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MODIFIED EXTENDED BLOCK BACKWARD

DIFFERENTIATION FORMULA FOR SOLVING

STIFF ODEs

ASMA IZZATI BINTI ASNOR

SCHOOL OF MATHEMATICAL SCIENCES

UNIVERSITI SAINS MALAYSIA

2015

MODIFIED EXTENDED BLOCK BACKWARD

DIFFERENTIATION FORMULA FOR SOLVING STIFF ODEs

by

ASMA IZZATI BINTI ASNOR

Dissertation submitted in partial fulfillment

of the requirements for the degree

of Master of Science in Mathematics

August 2015

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ACKNOWLEDGEMENT

In the Name of Allah, the Most Beneficient, the Most Merciful

Alhamdulillah, praise be to Allah. Peace and blessings be upon the Prophet of

Allah, Nabi Muhammad S.A.W. I am grateful to Allah that I have completed this

dissertation within the prescribed time. I would like to express my deepest gratitude to

my supervisor, Dr. Siti Ainor binti Mohd Yatim for her encouragement and guidance

throughout the learning process of this dissertation. Without her good supervision and

persistent help, this dissertation would never have been completed. I would like to thank

her for priceless advices, brilliant comments and valuable informations over this

semester. I am greatly appreciate it.

In addition, I also take this opportunity to offer my special thanks to the most

important people, my family members for their endless and continuous support. My

parents who have given their love and understanding towards me to get through this

period as a master student. Words cannot express how grateful I am for all the sacrifices

made by them. Alhamdulillah.

My special appreciation goes to all my friends who have supported me in

writing this project and assist me to achieve my goal and for those who always help me

during the completion of this dissertation. I will never forget your kindness. Thank you

so much.

Asma, 2015.

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TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT ii

TABLE OF CONTENTS iii

LIST OF TABLES vi

LIST OF FIGURES vii

LIST OF ABBREVIATIONS ix

ABSTRAK x

ABSTRACT xii

CHAPTER 1 : INTRODUCTION

1.1 Background 1

1.2 Objectives of the Study 6

1.3 Problem Statement 6

1.4 Scope of Study 7

1.5 Methodology 7

1.6 Outline of the Study 8

CHAPTER 2 : LITERATURE REVIEW

2.1 Introduction 10

2.2 Stiff initial value problem 10

2.3 Linear Multistep Method 12

2.4 Block method 13

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2.5 Review of Previous Work 15

CHAPTER 3 : FORMULATION OF THE METHOD

3.1 Introduction 18

3.2 Formulation of Predictor Method for VS-BBDF method 18

3.3 Formulation of Corrector Method for VS-BBDF method 21

3.4 Implementation of the Method 24

3.4.1 Newton’s Iteration 24

3.4.2 Choosing the step size 26

3.5 Stability of VS-BBDF Method 28

CHAPTER 4 : RESULTS AND DISCUSSIONS

4.1 Introduction 34

4.2 Test Problems 34

4.3 Numerical Results 38

4.4 Discussions 53

CHAPTER 5 : SUMMARY

5.1 Conclusions 55

5.2 Future Study 56

REFERENCES 57

APPENDIX

Appendix A : Algorithm for VS-BBDF method of order four

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LIST OF TABLES

Page

3.1 The formulae for predictor and corrector method for 23

two computed points

3.2 Local truncation error for the three step size ratios 27

3.3 Lists of stability polynomials and roots for the three step size ratios 30

iii

LIST OF FIGURES

Page

2.1 VS-BBDF method of order 4 (P4) 14

3.1 Stability region of for r = 1 31

3.2 Stability region of for r = 2 32

3.3 Stability region of for r = 5/9 32

3.4 Stability regions for r = 1, r = 2 and r = 5/9 33

4.1 Approximated solutions curves for variable step size ratios 39

for Test Problem 4.1

4.2 Graph of approximate solution and exact solution for 39

Test Problem 4.1

4.3 Total steps curves for Test Problem 4.1 40




Test Problem 4.2





Test Problem 4.3




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Test Problem 4.4





Test Problem 4.5





Test Problem 4.6


4.19 Graph of approximated solutions for variable step size ratios 51



Test Problem 4.7


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LIST OF ABBREVIATIONS

DE Differential Equation

ODE Ordinary Differential Equation

PDE Partial Differential Equation

IVP Initial Value Problem

BDF Backward Differentiation Formulae Method

BBDF Block Backward Differentiation Formulae Method

VS-BBDF Variable Step Block Backward Differentiation Formulae Method

LMM Linear Multistep Method

LTE Local Truncation Error

TOL Tolerance Limit

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PENGUBAHSUAIAN LANJUTAN FORMULA PEMBEZAAN

BLOK KEBELAKANG UNTUK MENYELESAIKAN PPB KAKU

ABSTRAK

Disertasi ini tertumpu kepada pengubahsuaian dan melanjutkan kaedah Formula

Pembezaan Blok Kebelakang (FPBK) yang sedia ada untuk menyelesaikan Persamaan

Pembezaan Biasa (PPB) kaku peringkat pertama. Kaedah baru ini akan diolah dengan

menggunakan saiz langkah 1.8. Kami memberi penekanan pada merumus kaedah blok

dua-titik berdasarkan kepada Langkah Berubah Formula Pembezaan Blok Kebelakang

(LB-FPBK) peringkat keempat dengan menggunakan pendekatan langkah berubah

untuk terbitan kaedah – kaedah tersebut. Strategi yang terlibat adalah pemilihan saiz

langkah. Jelasnya, matlamat kaedah blok dua-titik adalah untuk mengira dua nilai baru

dalam satu blok masa yang sama pada setiap langkah dan pada masa yang sama

menggunakan tiga nilai belakang blok sebelumnya. Beza Bahagi Newton digunakan

pada kaedah yang dicadangkan untuk menganggarkan masalah. Kemudian,

perbandingan prestasi untuk saiz langkah 1.8 dibuat dengan dua lagi saiz langkah, 1.6

dan 1.9 untuk menentukan kesan penyelesaian dengan menukar nilai nisbah saiz

langkah. Oleh yang demikian, rantau kestabilan kaedah LB-FPBK juga dibincangkan

dan illustrasi rantau kestabilan diplot di dalam graf. Kod program ditulis dalam

pengaturcaraan C. Keputusan berangka tersebut menunjukkan, sedikit perubahan pada

corak penyelesaian dengan menukar nilai nisbah saiz langkah. Kesimpulannya,

keputusan tersebut menunjukkan bahawa saiz langkah 1.8 memberi ketepatan yang

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paling baik untuk masalah terpilih yang diuji. Oleh itu, dengan meningkatkan nisbah

saiz langkah akan memberikan prestasi yang lebih baik dari segi ralat maksimum.

