solution of fractional-order differential equations based on the … · 2017. 2. 22. · (2007)...

$: Solution of fractional-order differential equations based on the … · 2017. 2. 22. · (2007) solved the differential equations by using Galerkin method based on the Bernstein polynomial$
Journal of King Saud University – Science (2017) 29, 1–18

King Saud University

Journal of King Saud University –

Sciencewww.ksu.edu.sa

www.sciencedirect.com

REVIEW ARTICLE

Solution of fractional-order differential equations

based on the operational matrices of new fractional

Bernstein functions

* Corresponding author.

E-mail address: [email protected] (M.H.T. Alshbool).

Peer review under responsibility of King Saud University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.jksus.2015.11.0041018-3647 � 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

M.H.T. Alshboola,*, A.S. Bataineh

b, I. Hashim

a, Osman Rasit Isik

c

aSchool of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, MalaysiabDepartment of Mathematics, Faculty of Science, Al Balqa Applied University, 19117 Salt, JordancDepartment of Mathematics, Faculty of Sciences, Mugla Sitki Kocman University, 48000 Mugla, Turkey

Received 20 August 2015; accepted 12 November 2015Available online 7 December 2015

KEYWORDS

Bernstein polynomials;

Fractional differential equa-

tion;

Error analysis

Abstract An algorithm for approximating solutions to fractional differential equations (FDEs) in

a modified new Bernstein polynomial basis is introduced. Writing x ! xa ð0 < a < 1Þ in the oper-

ational matrices of Bernstein polynomials, the fractional Bernstein polynomials are obtained and

then transformed into matrix form. Furthermore, using Caputo fractional derivative, the matrix

form of the fractional derivative is constructed for the fractional Bernstein matrices. We convert

each term of the problem to the matrix form by means of fractional Bernstein matrices. A basic

matrix equation which corresponds to a system of fractional equations is utilized, and a new system

of nonlinear algebraic equations is obtained. The method is given with some priori error estimate.

By using the residual correction procedure, the absolute error can be estimated. Illustrative exam-

ples are included to demonstrate the validity and applicability of the presented technique.� 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is

an open access article under the CCBY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Preliminaries and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3. Description of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. Error analysis and estimation of the absolute error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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2 M.H.T. Alshbool et al.

5. Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1. Introduction

Fractional differential equations have several applications inmathematical physics, fluid flow, engineering and other areas

of applications (Miller and Ross, 1993; Podlubny, 1999;Jafari and Momani, 2007; Daftardar and Jafari, 2007;

Figure 1 The absolute error, the estimated absolute error and the co

a ¼ 0:75.

Abdulaziz et al., 2008). In this paper, we consider the frac-tional differential equations (FDEs) of the form:

DayðxÞ ¼ gðxÞ þ qðxÞyðxÞ þ zðxÞðyðxÞÞr; 0 6 x 6 1: ð1Þunder the conditions

yð0Þ ¼ b1; yð1Þ ¼ b2: ð2Þ

rrected absolute error to Example 1, for the case n ¼ 3;m ¼ 9 and

Solution of fractional-order differential equations 3

Here, y(x) is an unknown function; g(x), q(x) and z(x) are the

functions that are defined in [0, 1].Numerical and analytical methods are applied to solve frac-

tional differential equation such as homotopy perturbation

method (Hosseinnia et al., 2008), Modification of homotopyperturbation methods (Odibat and Momani, 2008), Taylor col-location method (Cenesiz et al., 2010), Jacobi operationalmatrix method (Kazem, 2013), Variational iteration method

(Odibat and Momani, 2006), Legendre functions method(Kazem et al., 2013), Tau methods (Rad et al., 2014), A newoperational matrix method (Saadatmandi and Dehghan,


a ¼ 0:90.

2010). Yang et al. (2015) presented local fractional derivativeoperators and local integral transforms to solve fractionaldifferential equations. Rostamya et al. (2013) proposed anew

efficient basis to solve fractional partial differential equations.Fractional Riccati differential equations have been solved

by many efficient techniques such as Adomian’s decomposi-

tion method (Gejji and Jafari, 2007; Momani andShawagfeh, 2006) and new spectral wavelets methods (Abd-Elhameed and Youssri, 2014).

