

E−6, · · · , E−1 and EN+1, · · · , EN+6 are zero because they lie outside the interval

[0, 1]. Let us define error values at the end points as

Ej =

max0≤k≤4

{|Ek|}O(h6), −6 ≤ j < 0,

maxN−4≤k≤N

{|Ek|}O(h6), N < j ≤ N + 6.

(7.33)

By expanding (7.32), we get

(B+O(h5))E = 0.

These are equivalent to

(B1 +O(h5))E = O(h5) ∥ E ∥= O(h5),

190

Since for small h, the coefficient matrix B + O(h5) will be invertible and thus

using the standard result from linear algebra and effect of ||B−1||, we have

||E|| ≤(

||B−1||1−O(h6)

O(h5))= O(h3).

Hence ∥ E ∥= O(h3). The result is summarized in the following theorem.

Theorem 7.3.1. Let y be the exact solution of the system (7.1) and U , be the approxi-

mate solution of (7.1) then ∥ E ∥∞=∥ y − U ∥∞= O(h3).

7.4 Numerical results and discussion

In this section, we have solved four examples by using subdivision based nu-

merical algorithm to show the accuracy of our algorithm. Numerical results of

these examples are calculated by using MATLAB. We observed that accuracy

between the exact and approximate solutions is good.

Example 7.4.1. Consider the following singularly perturbed boundary value prob-

lem:

−εy(3) + y(x) = f(x), x ∈ [0, 1]

y(0) = 0, y(1) = 0, y(1)(0) = 0,(7.34)

where

f(x) = 6ε(1− x)5x3 − 6ε2[6(1− x)5 − 90x(1− x)4 + 180x2(1− x)3

−60x3(1− x)2].

The analytic solution of Example (7.34) is

y(x) = 6x3ε(1− x)5.

The numerical results of Example 7.4.1 for different values of N and ε are giv-

en in Tables 7.1 and 7.2. Graphical representation of these numerical results is

shown in Figures 7.1 and 7.2.

191

Example 7.4.2. Consider the following singularly perturbed boundary value prob-

lem:

−εy(3) + y(x) = f(x), x ∈ [0, 1]

y(0) = 0, y(1) = 0, y(2)(0) = 0,(7.35)

where

f(x) = 6ε(1− x)5x3 − 6ε2[6(1− x)5 − 90x(1− x)4 + 180x2(1− x)3

−60x3(1− x)2].

The analytic solution of (7.35) is y(x) = 6x3ε(1 − x)5. Numerical results of this

example is shown in Tables 7.3 and 7.4 for different values of N and ε . Graphical

representation of these numerical results is shown in Figures 7.3 and 7.4.

Example 7.4.3. Consider the following boundary value problems

−εy3(x) + y(x) = 81ε2 cos 3x+ 3ε sin 3x, x ∈ [0, 1]

y(0) = 0, y(1) = 3ε sin 3, y(1)(0) = 9ε,(7.36)

The analytic solution of the system (7.36)

y(x) = 3ε sin 3x.

The numerical results for Example 7.4.3 is given in Table 7.4. Graphical repre-

sentation of these numerical results is shown in Figures 7.5 and 7.6.

Example 7.4.4. Consider the following boundary value problems

−εy3(x) + y(x) = 81ε2 cos 3x+ 3ε sin 3x, x ∈ [0, 1]

y(0) = 0, y(1) = 3ε sin 3, y(2)(0) = 0.(7.37)

The analytic solution of the system (7.37)

y(x) = 3ε sin 3x.

Numerical results of (7.37) is shown in Table 7.5 . Graphical representation of

these numerical results is shown in Figures 7.7 and 7.8.

192

Table 7.1: Maximum absolute errors for N = 10 of Example 7.4.1

ε Our algorithm By Akram (2012)

116

9.5454E-04 2.9E-03132

4.2571E-04 9.2E-04164

1.7964E-04 1.4E-04

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

x

Sol

utio

n

Analytic solution YApproximate solution U

Figure 7.1: Comparability of analytic and approximate solutions of Example 7.4.1 for

N = 100 with ε = 10−4 .

193

Tabl

e7.

2:M

axim

umab

solu

teer

rors

ofEx

ampl

e7.