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ABSTRACT

This dissertation is concerned to modify and extend the existing block backward

differentiation formula (BBDF) method for solving first - order stiff IVP for ODE. The

new method will be modified by using step size 1.8. We emphasized on formulating

two-point block method based on fourth order variable step BBDF (VS-BBDF) method

by utilizing variable step approach for the derivation of the methods. The strategy for

choosing the step size are also involved. Apparently, the goal of two-point block

method is to compute two new values in a block simultaneously at every step and at the

same time using three back values of previous block. The Newton Divided Difference is

applied to the proposed method to approximate the problems. Then, the comparison for

performance step size 1.8 is made with the other two step sizes, 1.6 and 1.9 to determine

the effect of solutions by changing the value of step size ratios. Consequently, the

stability region of VS-BBDF method is also discussed and the illustration is plotted in a

graph. The source code is written in C language. The numerical results, shows a slight

change in the pattern of the solutions by changing the value of step size ratio. In

conclusions, the results show that step size 1.8 gives the best accuracy for the selected

tested problems. Thus, increasing the step size ratio will give better performance in

terms of maximum error.

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CHAPTER 1

INTRODUCTION

1.1 Background

Numerical analysis is such a very wide area of mathematics and computer science

study that deal with most of the problems in real-world. It is the study that creates,

analyzes, and implements algorithms for solving problems numerically of continuous

mathematics with finding approximation solutions. Mathematicians interpret the real-

world situations into mathematical formulation. The development of sophisticated

computer at this century, the computer is beneficial for researchers or mathematicians to

solve more complicated mathematical models of the real world.

Several natural phenomena in most real-life situations are represented or modelled by

functions. The functions may depend on one or more independent variables and usually

time and space (location) variables are choosen as the independent variables. For instance,

in real-life situation, the position of the earth changes with time, the area of a circle

changes with the size of radius and many more. For that reasons, any equation involving

an unknown function with some or all of its derivatives, either ordinary derivatives or

partial derivatives is known as differential equation (DE).

DE is applied to model problems in science such as physics, biology and chemistry

as well as many other branches of studies (engineering and economic) involve the change

of some variable with respect to another. The problems in real-life situations are

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complicated to be solved exactly. So, the easier way is to approximate the solutions.

Besides, it shows the relationship between physical quantities and the rates of change.

Physical quantities usually is defined as functions while the derivatives represent the rates

of change. There are two major kinds of DE, Ordinary differential equation (ODE) and

Partial differential equation (PDE).

Over the recent years, mathematical modelling problems has evolved in applied

science and engineering field of modern life. ODE is one of the problems arises in the

field. Many researchers and mathematicians have been solving ODE with various kind of

numerical methods from the earlier research with the intention to find the best

approximations for the solutions. Other than that, in 20th century the study on numerical

methods for the solution of initial value problem (IVP) ODE has become famous topic

and advanced in study as many researchers are interested in doing their researches. The

numerical methods used for solving ODE is to find the approximations to the solutions

of ODE where it provides an alternative way to some difficult problems. Some of the

numerical methods used in PDE convert the PDE to an ODE to solve it. The numerical

methods are classified as single-step methods or multi-step methods. In addition, the

methods can be divided into explicit methods or implicit methods. Since numerical

method has stability limitation on the step size, there are only few methods that can solve

stiff problems (Fatunla, 1990). Therefore, a commonly used method which is known as a

foundation method that still in use in the present for solving ODE is Backward

Differentiation Formula (BDF) (Suleiman et al., 2013) . It is also the most well-liked

implicit methods for solving stiff ODEs. This method has been proposed by Gear in 1970

who is also one of the well – known researcher in the study of stiff ODEs.

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By defining IVP as a differential equation together with a specified initial condition

of the function at a given point in the domain of the solution. IVP of first-order ODE

considered in this project is of the form

where 0x is initial value in the given interval ],[ 0 nxxx . An ODE equation (1.1) contain

a function and its ordinary derivative of dependent variable with respect to single

independent variable. Order of ODE is the highest order of the derivative of the function

that appears in the equation. Besides, there are many numerical techniques for the

numerical solution of stiff IVP and the techniques depend on many factors such as

computational expense, data-storage requirements, speed of convergence, accuracy and

stability.

Stiff ODE has been large development of study by researchers since the last

century and until now they are still interested to do some researches on solving stiff ODE.

The problem will be having some difficulties when standard numerical techniques are

applied to approximate the solution of a differential equation when the exact solution has

terms of the form te , where is complex number with negative real part. This term will

decay to zero as t increases. Besides, this problem occurs in wide variety of application

including the study of spring and damping systems, problems in chemical kinetics and

many more. The system is stiff if its solution contains components with both slowly and

rapidly decaying rates because of large difference of time scales exhibited by the system

(Suleiman, 2013). The idea is that, it can lead to rapid variation in the solution. However,

it is important to determine either the ODE is stiff problem or not by the presence of very

),( yxfy , 00 )( yxy (1.1)

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large negative eigenvalues of its Jacobian matrix, y

f

. However, the problem that is said

to be stiff requires very small step size to solve it. Otherwise, it will be unstable

(Mahayadin et al., 2014). For most stiff problems are difficult to solve since many

numerical methods have stability restriction on the step size unless the step size chosen

is very small to achieve the accuracy. Therefore, we must concern with choosing the

suitable numerical methods that solve stiff problem efficiently because if not the

numerical methods will become unbereably slow. Furthermore, stiff problem only works

with implicit method or in other words it can be said that explicit method cannot handle

the problem efficiently. Implicit methods on solving stiff ODEs are known to perform

better than explicit ones (Abasi, 2014). Theoretically, numerical methods that is suitable

for ODEs is usually implicit, which require repeated solutions of systems of linear

equations with coefficient matrix, JhI , here J is the Jacobian matrix (Ibrahim et al.,

2008).

BDF is linear multistep methods (LMM) that suitable for solving stiff initial value

problems. To improve the existing BDF method, Ibrahim et al. has proposed a new idea

in which the method generates block approximation knnn yyy ,...,, 21 (Ibrahim et al.,

2007). The reason is to produce better approximations in terms of computation time and

precision on solving first order stiff ODE. Hence, the most recent study on this new

method is known as Block Backward Differentiation Formula (BBDF). Over the years, a

block method has been discussed by a few researchers since the earlier research. This

block approximations has been used in different methods. For instance, block implicit

one step methods was proposed by Shampine and Watts (1969) is the earliest research on

block methods. Other studies done by Chu and Hamilton (1987) with multi-block

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methods, Voss and Abbas (1997) with block predictor-corrector schemes and Ibrahim et

al. (2007) with block method based on BBDF method for first order stiff ODEs. The rapid

growth of the studies on the block methods for solving ODEs contribute to the

competition in developing and deriving an accurate method for solving many types of

ODEs by Zawawi et al. (2012). Block methods also have some advantages such as it

approximates the solution at more than one point and the number of point depends on the

structure of the block methods. So from the advantage, the execution time and the total

number of iterations can be reduced because it gives faster solutions. Therefore, this

method shows that it is more efficient than BDF method.