One of the most important numerical methods for solvinglinear and nonlinear equations is Bernstein polynomials which



have been used by many authors. Pandey and Kumar (2012)used Bernstein operational matrix of differentiation to solveLane–Emden type equation. Besides that Bhatti and Bracken

(2007) solved the differential equations by using Galerkinmethod based on the Bernstein polynomial basis. Yousefiand Behroozifar (2010) presented an operational matrix

method based on Bernstein polynomials for the differentialequations. Recently, Yiming et al. (2014) solved a variableorder time fractional diffusion equation using Bernstein poly-

nomials. Alshbool and Hashim (2016) presented multistageBernstein polynomials which is a new modification of


a ¼ 0:75.

Bernstein polynomials to find the solutions of fractional orderstiff systems.

In the present paper, we utilize a new operational matrices

method based on the Bernstein polynomials to solve linear andnon-linear fractional differential equations. We generalize r forr 2 N0, also the absolute error is estimated and corrected, as

well as the addition of the roots of Chebyshev polynomials isused to reduce the interpolation error. The suggested methodshows that this approach can solve the problem effectively.

This article is structured as follows. In Section 2, the defini-tions and properties of the fractional calculus are given. In



Section 3, our method and its applications are explained. InSection 4, an error analysis of the method and estimation ofthe error are presented, as well as the corrected absolute error.

In Section 5, our numerical findings, exact solution anddemonstration of the validity, accuracy and applicability ofthe operational matrices by considering numerical examples

are reported. Section 6, consists of brief summary andconclusion.

2. Preliminaries and notations

In this section, we provide some definitions and properties ofthe fractional calculus (Diethelm et al., 2005; Podlubny, 1999).


a ¼ 0:90.

Definition 2.1. A real function fðxÞ; x > 0, is said to be in the

space Cl; l 2 R, if there exists a real number p > l, such that

fðxÞ ¼ xpf1ðxÞ, where f1ðxÞ 2 Cð0;1Þ, and it is said to be in

the space Cnl if and only if fðnÞ 2 Cl, n 2 N.

Definition 2.2. The Riemann–Liouville fractional integraloperator ðJaÞ of order a P 0, of a function f 2 Cl; l P �1,

is defined as

JafðxÞ ¼ 1

CðaÞZ x

0

x� sð Þa�1fðxÞds ða > 0Þ;

J0fðxÞ ¼ fðxÞ; ð3Þ


Figure 5 The absolute error, the estimated absolute error and the corrected absolute error to Example 1, for the case n ¼ 5;m ¼ 9 and

a ¼ 1.


where CðaÞ is well-known gamma function. Some of theproperties of the operator Ja, which we will need here, are asfollows: For f 2 Cl; l P �1; a; b P 0 and c P �1:

1. J aJbf ðxÞ ¼ J aþbf ðxÞ,2. J aJbf ðxÞ ¼ J bJ af ðxÞ,3. J axc ¼ Cðcþ1Þ

Cðaþcþ1Þ xaþc.

Definition 2.3. The fractional derivative ðDaÞ of fðxÞ, in theCaputo sense is defined as

DafðxÞ ¼ 1

Cðn� aÞZ t

0

x� sð Þn�a�1fðnÞðxÞds; ð4Þ

for n� 1 < a < n; n 2 N; x > 0; f 2 Cn�1.

The following are two basic properties of the Caputo frac-tional derivative (Miller and Ross, 1993):

1. Let f 2 Cn�1; n 2 N . Then Daf ; 0 6 a 6 n is well defined and

Daf 2 C�1.2. Let n� 1 6 a 6 n; n 2 N and f 2 Cn

l; l P �1. Then

ðJaDaÞfðxÞ ¼ fðxÞ �Xn�1

k¼0

fðkÞð0þÞxk

k!: ð5Þ

For the Caputo derivative we have

Figure 6 Compare between our present solution with terms 9 and Exact solution with 1000 terms.