4.1

εN

=10

N=

50N

=100

N=

150

N=

200

N=

250

0.1

1.60

53E

-03

7.48

32E

-04

7.14

46E

-04

7.09

10E

-04

7.07

44E

-04

7.06

94E

-04

0.01

1.00

97E

-04

4.73

93E

-05

4.68

96E

-05

4.74

64E

-05

4.78

77E

-05

4.81

67E

-05

0.001

9.86

78E

-06

1.56

62E

-06

1.19

65E

-06

1.38

00E

-06

1.49

73E

-06

1.57

54E

-06

0.0001

1.06

34E

-04

2.39

26E

-07

1.04

01E

-07

6.18

03E

-08

4.32

21E

-08

3.34

48E

-08

194

Tabl

e7.

3:M

axim

umab

solu

teer

rors

ofEx

ampl

e7.

4.2

εN

=10

N=

50N

=100

N=

150

N=

200

N=

250

0.1

1.61

90E

-02

7.33

71E

-04

6.44

63E

-04

6.36

71E

-04

6.34

96E

-04

6.34

46E

-04

0.01

5.47

77E

-04

3.53

02E

-05

3.27

08E

-05

3.30

05E

-05

3.33

31E

-05

3.35

79E

-05

0.001

4.38

14E

-05

2.41

50E

-06

1.39

66E

-06

1.15

44E

-06

1.23

48E

-06

1.29

13E

-06

0.0001

7.56

23E

-06

2.43

29E

-07

1.12

23E

-07

7.63

23E

-08

6.15

21E

-08

5.38

11E

-08

195

Table 7.4: Maximum absolute errors for N = 10 of Example 7.4.2

ε Our algorithm By Akram (2012)

116

6.2854E-03 1.3E-02132

1.9707E-03 3.2E-03164

3.9065E-04 3.4E-04

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

x

Sol

utio

n



N = 200 with ε = 10−4 .

196

Tabl

e7.

5:M

axim

umab

solu

teer

rors

ofEx

ampl

e7.

4.3

εN

=10

N=

50N

=100

N=

150

N=

200

N=

250

0.001

6.38

64E

-04

9.24

76E

-04

1.00

85E

-03

1.03

71E

-03

1.05

11E

-03

1.05

93E

-03

0.0001

4.74

79E

-04

3.25

67E

-05

4.14

41E

-05

4.44

05E

-05

4.58

84E

-05

4.67

75E

-05

0.00001

···

2.30

18E

-06

1.43

49E

-06

1.72

61E

-06

1.88

11E

-06

1.97

31E

-06

197

Tabl

e7.

6:M

axim

umab

solu

teer

rors

ofEx

ampl

e7.

4.4

εN

=10

N=

50N

=100

N=

150

N=

200

N=

250

0.001

2.52

76E

-03

1.75

08E

-04

7.01

00E

-05

3.67

47E

-05

2.03

55E

-05

1.06

33E

-05

0.0001

1.99

38E

-03

2.52

43E

-05

1.14

40E

-05

7.34

69E

-06

5.37

22E

-06

4.20

54E

-06

0.00001

···

2.04

15E

-05

1.33

33E

-06

8.32

94E

-07

6.09

41E

-07

4.80

45E

-07

198

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

X

Sol

utio

n



N = 100 with ε = 10−4 .

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−6

X

Sol

utio

n



N = 200 with ε = 10−4 .

199

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

−5

x

Solu

tion



N = 250 with ε = 10−5 .

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

−5

x

Solu

tion



N = 300 with ε = 10−5 .

200

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

−5

x

Solu

tion



N = 250 with ε = 10−5 .

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

−5

x

Solu

tion



N = 300 with ε = 10−5 .

201

7.5 Concluding remark

A binary interpolating subdivision scheme is used to construct a numerical al-

gorithm for solving third order singularly perturbed boundary value problem.

The algorithm is third order convergent. The numerical illustration shows that

the developed algorithm maintains a very remarkable high accuracy that makes

it very encouraging for dealing with the solution of singularly perturbed bound-

ary value problems. We compare numerical results with Akram (2012) algorith-

m and observed that our results are better than their results.

202

Chapter 8

Comparison, Conclusions,

Limitations and Future Work

In this chapter, we present the comparison, conclusion, limitations of proposed

algorithms and future research directions.

In this dissertation, we have presented the subdivision schemes based algo-

rithms for numerical solutions of the following types of problems:

• The linear boundary value problems of order three and four.

• The nonlinear third and fourth order boundary value problems.