Basically, a block method is a method of a block of new values is obtained

simultaneously. It is called a block method because it compute previous k blocks and

calculate the current block where each block contain r points. In this dissertation, we

would only emphasize on two-points block method where two values are computed

simultaneously in a block by using the values of previous block with each block

containing two points. The solution of 1ny and 2ny are computed using three back

values, nnn yyy ,, 12 . There are five points for fourth order where two points will be

calculated and the rest three points are the previous points. Initially, this study is supposed

to formulate the BBDF method by using variable step size approach. The strategy

involved based on the value of Local Truncation Error (LTE) for choosing step size of

the method. The LTE is also depends on error tolerance limit where LTE will be less or

greater than error tolerance limit. The step size involve are constant step size, half of the

step size and increment of the step size. The doubling step size is not considered due to

zero instability. After that we can plot and investigate the stability region for BBDF

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method. For a better performance, Newton iteration has been implemented to BBDF

method.

Nowadays, the algorithms are implemented in variety of programming languages for

numerical analysis. Some of the popular languages that are used to implement the

algorithms of numerical methods such as Fortran, C++ and Java. They are utilized to

make the complexity of the things for making more uncomplicated and efficient as well

as productive. In this dissertation, we only apply Microsoft Visual C++ to implement

BBDF method while Maple15 software is used to derive the formulae for predictor and

corrector of VS-BBDF methods.

1.2 Objectives

The main objective of the dissertation is to construct block BDF method by using variable

step size approach for solving first-order stiff ODE. The aim can be accomplished by :

• Derivation of the fourth order Variable Step Block Backward Differentiation

Formula (VS-BBDF) method.

• Solving numerically IVP of first order stiff ODE by applying Variable Step Block

Backward Differentiation Formula (VS-BBDF) method.

• Investigating the effect of the solutions by changing the value of step size ratio, r.

1.3 Problem Statement

An ordinary differential equation (ODE) that has a function and its ordinary derivative.

For this dissertation, we consider a system of first order stiff initial value problems (IVPs)

ODE. The problem to be considered in general form

)),(,()( xYxfxAYy iii )(aY ni ,...,2,1 (1.2)

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where

x is the interval from a to b, ],[ bax ,

),,...,,,()( 211 nyyyyxY

),...,,( 21 n ,

A is mm matrices with large negative eigenvalues.

1.4 Scope of the Study

This dissertation is emphasized only on numerical solution of IVP of first - order stiff

ODE. We are interested to solve the problem by using two-point block method based on

BBDF method where two new values in a block will be computed simultaneously using

three back values. This study is focused on fourth order variable step BBDF (VS-BBDF)

method. We utilize variable step size approach to derive the formula for the predictor and

corrector block methods where the variable step size involve in this dissertation are

constant step size, half the step size and increment the step size by a factor of 1.8 at which

all of these satisfy the zero stability. The strategy for choosing the step size is based on

Local Truncation Error (LTE). In this dissertation, we only compare the numerical results

for step size of the proposed method within the selected test problems with the results of

the previous researches on step size 1.6 and 1.9. In this dissertation, we will use Microsoft

Visual C++ 6.0 as platform to implement the VS-BBDF method with Newton iteration

algorithm. The program will be written in C language.

1.5 Methodology

Next section is methodology. Basically, there are a few methods that are needed to

accomplish this dissertation in order to find the numerical results for the problems that

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have been tested. Firstly, we formulate the predictor and corrector methods formulae for

VS-BBDF method using Maple 15 software. Then, we apply Newton’s iteration to the

corrector method. Microsoft Visual C++ is used as a compiler for the implementation of

the method. In this dissertation, we consider the variable step size approach where the

appropriate step size is chosen based on the Local Truncation Error (LTE). In addition,

the stability of the proposed method is investigated by applying the VS-BBDF method to

the test equation. We then check for the zero-stability and plot the stability region in the

complex plane.

1.6 Outline of the Project

This dissertation is organized as follows.

Chapter 1 is a brief introduction of this dissertation regarding the ODE and the

background of the method used for solving the problem. Besides, this chapter also

includes objectives of the study to achieve the goals for this dissertation. Then, provides

also problem statement, scope of the study and also methodology. Lastly, the outline of

the later chapters will be in the last part for this chapter.

In Chapter 2, we explain in detail for each of the basic concepts, theories and

definitions of the related study to support the completion of the current research. In the

same chapter, we provide some literature review where it discussed in detail about the

previous researches and overview from the early researches on background of existing

BBDF method. In this chapter, there are also review on other block methods that are used

to solve stiff ODE by previous researchers.

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Chapter 3 provides the formulation of the method of this project especially further

explanation for the development of VS-BBDF using Lagrange polynomial as the

interpolating polynomial. The stability region is presented and the zero stability is

investigated. In addition, we develop the algorithm of VS-BBDF method that would solve

the stiff ODE problems by applying Newton iteration to the method. The source code is

written in C language and using Microsoft Visual C++ platform to implement the method.

The results of this dissertation are presented in Chaper 4. There are seven selected

tested problems of stiff IVP ODE that are tested. The numerical results for step size 1.8

is compared with the step size 1.9 and 1.6. Then, the comparisons are made between the

computed results and the existing results for step size 1.9 and 1.6. Last but not least, the

results are represented in figures.

Finally, the summary of this dissertation is presented in Chapter 5. We also made

some conclusions and a few recommendations for future research.

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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

This chapter discusses the review of the previous researches on stiff ODE and block

methods. We discuss in detail for every main points in this chapter by providing some

definitions and theorems to strengthen the main points.

2.2 Stiff initial value problems

Stiff IVP were first encountered when varying stiffness occurs in the study of the

motion springs. The IVPs is considered as the following general linear systems with

constant coefficients,

),(xdAyy ,)( 0yay ,bxa (2.1)

where A is mm with real entries.

There are no universal definition of stiffness but the idea is more or less similar to one

another. However, a few experts have defined the definition of stiffness according to their

own way.

Definition 2.1 (Curtiss and Hirschfelder, 1952)

Stiff equations are equations where certain implicit methods, in particular BDF, perform

better, usually tremendously better, than explicit ones. The eigenvalues of the Jacobian

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y

f

play certainly a role in this decision, but quantities such as the dimension of the

system, the smoothness of the solution or the integration interval are also important.

Definition 2.2 (Lambert, 1973;1991)

The system of IVP ODE is said to be stiff if

i. Re )( t < 0, t = 1, 2, …, m and

ii. |)Re(|min|)Re(|max tttt where t are the eigenvalues of the Jacobian

matrix, y

f

If the numerical method with a finite region of absolute stability, applied to a system

with any absolute stability, applied to a system with any initial conditions, is forced to

use in a certain interval of integration a step length which is excessively small in relation

to the smoothness of the exact solution in that interval, then the system is said to be stiff

in that interval.

There are other characteristics that exhibit by many examples of stiff problems, but for

each there are counter examples, so these characteristics do not make good definitions of

stiffness. Lambert refers to these as ‘statements’ rather than definitons. A few of these

are:

i. A linear constant coefficient system is stiff if all of its eigenvalues have negative

real part and the stiffness ratio is large.

ii. Stiffness occurs when stability requirements, rather than those of accurancy,

constrain the steplength.

iii. Stiffness occurs when some components of the solution decay much more rapidly

than others.

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Definition 2.3 (Fatunla, 1987)

he stiffness ratio S of the system (2.1) is given as

,)ln(

)(||max

TOL

abS i

i

where )ln(TOL is the exponential logarithm of TOL.

Definition 2.4 (Dahlquist, 1974)

Systems containing very fast components as well as very slow components.