Da�c ¼ 0; ðc constantÞ; ð6Þ

Da�x

b ¼0; for b2N0 and b< dae;Cðbþ1Þ

Cðbþ1�aÞxb�a; for b2N0 and bP dae or b> bac:

(

ð7ÞWe note that the approximate solutions will be found by usingthe Caputo fractional derivative and its properties in thisstudy.

3. Description of the method

To solve (1), (2) by developing the Bernstein polynomial

approximation with the help of the matrix operations, the col-location method and the caputo fractional derivative are used.

We obtain an approximate solution of the problem (1), (2) inthe form of

yn;aðxÞ ¼Xnk¼0

ckBan;kðx� cÞ: ð8Þ

Here 0 < alpha < 1; ck; k ¼ 0; 1; 2; . . . ; n are the unknown

Bernstein coefficients, n is chosen for any positive integers,and Ba

k;nðxÞ are obtained by putting x ! xa in Bernstein poly-

nomials as defined by

Bk;nðxÞ ¼n

k

� �xkðR� xÞn�k

Rn ; k ¼ 0; 1; 2; . . . ; n x 2 ½0;R�;

i ¼ 0; 1; . . . ; n

it becomes

Figure 7 Compare between a ¼ 0:75; a ¼ 0:90 and a ¼ 1 to Examples 1, for the case n ¼ 3, and n ¼ 5.


Bak;nðxÞ ¼

n

k

� �xkaðR� xaÞn�k

Rn ; k ¼ 0; 1; 2; . . . ; n x 2 ½0;R�;

i ¼ 0; 1; . . . ; n ð9Þso that

yn;aðxÞ ¼Xnk¼0

ckBan;kðx� cÞ ¼ CT/�ðxÞ: ð10Þ

where

/�ðxÞ ¼ ½Ba0;nðxÞ;Ba

1;nðxÞ; . . . ;Ban;nðxÞ�T; CT

¼ ½c0; c1; . . . ; cn�: ð11ÞHere, the expression (/�ðxÞ) in (11) can be written as

/�ðxÞ ¼ AX ð12Þ

Where

A ¼

a00 a01 . . . a0n

a10 a11 . . . a1n

..

. ...

. . . ...

an0 an1 . . . ann

0BBBB@

1CCCCA; X ¼

1

xa

x2a

..

.

xna

0BBBBBB@

1CCCCCCA: ð13Þ

and

aij ¼ð�1Þj�i

R j

n

i

� �n� i

j� i

� �; i 6 j

0; i > j

8<:

So that

A�1/�ðxÞ ¼ X: ð14Þ

By using the Caputo fractional derivative (7) and applying it

on Eq. (12), the ath order fractional derivative of /�ðxÞ canbe written as

Da/�ðxÞ ¼ Ada

dxa X: ð15Þ

Da/�ðxÞ ¼ AX�: ð16Þwhere

X� ¼

0

Cðaþ 1ÞCð2aþ1ÞCðaþ1Þ x

a

..

.

Cðnaþ1ÞCððn�1Þaþ1Þ x

ðn�1Þa

0BBBBBBBB@

1CCCCCCCCA:

We can write (16) as

Da/�ðxÞ ¼ AXX: ð17ÞWhere

X ¼

0 0 0 0

0 Cðaþ1ÞCð1Þ 0 0

0 0 Cð2aþ1ÞCðaþ1Þ 0

..

. ... ..

. ...

0 0 0 Cðnaþ1ÞCððn�1Þaþ1Þ

0BBBBBBBB@

1CCCCCCCCA; X ¼

0

1

xa

..

.

xðn�1Þa

0BBBBBB@

1CCCCCCA:

ð18ÞSimilarly, the Da fractional derivative of /ðxÞ is given by therecurrence relation

Da/�ðxÞ ¼ AXWX: ð19Þ

Figure 8 The error function, the absolute of error function, the corrected error function and the absolute corrected error function to

Example 2, for the case n ¼ 3;m ¼ 9 and a ¼ 0:75.


Where

W ¼

0 0 0 0

1 0 0 0

0 1 0 0

..

. ... ..

. ...