• The second and third order singularly perturbed boundary value prob-

lems.

8.1 Comparison and Conclusion

In this section, first we present a comparison of proposed algorithms with other

numerical methods in literature and then we give concluding remarks about

our results.

203

8.1.1 Comparison with existing methods

In this section, we present comparison of our numerical algorithms with other

existing methods.

We compared numerical results by proposed algorithms for third order lin-

ear and nonlinear BVP’s with the numerical method based on fourth degree B-

splines of Caglar et al. (1999). We observed that our algorithm is more efficient

than that of Caglar et al.’s method. The order of approximation of proposed

algorithm and modified Adomian decomposition method of Hasan (2012) is

same. We have presented two collocation algorithms based on interpolating

and approximating subdivision schemes for the solution of linear fourth order

boundary value problems. From the numerical results, we observed that ap-

proximating subdivision scheme based collocation algorithm gives better result-

s comparative to the second order finite difference method. It is also observed

that, approximating subdivision scheme based collocation algorithm and quin-

tic spline based collocation algorithm have same order of approximation. The

proposed iterative algorithms for nonlinear fourth order BVP’s have been ap-

plied on different problems. Numerical results show that the accuracy of ap-

proximate solution is O(h3). It has been observed that proposed algorithm for

the solution of second order singularly perturbed boundary value problems is

better than spline based methods by Aziz and Khan (2002), Bawa and Natesan

(2005), Khan and Aziz (2005), Kadalbajoo and Aggarwal (2005) , finite difference

methods by Lubuma (2006), Miller (1979), Niijima (1980a), Niijima (1980b) and

Haar wavelet methods by Kumar and Mehra (2002), Pandit and Kumar (2014).

Numerical results of singulary third order boundary value problems by pro-

posed algorithm showed that our results are better than the results of Akram

(2012).

204

8.1.2 Conclusion

It is concluded that the subdivision based collocation algorithm is an ideal tool

to attain the numerical solutions of boundary value problems. The proposed al-

gorithms are computationally efficient and can be easily implemented on com-

puter. The numerical results showed that the adjustment of boundary condi-

tions at the end points influence the accuracy of the approximate solution. That

is, the accuracy of the solution can be improved by the proper adjustment of

boundary conditions. So our algorithm has flexibility to improve the results

by adjusting boundary conditions. The automatic selection and adjustment of

the boundary conditions to improve the approximation order of the solution is

possible future research direction.

8.2 Limitations of algorithms

In this section, we presented limitations of our work. These are the followings

• The proposed algorithms are applicable only for two point boundary val-

ue problems.

• We used only primal symmetric binary subdivision schemes, either they

are interpolating or approximating, to construct a subdivision based nu-

merical collocation algorithms.

• Our algorithms are applicable for all types of boundary value problems

with either the coefficients are polynomial or trigonometric functions.

• Our proposed algorithms are not applicable when the boundary condi-

tions are defined other than the interval 0 6 x 6 1.

205

8.3 Future Work

The proposed subdivision based collocation algorithms are limited to two point

boundary value problems. So the extension of these algorithms for the numer-

ical solutions of n-point nth order boundary value problems is possible future

research direction.

206

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Publications of Syeda Tehmina Ejaz

1. Numerical solution of two-point boundary value problems by interpolat-

ing subdivision schemes, Abstract and Applied Analysis, Vol. 2014, Article

ID 721314, 13 pages, (2014).

2. Subdivision schemes based collocation algorithms for solution of fourth

order boundary value problems, Mathematical Problems in Engineering, Vol.

2015, Article ID 240138, 18 pages, (2015). Impact factor = 0.644

3. A subdivision based iterative collocation algorithm for nonlinear third or-

der boundary value problems, Advances in Mathematical Physics, Vol. 2016,

Article ID 5026504, 15 pages, (2016). Impact factor = 1.12

4. A numerical approach based on subdivision schemes for solving nonlinear

fourth order boundary value problems, Journal of Computational Analysis

and Applications, Vol. 23(4), page no. 607-623, (2017). Impact factor = 0.481

5. Subdivision based collocation method for singularly perturbed boundary

value problems. Article submitted.

6. A subdivision collocation method for solving two point boundary value

problems of order three. Accepted in Journal of Applied Analysis and Com-

putation, 2017. Impact factor = 0.844

222

subdivision schemes and their applications to solve differential...

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