Definition 2.5 (Shampine, 1981)

A major difficulty is that stiffness is a complex of related phenomena, so that it is not

easy to say what stiffness is.

2.3 Linear Multistep Method

Basically, multistep method use the approximation values at more than one

previous value to approximate the subsequent value. Furthermore, multistep method is

more accurate than one-step method because it use more information about the known

portion of the solution than one-step. One of the category of multistep is Linear Multistep

Method (LMM). It can be written as linear combination of the value of solution and the

value of function at previous points.

The general LMM is

k

j

jnj

k

j

jnj fhy00

(2.2)

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where j and j are constant, k is the number of steps used in multistep and h is the step

size. Coefficients are presumed to be real and satisfy the conditions 1k and

0|||| 00 . If 00 , so the method is explicit, otherwise it is an implicit method.

2.4 Block method

Apparently, block method has become one of the available solvers for stiff ODE.

Generally, block method is a method that obtain concurrently a block of new values by

computing k number of blocks. It uses values from the preceeding block to compute the

values for current block. Apart from that, block method will have r block size where it

will become r by r matrices of k blocks all together.

This method has some benefits in particular that can reduce the execution time

and total number of iterations where this block method generates new values at r points

concurrently at each step.

For instance, two-point block method introduced by Ibrahim et al. (2007), two

new values that are 1ny and 2ny are computed simultaneously by using the values from

previous block whereby each block will have maximum of two points.

In addition, the orders of the method are determined by the number of back values

contained in total blocks (Yatim et al., 2013). Step size ratio, r is defined as the ratio

distance between current step, nx and previous step, 1nx . For computed block, the step

size is 2h and 2rh is the step size for previous block.

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The figure below shows the illustration of the block method.

Previous block Current block

rh rh h h

2nx 1nx

nx 1nx

2nx

p4

Figure 2.1 VS-BBDF method of order 4 (p4) (Yatim et al., 2013)

2.5 Review of previous work

Shampine and Watts (1969) has discussed on block implicit one-step methods.

They studied a class of one-step methods for solving ODE at which they obtained block

of r new values at each step for solving linear and nonlinear ODE. In addition, they also

studied the stability and convergence for the particular method. The results showed that

Block one-step (BOS) method and Block one-step (BOSLS) method (as applied to linear

problems) were competitive with the other methods and BOS was more accurate than

BOSLS when small step size was used.

BBDF method was proposed by Ibrahim et al. (2007a) which was similar to

standard form of BDF method. But the difference was this method computed new values

in a block at the same time. The block method allowed us to store the coefficient of y

values. For that reason, it will avoid us to repeat the calculation of differentiation

coefficients at each step. In this research, they also presented the regions of absolute

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stability for the method. They compared the efficiency of 2-point BBDF method with

conventional variable step variable order BDF (VSVOBDF) method. Hence, the results

showed that BBDF method gave more accurate results than VSVOBDF method with

lesser total steps and lesser computational time. In conclusion, this method was suitable

to solve stiff ODE.

According to Ibrahim et al. (2007b), they focused on implicit 2-point block

method based on BDF method. Therefore, two new values will be generated concurrently.

In this study, they also used variable step approach on variable step block backward

differentiation formula (VS-BBDF) method. Hence, the variable step size were constant

step size, half of the step size and increment the step size to 1.6. Then, they also have

plotted the absolute stability region for the method. The method applied to the selected

test problems were compared with variable step variable order non block BDF method

(NBDF). So, the results showed that 2-point BBDF method gave better performances than

non-block BDF (NBDF) with reduction of total step and lesser computational time.

In the same year, Ibrahim et al. (2007c) has released a new idea where they solved

ODE by using implicit r-point BBDF method. They derived a block of r new values at

each step and make a comparison between r-point block methods with the existing BDF

method. Hence, the results indicated that the r-point BBDF method was more efficient

than BDF method with reduction of the number of integration step and improve CPU

time.

Three years later Yatim et al. (2010) has extended the study by Ibrahim et. al in

2007. A few selected test problems have been tested with the increment of step size to

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1.9. The main idea of this study was quite similar to Ibrahim et. al (2007b) with the

intention to optimize the performance of the method to produce better approximations

and good computation time. They derived the implicit block methods based on BDF

method for the solution stiff IVP while they also used variable step size approach.

Besides, the construction of variable step size block methods will store all the coefficients

of the method. Then they compared the results with the step size 1.6. In conclusion, this

study gave better results because the increment of step size can reduce the number of total

steps and also lesser computational time. Therefore, step size 1.9 was more efficient than

step size 1.6.

The research of Nasir et al. (2011) has presented a new method which called as

fifth order 2-point BBDF method. They derived the formula of the method where two

new values will be produced concurrently at each step using four back values. This new

method was used for solving first order ODE. Other than that, the study has shown that

this method was stiffly stable and satisfied the conditions to solve stiff problems. They

also compared the new method with classical BDF Method and ode15s in MATLAB.

Lastly, the study has concluded that the methods performed competitively and the

efficiency was over the BDF method and ode15s.

Implicit Continuous BBDF (CBBDF) method has been proposed by Akinfenwa

et al. (2013). This method was applied to solve ODE which they derived a block of p new

values at each step which simultaneously provided the approximate solutions for the stiff

ODEs. The performance of CBBDF method was compared with BDF method. So from

the numerical results, the proposed method was more efficient because it produced

accurate results and fewer number of function evaluations and computational steps.

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Then, Yatim et. al. (2013) has discussed ‘A numerical algorithm for solving stiff

ordinary differential equations’. In this research, they applied variable step variable order

approach to the BBDF (VSVO-BBDF) method. Other than that, they also formulated

VSVO-BBDF method of order 3 to 5 and investigated the stability region of the particular

method. A comparison for the results was made between the proposed method and

MATLAB’s suite of ODEs solvers namely, ode15s and ode23s. Therefore, VSVO-BBDF

method outperformed ode15s and ode23s at which it managed to reduce the number of

total steps taken as well lesser computational time.

The recent study of Abasi et al. (2014) has introduced a formula for 2-point block

method with two off step of order 5 based on BDF method. The method was utilized for

finding the solution for stiff ODE. Furthermore, a strategy of the method was to produce

two new values ( 1ny and 2ny ) with step size h and two off step points 2

1n

y and 2

3n

y

with step size is halved simultaneously at each step. The formulae were computed using

two back values ny and 1ny with step size h in the previous block. This paper also

generated the stability region and convergence of the proposed methods. The method was

shown to be A-stable and convergent. In addition, they compared the proposed method

with the existing fifth oder BBDF method and the results showed that the performance of

methods were competitive in terms of accuracy and execution time.

18

CHAPTER 3

FORMULATION OF THE METHOD

3.1 Introduction

This chapter involve the derivation of predictor and corrector methods for VS-BBDF of

order four that are derived using Maple 15. We describe in detail one by one step that

include in the derivation of the methods. We only need three backvalues for predictor

methods whilst, five points are needed for corrector method as it is of order four. Then

for implementation of the method, we apply Newton’s iteration to the methods which we

derive later in this chapter. At the end of this chapter, we will present the further

implementation of the method and the strategy used for choosing the suitable step size.