0 0 1 0

0BBBBBB@

1CCCCCCA

We place relation (14) into Eq. (19) and then we get

Da/�ðxÞ ¼ AXWA�1/�ðxÞ: ð20ÞFinally, we obtain Da as

Da ¼ AXWA�1: ð21ÞNote that every matrix is in the dimension of ðnþ 1Þ � ðnþ 1Þ.

Substituting (Da) in Eq. (10) we obtain

DayðxÞ � CTDa/�ðxÞ: ð22ÞTo obtain the solution of (1), the methods that are applied are:

Firstly, Eq. (10) is applied to approximate ðyn;aðxÞÞr and

gðxÞ asðyn;aðxÞÞr ¼ ðCT/�ðxÞÞr ð23Þ

gðxÞ � GT/�ðxÞ; ð24Þ




where the vector GT ¼ ½g0ðxÞ; g1ðxÞ; . . . ; gnðxÞ�T, represents thenon-homogenous term. By substituting the Eqs. (22)–(24), inEq. (1) we obtain

CTDa/�ðxÞ ¼ GT/�ðxÞ þ qðxÞCT/�ðxÞþ zðxÞ CT/�ðxÞ� �r

; ð25Þ

it can be written as

CTDa/�ðxÞ � qðxÞCT/�ðxÞ � zðxÞ CT/�ðxÞ� �r � GT/�ðxÞ ¼ 0;

ð26ÞThe interpolation error may be reduced by using the roots ofChebyshev polynomials

xi ¼ 1

2þ 1

2 cosðð2iþ 1Þ p2nÞ ; i ¼ 0; 1; . . . ; n� 1; ð27Þ

By substituting these roots in Eq. (26), we obtain n� 1 ofequations where the unknowns are ci and each equation equalszero. Nonlinear equations can be solved by using Newton’s

iteration method. Consequently y(x) which is given in Eq.(10) can be calculated, and this method (collocation method)is used to avoid the difficulty of integration.

4. Error analysis and estimation of the absolute error

In this section,we present the error analysis of the method

used. Residual correction procedure which may estimate theabsolute error will be assigned for the problem.




Let yn;aðxÞ and y(x) be the approximate solution and the

exact solution of (1), respectively. The following procedure,residual correction can be assigned to the estimation of the

absolute error (Celik, 2006).First, adding and subtracting the term

R :¼ Dayn;aðxÞ þ qðxÞyn;aðxÞ þ zðxÞðyn;aðxÞÞr þ gðxÞ; ð28Þto (1) yield the following differential equation

Daen;aðxÞ þ qðxÞen;aðxÞ þ zðxÞðen;aðxÞÞr ¼ gðxÞ � R; ð29Þ

with the conditions

eð0Þ ¼ 0; eð1Þ ¼ 0; or e0ð0Þ ¼ 0; eð0Þ ¼ 0: ð30Þ

where enðxÞ ¼ yðxÞ � yn;aðxÞ. For a given value m let e�mðxÞ bethe approximate solution of (29), where m is a polynomialsdegree of (29) and m P n.

Corollary 1. Let yn;aðxÞ be the approximate solution of (1) and

e�mðxÞ is the approximate solution of (29). Then yn;aðxÞ þ e�mðxÞ




is also an approximate solution of (1) and its error functions isen;aðxÞ � e�mðxÞ.

We term the approximate solution yn;aðxÞ þ e�mðxÞ as the

corrected approximate solution. Note that if kenðxÞ � e�mðxÞk< �, then the absolute error can be estimated by e�mðxÞ.Moreover, if ken;aðxÞ � e�mðxÞk < kyðxÞ � yn;aðxÞk, then

yn;aðxÞ þ e�mðxÞ is a more accurate solution than yn;aðxÞ in any

given norm.

5. Illustrative examples

To illustrate the effectiveness of the presented method, the fol-

lowing examples of linear and non-linear fractional differentialequations (FDEs) are provided.

Example 1. Consider the fractional differential equation whichhas been considered by Saadatmandi and Dehghan (2010).