We will use environment of Microsoft Visual C++ 6.0 to implement the algorithm in this

dissertation.

3.2 Formulation of predictor method for VS-BBDF method

The derivation of predictor method for VS-BBDF method presented here is conducted

using Maple 15 with ‘CurveFitting’ package. Therefore, there are three points, 12 , nn yy

and ny as the backvalues used to predict the values for points 1ny and 2ny . The

backvalues will be the interpolating points. Thus, we interpolates the points in Lagrange

form where Lagrange polynomial is the polynomial )(xPk degree k that passes through

the points ),(),,( 1122 nnnn yxyx and ),( nn yx .

19

The Lagrange basis polynomial is

k

jii injn

jn

jkxx

xxxL

0 11

1

,)(

)()( for each kj ,...,1,0 . (3.1)

So, the general form of Lagrange polynomial defined as follows

k

j

jkjnk xLxyxP0

,1 )()()( (3.2)

or expansion of the product (3.2) will be as below

))...((

))...((...

))...((

))...(()(

1

1

1

1

kknnkn

knkn

knnnn

knnnk

xxxx

xxxxy

xxxx

xxxxyxP

(3.3)

Start the derivation by using Lagrange’s interpolation formula to find the interpolating

polynomial through the points ),(),,( 1122 nnnn yxyx and ),( nn yx and obtained the

polynomial as follows

n

nnnn

nnn

nnnn

nnn

nnnn

nn yxxxx

xxxxy

xxxx

xxxxy

xxxx

xxxxxP

))((

))((

))((

))((

))((

))(()(

12

121

121

22

212

1

(3.4)

20

Then, define h

xxs n 1 and substitute 1)( nxhsx into (3.4) yield equation (3.5),

n

nnnn

nnnn

n

nnnn

nnnn

n

nnnn

nnnn

yxxxx

xxhsxxhs

yxxxx

xxhsxxhs

yxxxx

xxhsxxhsxP

))((

))((

))((

))((

))((

))(()(

12

1121

1

121

121

2

212

111

(3.5)

Next, replace ),( yxf in (1.1) by polynomial (3.5). Substitute 0s at point 1 nxx and

1s at point 2 nxx into P(x). So, for latest values of P(x), we replace step size ratio, r

by three distinct values that are 1, 2 and 5/9. Subsequently, the last part of derivation gives

the formulae for predictor method for fourth order VS-BBDF method. Step size ratio, r

is defined as the ratio distance between current step, nx and previous step, 1nx .

Let old

new

h

hr .

The predictor formulae for first and second points are

i. For r = 1 gives

nnnn

nnnn

yyyy

yyyy

683

33

122

121

(3.6)

ii. For r = 2 gives

nnnn

nnnn

yyyy

yyyy

33

8

15

4

5

8

3

122

121

(3.7)

21

iii. For r = 5/9 gives

nnnn

nnnn

yyyy

yyyy

25

322

25

504

25

207

25

133

25

171

25

63

122

121

(3.8)

3.3 Formulation of corrector method

The steps involve are quite similar to predictor method where the derivation of

corrector method are constructed using Maple 15 environment with “CurveFitting”

package as well as ‘PolynomialInterpolation’ command. The different from predictor is

corrector method need five points as interpolating points. Thus, we interpolates the points

in Lagrange form where Lagrange polynomial is the polynomial )(xPk degree k that

passes through the points ),(),,(),,(),,( 111122 nnnnnnnn yxyxyxyx and ),( 22 nn yx .

We begin our derivation by using Lagrange’s interpolation formula to find the

interpolating polynomial through the points ),(),,(),,(),,( 111122 nnnnnnnn yxyxyxyx

and ),( 22 nn yx and obtained the polynomial as follows

2

1221222

112

1

2111121

212

2112

2112

1

2111121

212

2

2212212

211

))()()((

))()()((

))()()((

))()()((

))()()((

))()()((

))()()((

))()()((

))()()((

))()()(()(

n

nnnnnnnn

nnnn

n

nnnnnnnn

nnnn

n

nnnnnnnn

nnnn

n

nnnnnnnn

nnnn

n

nnnnnnnn

nnnn

yxxxxxxxx

xxxxxxxx

yxxxxxxxx

xxxxxxxx

yxxxxxxxx

xxxxxxxx

yxxxxxxxx

xxxxxxxx

yxxxxxxxx

xxxxxxxxxP

(3.9)

22

Then, define h

xxs n 1 and substitute 1)( nxhsx into (3.9) yield equation (3.10)

2

1221222

1111121

1

2111121

2111121

2112

21111121

1

2111121

2111121

2

2212212

2111111

))()()((

)))(())(())(()((

))()()((

)))(())(())(()((

))()()((

)))(())(())(()((

))()()((

)))(())(())(()((

))()()((

)))(())(())(()(()(

n

nnnnnnnn

nnnnnnnn

n

nnnnnnnn

nnnnnnnn

n

nnnnnnnn

nnnnnnnn

n

nnnnnnnn

nnnnnnnn

n

nnnnnnnn

nnnnnnnn

yxxxxxxxx

xxhsxxhsxxhsxxhs

yxxxxxxxx

xxhsxxhsxxhsxxhs

yxxxxxxxx

xxhsxxhsxxhsxxhs

yxxxxxxxx

xxhsxxhsxxhsxxhs

yxxxxxxxx

xxhsxxhsxxhsxxhsxP

(3.10)

Next, replace ),( yxf in (1.1) by polynomial (3.10). Later, we differentiate (3.10) with

respect to s as well as substitute 0s at point 1 nxx and 1s at point 2 nxx into

P’(x). So, for latest values of P’(x), we replace step size ratio, r by three distinct values

that are 1, 2 and 5/9. Subsequently, the last part of derivation gives the formulae for

corrector methods for fourth order VS-BBDF method.

Let old

new

h

hr .

The formulae obtained are

i. Formula for r = 1 gives

nnnnnn

nnnnnn

yyyyhfy

yyyyhfy

25

36

25

16

25

3

25

48

25

12

5

9

5

3

10

1

10

3

5

6

12122

12211

(3.11)

23

ii. Formula for r = 2 gives

nnnnnn

nnnnnn

yyyyhfy

yyyyhfy

23

18

23

3

115

2

115

192

23

12

128

225

128

25

128

3

128

75

8

15

12122

12211

(3.12)

iii. Formula for r = 5/9 gives

nnnnnn

nnnnnn

yyyyhfy

yyyyhfy

35625

103684

11875

27216

225625

128547

27075

59248

1425

644

7425

17689

6325

9747

550

189

13662

2527

297

266

12122

12211

(3.13)

Table 3.1 below represents the formulae for predictor and corrector for the two computed

points.