DayðxÞ þ yðxÞ ¼ 0; yð0Þ ¼ 1; 0 < a < 1: ð31Þthe exact solution of this problem is

yðxÞ ¼X1k¼0

ð�xaÞkCðakþ 1Þ :

In this problem qðxÞ ¼ �1; zðxÞ ¼ 0; gðxÞ ¼ 0 and r ¼ 1, weapproximate the solution as

yn;aðxÞ ¼Xnk¼0


By applying the method which is developed in Section 3 withn ¼ 3 and n ¼ 5 for a ¼ 0:75; a ¼ 0:9 and a ¼ 1, a good result


Example 2, for the case n ¼ 5;m ¼ 9 and a ¼ 1.






with a tiny error is obtained. We use the method in Section 4 toestimate the error and correct our approximate solution. Theabsolute errors and their estimations obtained from residual

correction procedure for m ¼ 9 are plotted in Figs. 1–6. Inaddition, the corrected approximate solutions yn;aðxÞ þ e�mðxÞare represented in the same figures. It is noticed from thefigures that residual correction procedure works well and the

corrected approximate solutions are better than the approxi-mate solutions. Note that when a ¼ 1, the exact solution is

given as yðxÞ ¼ expð�xÞ. Comparisons betweena ¼ 0:75; a ¼ 0:90 and a ¼ 1 to Examples 1, for the casen ¼ 3, and n ¼ 5 are plotted in Fig. 7.

Example 2. Consider the nonlinear fractional differentialequation which has been considered by Yuzbası (2013).

dayðxÞdxa þ y2ðxÞ ¼ 1; yð0Þ ¼ 0; 0 < a < 1: ð33Þ




In this problem qðxÞ ¼ 0; zðxÞ ¼ �1; gðxÞ ¼ 1 and r ¼ 2, by

applying the method developed in Section 3 with n ¼ 3 andn ¼ 5 for a ¼ 0:75; a ¼ 0:9 and a ¼ 1, good result is obtainedwith a slight error. The method is used in Section 4 to estimatethe error and correct our approximate solution. In this

example there is no exact solution, so we find error functionand the corrected error function by residual correction proce-dure for m ¼ 9, and all the results are plotted in Figs. 8–12.

Comparisons between a ¼ 0:75; a ¼ 0:90 and a ¼ 1 to Exam-ples 1, for the case n ¼ 3, and n ¼ 5 are plotted in Fig. 13.

Example 3. Consider the nonlinear fractional differential

equation which has been considered by Yuzbası (2013).

dayðxÞdxa ¼ 2yðxÞ � y2ðxÞ þ 1; yð0Þ ¼ 0; 0 < a < 1: ð34Þ

The exact solution for the problem for a ¼ 1 is given by

yðxÞ ¼ 1þffiffiffi2

ptanh

ffiffiffiffiffiffi2x

pþ 1

2log

ffiffiffi2

p � 1ffiffiffi2

p þ 1

! !: ð35Þ




In this problem qðxÞ ¼ 2; zðxÞ ¼ 1; gðxÞ ¼ 1 and r ¼ 2, weapproximate the solution as

yn;aðxÞ ¼Xnk¼0


We present the solution with n ¼ 3 and n ¼ 5 for a ¼ 0:9 and

a ¼ 1. We use the method in Section 4 to estimate the errorand correct our approximate solution. In this example thereis no exact solution, so we find the error function and the cor-

rected error function by residual correction procedure form ¼ 9. All the results are plotted in Figs. 14–17. Comparisonsbetween a ¼ 0:75; a ¼ 0:90 and a ¼ 1 to Examples 1, for the

case n ¼ 3, and n ¼ 5 are plotted in Fig. 18.

6. Conclusions

In this paper, the Bernstein operational matrix of derivativewas applied to solve linear and non-linear fractional differen-tial equations. Different from other numerical techniques,

the few Da matrices are obtained. These matrices are used toapproximate the numerical solution of fractional differentialequations. It can be clearly noticed that the proposed method

performs well even on a few number of terms of the Bernsteinpolynomials. The method is presented with some error analy-sis, and residual correction procedure is extended for this prob-

lem. The subjects of our future works can be exemplified byapplying the presented technique for solving system of frac-


tional differential equations and fractional partial differentialequations.

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