Table 3.1 The formulae for predictor and corrector method for two computed points

Step

size

ratio

Points Coefficients of the points

r = 1

1ny

Predictor nnn yyy 33 12

Corrector nnnnn yyyyhf5

9

5

3

10

1

10

3

5

61221

2ny


Corrector nnnnn yyyyhf25

36

25

16

25

3

25

48

25

121212

r = 2 1ny

Predictor nnn yyyy8

15

4

5

8

312

Corrector

nnnnn yyyyhf128

225

128

25

128

3

128

75

8

151221

24

2ny


Corrector

nnnnn yyyyhf23

18

23

3

115

2

115

192

23

121212

r = 5/9

1ny

Predictor nnn yyy25

133

25

171

25

6312

Corrector

nn

nnn

yy

yyhf

7425

17689

6325

9747

550

189

13662

2527

297

266

1

221

2ny

Predictor nnn yyy25

322

25

504

25

20712

Corrector

nn

nnn

yy

yyhf

35625

103684

11875

27216

225625

128547

27075

59248

1425

644

1

212

3.4 Implementation of the method

There are many compiler that can be used to implement the algorithm of numerical

methods. In this dissertation, we choose to use Microsoft Visual C++ 6.0 as platform to

implement the VS-BBDF method with Newton iteration algorithm. The program will be

written in C language.

3.4.1 Newton’s iteration

Further explanation on implementation of the method will be discussed in this section.

Newton’s iteration is applied to fourth order VS-BBDF method for better performance in

finding the approximation solutions of 1ny and 2ny . From the derivation above, the

general form of corrector formulae (3.11-3.13) can be written in matrix form as

25

2

2

1

1

2

1

2

1

2

2

1

1

1

2

2

1

21

0

0

0

01

0

0

1

nnnnnnnnn yyyhfhfyyyy

(3.14)

where 2121212121 ,,,,,,,,, are the coefficients of the points. So, matrix form

in (3.14) can be simplified in this way with representation 21, as the backvalues

nnn yyy ,, 12 .

2

1

2

1

2

1

2

1

1

2 0

0

1

1

n

n

nnf

fhyy (3.15)

Thus, in simpler way we have (3.15) as

hBFYAI )(

(3.16)

where

2

1

2

1

2

1

2

1

2

1,,

0

0,,

0

0,

10

01

n

n

n

n

f

fFB

y

yYAI

Let the system equivalent to 0, we have

0)(ˆ hBFYAIF

(3.17)

Newton’s iteration is performed to the system (3.17) and we get a new equation in the

form of Newton’s iteration

)(

)()(

)2,1(

)(

)2,1()(

)2,1(

)1(

)2,1( i

nn

i

nni

nn

i

nnYF

YFYY

(3.18)

26

To approximate the solution, apply Newton’s iteration (3.18) to the system (3.17). The

new equation for approximating the solutions contain the Jacobian matrix of F with

respect to y shows in (3.19) as below

)(

)2,1(

)(

)2,1(

)(

)2,1()(

)2,1(

)1(

)2,1(

)(

)(

i

nn

i

nn

i

nni

nn

i

nn

Y

FhBAI

hBFYYAIYY

(3.19)

Hence, equation (3.19) is used to approximate the solutions.

3.4.2 Choosing the step size

Apart from that, there are three strategies that very significant in choosing

appropriate step size at each step of iteration. Two possibilities of adjustment the step size

for every successful step and only one possibilities for every failure step. Therefore, the

possibilities are remain the step size (r = 1), decrease the step size to half (r = 2) and

increase the step size to a factor 1.8 (r = 5/9). For every successful step, the chosen step

size is either maintained or increased the step size to a factor 1.8 whereby for failure step

the chosen step size is halved the current step size.

Furthermore, the values of 1ny and 2ny are accepted if the current step is successful and

otherwise the values of 1ny and 2ny are rejected. Consequently, cases for the step are

determined by checking the local truncation error (LTE) either it is less than or greater

than tolerance limit (TOL). The user will provide the TOL on any given step. The selected

test problems are solved with error tolerances limit of 10-2, 10-4, 10-6 and 10-8.

27

The formula of determining the LTE is given as

i

n

i

n yyLTE 2

1

2

(3.20)

where 1

2

i

ny is (i+1)-th order method and i

ny 2is the i-th order method.

Here are the lists of LTE for the proposed method. The lists are shown in the table below.

Table 3.2 Local truncation error for the three step size ratios

Step size

ratio Local Truncation Error, LTE

r = 1 nnnnnnn yyyyhfyy275

171

275

126

25

3

275

78

275

181212

3

2

4

2

r = 2 nnnnnnn yyyyhfyy161

34

483

40

115

2

2415

352

161

81212

3

2

4

2

r = 5/9

nn

nnnnn

yy

yyhfyy

344375

583487

2410625

4370598

225625

128547

1832075

826298

13775

1058

1

212

3

2

4

2

Basically, there are two possibilities in choosing the appropriate step size.

Case 1 : Successful step (LTE < TOL)

In this case, accept the values of 1ny and 2ny . But two possibilities of choosing the

suitable step size if we need to maintain the current step size (r = 1) or increase the step

size by a factor of 1.8 (r = 5/9). So, for this case it is significant to ensure the previous

28

step that it is successful or not. Let say the previous step is failed, then current step size

should be maintained. Or else the step size should be increase by a factor of 1.8. The step

size increment is given by

p

oldnewLTE

TOLhch

1

and if oldnew hh 8.1 then oldnew hh 8.1

where c is safety factor and we set it to 0.8, p is the order of the method, oldh is previous

step size and newh is current step size.

Case 2 : Failure step (LTE > TOL)

In this case, reject the values of 1ny and 2ny . Thus, reiterate the current step by halving

the current step size (r = 2).

3.5 Stability of VS-BBDF method

A method to be of practical importance it must have a region of absolute stability to ensure

that the method will be able to solve at least for the mildly stiff problems (Majid &

Suleiman, 2006). Absolute stability can indicate the stability of numerical methods. The

stability region is the region enclosed by the set of points determined by replacing

20 ,cossin iet i in stability polynomial. In order to determine the

absolute stability, we have to apply VS-BBDF method formula to the test equation. The

stability region is determined by finding the region for which |t| < 1.

Definition 3.1 (Lambert, 1991)

A method is said to be absolute stable in a region R for a given h , all the roots sr of the

stability polynomial 0)()(:);( rhrhr , satisfy 1|| sr where ks ,...,2,1 .

29


A numerical method is said to be A stable if its region of absolute stability contains the

whole of the left-hand half-plane 0)Re( h .


The LMM is said to be zero stable of no root of the first characteristic polynomial )(rp

has modulus greater than one, and if every root with unit modulus is simple.

Apply equation (1.2) to the the test equation, 𝑦′ = 𝑦 and then obtain

k

j

k

j

jnijjnij yhy0 0

22 (3.21)

or

k

j

jnijij yH0

2 0 (3.22)

where 𝐻 = ℎ and i = 1 and 2.

The equation (3.22) can be simplify in other form as

k

j

jjYA0

,0 (3.23)

where

],...,[ 0 rj AAA equivalent to

)12(,2)12(,2)12(,2

)12(,1)12(,12,1

jjj

jjj

j h

hA

,

30

rj YYY ,...,0 equivalent to

jn

jn

j y

yY

21

22 .

The stability polynomial of the method is

0det);(0

r

j

j

j tAHtR (3.24)

Then, substitute H = 0 of equation (3.24) to obtain the roots for stability polynomial. The

stability polynomial for the three step size ratios are listed in the table below.

Table 3.3 Lists of stability polynomials and roots for the three step size ratios

Step size

ratio, r

Stability polynomial, );( HtR Roots

r = 1

HtHtHt

Httttt

2324

4234

125

18

125

252

125

72

25

42

125

1

25

9

125

153

125

197

1,

0.0207917599,

-0.2441420137

r = 2

HtHtHt

Httttt

2324

4234

92

3

736

1155

46

45

184

441

2944

1

2944

289

93

173

46

91

1,

0.00325762197,

-0.05270817143

31

r = 5/9 HtHtHtHt

tttt

23244

234

130625

66654

556875

1839404

22275

9016

141075

190106

653125

59049

653125

755829

10580625

3575389

22275

31291

1,

0.0769489074

-0.8363962010

In order to determine the stability region, solve t in stability polynomial as stated in the

table by replacing the value of H. As we have mention earlier, the method is stable if the

absolute value of t that has been solved is less than 1. Therefore, the region of stability of

VS-BBDF method is plotted using MAPLE 15. Hence, figure 3.1- 3.4 show the region of

stability for VS-BBDF method.

Figure 3.1 Stability region for r = 1

32

Figure 3.2 Stability region for r = 2

Figure 3.3 Stability region for r = 5/9

33

Figure 3.4 Stability regions for r = 1, r = 2 and r = 5/9

From the figures, the stability region for VS-BBDF method lies outside the closed region.

Since all the roots for the step size ratios have modulus less than or equal to 1, thus the

proposed method are satisfied zero stability. The figures also show that the absolute

stability region for step size ratios 1 and 5/9 are almost A-stable while the stability region

for step size ratio 2 is A-stable since it contains the entire left half-plane of the complex

plane, 0)Re( h .

34

CHAPTER 4

RESULTS AND DISCUSSIONS

4.1 Introduction

The results of this dissertation will be presented in this chapter. Fourth order VS-BBDF

method with Newton’s iteration is used to solve first order stiff IVP ODEs problem. There

are seven selected test problems with different values of eigenvalues, . The results that

are obtained will be compared with the results of step size ratios 1.6 and 1.9 which have

been solved by the previous researchers. The results for each problem are illustrated in

three figures. The first figure is the approximated solutions curve for variable step size

ratio. The second figure is the graph of approximate solution and exact solution while the

third figure is total steps curve for every test problems considered in this dissertation.

4.2 Test Problems

There are seven test problems in this section. The following test problems are stiff IVP

ODEs. The tested problems of stiff ODEs are solved using the fourth order VS-BBDF

method with Newton’s iteration. The numerical results for step size 1.8 will be compared

with step size 1.6 and 1.9 in term of maximum errors.

35

The formula for maximum error is defined as

MAXE =

errorniTSi

maxmax11

,

where TS is the total steps, n is the number of equations and

))((

)(

i

ii

xyBA

xyyerror

with A = 1, B = 1 for mixed error test.

Problem 4.1

212

211

2019

1920

yyy

yyy

Interval : 200 x

Exact solution : xx

xx

eexy

eexy

39

2

39

1

)(

)(

Initial conditions : 0)0(

2)0(

2

1

y

y

Eigenvalues : 39

1

2

1

Source : Cheney and Kincaid (1999)

36

Problem 4.2

)1(

10001002

2212

2

211

yyyy

yyy

Interval : 200 x

Exact solution : x

x

exy

exy

)(

)(

2

2

1


1)0(

2

1

y

y

Eigenvalue : 1002

Source : Kaps and Wanner (1981)

Problem 4.3

1)(20 xyy

Interval : 100 x

Exact solution : xexy x 20)(

Initial conditions : 1)0( y

Eigenvalues : 20

Source : Artificial problem

37

Problem 4.4

23 3)(100 xxyy

Interval : 100 x

Exact solution : 3)( xxy


Eigenvalues : 100

Source : Gear (1971)

Problem 4.5

212

211

1999999

1998998

yyy

yyy

Interval : 200 x

Exact solution : xx

xx

eexy

eexy

1000

2

1000

1

)(

2)(


1)0(

2

1

y

y

Eigenvalues : 1000

1

2

1


38

Problem 4.6

1)(100 xyy

Interval : 100 x

Exact solution : xexy x 100

1 )(


Eigenvalues : 100


Problem 4.7

212

211

19971197

19951195

yyy

yyy

Interval : 200 x

Exact solution : xx

xx

eexy

eexy

8002

2

8002

1

86)(

810)(


2)0(

2

1

y

y

Eigenvalues : 800

2

2

1

Source : Gerald and Wheatley (1989)

39

4.3 Numerical Results

In this section, we discussed the results for tested problem of first-order stiff IVP ODEs given

in the previous section. In this dissertation, we considered constant step size (r = 1), half the

step size (r = 2) and increment the step size to a factor 1.8 (r = 5/9). The results obtained are

then compared with the results of step size 1.6 and 1.9. Numerical results are presented in

the figures 4.1 until 4.21 where each problem have three different figure. Basically, the first

figure represented the approximated solutions for variable step size ratio, the second figure

is the approximate solutions and exact solutions for every tested problems and the third figure

is the total steps curve for each problem. Then, we make conclusions based on the results

obtained in those figures.

Figures below represent the results for every tested problems considered in this dissertation.

4.3.1 Results and Discussions for Test Problem 4.1

Figure 4.1 Approximated solutions curves for variable step size ratios


-25

-20

-15

-10

-5

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

40

Figure 4.2 Graph of approximate solution and exact solution


Figure 4.3 Total steps curves for Test Problem 4.1

-16

-14

-12

-10

-8

-6

-4

-2

0

10e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

50

100

150

200

250

300

350

400

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

41

The three figures represent the results for the test problem 4.1. The first figure represents an

approximated solutions curves for variable step size ratios. As we can observe clearly from

the figure, the maximum errors of step size 1.8 were reducing as the value of tolerances were

decreasing. For the second figure, it can be seen that the approximate solution of step size

1.8 was lying almost in the same line with the exact solution. Besides, the third figure

represents as the total steps taken during the computation of the solution. Therefore, it was

obviously seen that the total steps taken were increasing as the value of tolerances were

decreasing.




-20

-15

-10

-5

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

42




-7

-6

-5

-4

-3

-2

-1

0

10e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

50

100

150

200

250

300

350

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

43

The three figures represent the results for the test problem 4.2. The first figure shows an

approximated solutions curves for variable step size ratios. As we can observe clearly from

the figure, the maximum errors of step size 1.8 were also reducing as the value of tolerances

were decreasing. Next, for second figure, it can also be seen that the approximate solution of

step size 1.8 was lying almost in the same line with the exact solution. Then, it was obviously

seen that the total steps taken were also increasing as the value of tolerances were decreasing.




-25

-20

-15

-10

-5

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

44




-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

10e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

50

100

150

200

250

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

45

The three figures represent the results for the test problem 4.3. The first figure represents

an approximated solutions curves for variable step size ratios. Similarly to those

problems, the observation showed that the maximum errors of step size 1.8 were also

reducing when the tolerances value were decreasing. The approximate solution of step

size 1.8 was also located almost in the same line with the exact solution for the second

figure. Then, the third figure also shows that the total steps taken were increasing as the

value of tolerances were decreasing.




-45

-40

-35

-30

-25

-20

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

46




-40

-35

-30

-25

-20

-15

-10

-5

0

10e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

10

20

30

40

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

47

The three figures represent the results for the test problem 4.4. So, the results was the

same as previous results where the maximum errors of step size 1.8 were reducing as the

value of tolerances were decreasing. The second figure also represents the approximate

solution of step size 1.8 and the exact solution. Then, the results also showed the

approximate solution of step size 1.8 was lying almost in the same line with the exact

solution. The third figure represents the total steps taken during the computation.

Consequently, the figure showed that the total steps taken were also increasing as the

value of tolerances were decreasing.




-25

-20

-15

-10

-5

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

48




-2.5

-2

-1.5

-1

-0.5

0

10e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

50

100

150

200

250

300

350

400

450

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

49

The three figures represent the results for the test problem 4.5. We can observe the

maximum errors of step size 1.8 were also decreasing as the value of tolerances were

decreasing. Besides, we can also see on the second figure where the approximate solution

of step size 1.8 was also lying almost in the same line with the exact solution. Results for

the third figure was similar to the previous results at which the total steps taken during

the computation were also increasing as the value of tolerances were decreasing.




-25

-20

-15

-10

-5

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

50




-12

-10

-8

-6

-4

-2

0

10e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

50

100

150

200

250

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

51

The three figures represent the results for the test problem 4.6. So, the results was similar

to the previous results where the maximum errors of step size 1.8 were reducing as the

value of tolerances were decreasing. The second figure was also similar to the previous

results whereby the approximate solution of step size 1.8 was lying almost in the same

line with the exact solution. Lastly, the third figure were also increasing as the value of

tolerances were decreasing.




-25

-20

-15

-10

-5

10e-2 10e-4 10e-6 10e-8

log|

Max

imu

m E

rro

r|

TOL

r = 1.6

r = 1.8

r = 1.9

52

Figure 4.20 Graph of approximated solutions for variable step size ratios



-12

-10

-8

-6

-4

-2

0

2

410e-2 10e-4 10e-6 10e-8

log|

Solu

tio

n|

TOL

Exact solution

Approximatesolution

0

100

200

300

400

500

600

10e-2 10e-4 10e-6 10e-8

T

o

t

a

l

s

t

e

p

s

TOL

r = 1.6

r = 1.8

r = 1.9

53

The three figures represent the results for the test problem 4.7. The results for all the

figures were also similar to those results of previous tested problems where the results for

first figure was the maximum errors of step size 1.8 were also reducing as the value of

tolerances were decreasing. Then, the second figure also shows the approximate solution

of step size 1.8 was lying almost in the same line with the exact solution and last but not

least, the results for the third figure were also increasing as the value of tolerances were

decreasing.

4.4 Discussions

Generally, the first figure for each problem represented the approximate solutions for

every tested problems at three different values of step sizes (1.6, 1.8 and 1.9) in term of

maximum errors. From the figure, he maximum errors were also reduced when the value

of tolerances were reduced. Therefore, it can be seen that the performance was getting

better when step size ratio was increased. In addition, the second figure showed the results

for approximate solutions and exact solutions of all the selected tested problems. As we

can observed from the figures, it was clearly shown that the approximate solutions were

almost on the same line with the actual solutions in those figures. Subsequently, the

approximate solutions were converging to the exact solutions. Then, the third figures

represented all the total steps that were involved in computing the solutions. The total

steps were increasing as the value of tolerances were decreasing. As we can also see in

the same figure, mostly the step size 1.9 gave the least total steps for tolerances 10-2, 10-

4 and 10-6. However, mostly the step size 1.6 gave the least total steps for tolerance 10-8.

As a conclusion, we can conclude that the step size 1.9 gave the best results for all those

selected problems considered in this dissertation. However, the step size 1.8 was also

reliable to solve those selected problems of stiff ODEs based on the accuracy obtained

54

from the results. So, this showed that the proposed methods used were relevant for solving

stiff ODEs. Therefore, in conclusion, increasing the value of step size ratio will give more

accurate results.

57

CHAPTER 5

SUMMARY

5.1 Conclusions

The aim of this dissertation is to derive the fourth order Variable Step Block Backward

Differentiation Formula method. The aim is accomplished by applying the derived

method to the first order stiff ODE based on variable step size approach. In this

dissertation, we derived the method by using step size 1.8 where the proposed method

inspired from the fourth order Variable Step Block Backward Differentiation Formula of

step sizes 1.6 and 1.9. On the whole dissertation, the main objective is to investigate the

effect of solutions by changing the values of step size ratio, r where all the step size ratios

are satisfying zero stability.

There are seven selected tested problems that have been solved in this dissertation. Each

of the problems are tested at variable step sizes depending on the LTE. The numerical

results showed that the effectiveness of step size ratio 1.8 is relevant to solve the restricted

problems studied as the errors produced are within the tolerance given at each step of the

iteration. Hence, it also gave an accurate solutions. The conclusions made are restricted

only to the problems studied based on the results and discussions earlier. As the step size

ratio increased, the results produced better approximated solutions as the error goes to

zero and the solutions converged to the exact values. Therefore, by increasing the step

size ratio will

58

give better performance in terms of maximum error and reduction of total steps involved

at each iteration for all tested problems.

5.2 Future Study

A few recommendations and outlines stated for future study in this area as follows:

1. The first order stiff ODE has been solved in this dissertation. Therefore, it would

be interesting to solve higher order of stiff problems.

2. For next study, it would be exciting for solving first order stiff ODE by increasing

the number of points in the block for further improvement in performance of the

method by the aim of reducing the total number of steps.

3. For future research, it would be challenging to investigate the efficiency of the

method in terms of computation time.

56

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APPENDIX

Appendix A : Algorithm for VS-BBDF method of order four

Start

Input: Starting point a,

End point b,

Step size h,

Step size ratio r,

Initial value of X ]0[X ,

Point = 2,

Tolerance TOL,

IND2 = 1.

Step 1 : Compute backvalues points, ny , 1ny and 2ny (Euler method).

Step 2 : Compute predictor values for points 1ny and 2ny (Predictor method).

Step 3 : Compute Jacobian matrix

Step 4 : Compute Newton iteration matrix

Step 5 : Compute Error

Estimate the error (LTE)

Do Step 6 (Test for convergence) : Check

If LTE < =TOL & convergence

Accept values for 1ny and 2ny .

Step = Step + 1,

Compute maximum error, MAXE,

If IND2 = 0,

h = hold,

r = 1,

If IND2 = 1,

Calculate hnew for new step size

p

oldLTE

TOLhc

1

hacc

and if then

If oldh 8.1h acc then

oldhh 8.1

r = 5/9,

else

Reject values for 1ny and 2ny .

Failure step = Failure step + 1,

oldhh 2

1,

r = 2,

Step 7:

Output ),( 21 nn yy ,

Total steps,

Maximum error.

modified extended block backward differentiation formula … · modified extended block backward...

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