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General Letters in Mathematics Vol. 3, No. 2, Oct 2017, pp.102-111 e-ISSN 2519-9277, p-ISSN 2519-9269 Available online at http:// www.refaad.com Numerical Solution for Solving Fractional Differential Equations using Shifted Chebyshev Wavelet Mohamed Elarabi Benattia *1 and Belghaba Kacem 2 1,2 Laboratory of Mathematics and its Applications (LAMAP) University of Oran 1, Ahmed Ben Bella 1 [email protected] 2 [email protected] Abstract. In this paper, we are interested to develop a numerical method based on the Chebyshev wavelets for solving fractional order differential equations (FDEs). As a result of the presentation of Chebyshev wavelets, we highlight the operational matrix of the fractional order derivative through wavelet-polynomial matrix transformation which was utilized together with spectral and collocation methods to reduce the linear FDEs, to a system of algebraic equations. This method is a more simple technique of obtaining the operational matrix with straight forward applicability to the FDEs . The main characteristic behind the approach using this technique is that only a small number of shifted Chebyshev polynomials is needed to obtain a satisfactory results. Illustrative examples reveal that the present method is very effective and convenient for linear FDEs. Keywords: Operational matrix, shifted Chebyshev wavelet, fractional derivatives, shifted Chebyshev polynomials, Caputo fractional derivative. MSC2010 34A08. 1 Introduction Over the last decades several analytical or approximate methods have been developed to solve fractional differential equations in many scientific areas. A particular attention has been given to solution studies of fractional ordinary differential equations , integral equations and fractional partial differential equations. Approximations and numerical methods are extensively used for the fractional differential equations do not have exact analytic solutions. The fractional derivative has generated mathematics tools perfectly adapted to different scientific fields. The fractional derivatives express very often the modeled and simulated properties (viscoelasticity for example) of different research domains such as fluid dynamics, heat and mass transfer, elasticity, etc[10, 13, 1, 16]. . The operational matrices of fractional order integration for Haar wavelets, Legendre wavelets, Chebyshev wavelets and Bernoulli wavelets have been developed in [2, 4, 5, 6, 15] respectively, to solve FDEs. A variant of theses methods, using shifted Legendre polynomial has been studied by [3]. It is natural to try to relate this work using described with shifted Chebyshev polynomials. Another motivation is the direct solution techniques for solving the derivatives forms of FDEs using shifted Chebyshev Tau method based on operational matrix of fractional derivative. The outline of this paper is organized as follows. In section 1, we describe some basic definition and properties of fractional calculus. In section 2, we first defined shifted Chebyshev polynomial then, we describe some properties of Chebyshev wavelets. In Section 3, the shifted Chebyshev operational matrix of fractional order derivative and * Corresponding author. Mohamed Elarabi Benattia 1 [email protected]
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Page 1: Numerical Solution for Solving Fractional Di erential ... · Numerical Solution for Solving Fractional Di erential Equations 103 Chebyshev wavelet operational matrix of fractional

General Letters in Mathematics Vol. 3, No. 2, Oct 2017, pp.102-111

e-ISSN 2519-9277, p-ISSN 2519-9269

Available online at http:// www.refaad.com

Numerical Solution for Solving Fractional Differential

Equations using Shifted Chebyshev Wavelet

Mohamed Elarabi Benattia ∗1 and Belghaba Kacem 2

1,2 Laboratory of Mathematics and its Applications (LAMAP)

University of Oran 1, Ahmed Ben Bella1 [email protected] 2 [email protected]

Abstract. In this paper, we are interested to develop a numerical method based on the Chebyshev wavelets for solving

fractional order differential equations (FDEs). As a result of the presentation of Chebyshev wavelets, we highlight the

operational matrix of the fractional order derivative through wavelet-polynomial matrix transformation which was utilized

together with spectral and collocation methods to reduce the linear FDEs, to a system of algebraic equations. This method

is a more simple technique of obtaining the operational matrix with straight forward applicability to the FDEs . The main

characteristic behind the approach using this technique is that only a small number of shifted Chebyshev polynomials is needed

to obtain a satisfactory results. Illustrative examples reveal that the present method is very effective and convenient for linear

FDEs.

Keywords: Operational matrix, shifted Chebyshev wavelet, fractional derivatives, shifted Chebyshev polynomials,Caputo fractional derivative.

MSC2010 34A08.

1 Introduction

Over the last decades several analytical or approximate methods have been developed to solve fractional differentialequations in many scientific areas. A particular attention has been given to solution studies of fractional ordinarydifferential equations , integral equations and fractional partial differential equations. Approximations and numericalmethods are extensively used for the fractional differential equations do not have exact analytic solutions. Thefractional derivative has generated mathematics tools perfectly adapted to different scientific fields. The fractionalderivatives express very often the modeled and simulated properties (viscoelasticity for example) of different researchdomains such as fluid dynamics, heat and mass transfer, elasticity, etc[10, 13, 1, 16].

. The operational matrices of fractional order integration for Haar wavelets, Legendre wavelets, Chebyshevwavelets and Bernoulli wavelets have been developed in [2, 4, 5, 6, 15] respectively, to solve FDEs. A variantof theses methods, using shifted Legendre polynomial has been studied by [3]. It is natural to try to relate thiswork using described with shifted Chebyshev polynomials. Another motivation is the direct solution techniques forsolving the derivatives forms of FDEs using shifted Chebyshev Tau method based on operational matrix of fractionalderivative.

The outline of this paper is organized as follows. In section 1, we describe some basic definition and propertiesof fractional calculus. In section 2, we first defined shifted Chebyshev polynomial then, we describe some propertiesof Chebyshev wavelets. In Section 3, the shifted Chebyshev operational matrix of fractional order derivative and

∗Corresponding author. Mohamed Elarabi Benattia 1 [email protected]

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Numerical Solution for Solving Fractional Differential Equations 103

Chebyshev wavelet operational matrix of fractional order derivative, in section 4 the proposed method is used toapproximate solution of the problem. In section 5, application of the Chebyshev wavelets operational matrix offractional order derivative. In section 6, the numerical examples of applying the method of this article are presented.Finally in section 7, we achieve our work by some commentaries.

2 Preliminaries

2.1 Fractional Derivative and Integral

Here, we recall some basic definitions and properties The Riemann - Liouville integral I of fractional order α of f (t)is given by

Iαf (t) =1

Γ (α)

t∫0

(t− s)α−1f(s)ds, t > 0, α > 0, (1)

its fractional derivative of order α > 0 is given by

(Dαl f)(x) = (

d

dx)m(Im−αf)(x), (α > 0, m− 1 < α < m),

where Γ (:) is the gamma function which achieve the following properties:{Γ(n+ 1) = n!

Γ(n+ 12 ) = (2n)!

22nn!

√π.

, ∀n ∈ N.

Some proprieties of Iα are as follows:

IαIβf(t) = Iα+βf(t), α > 0, β > 0. (2)

Iαtβ =Γ(β + 1)

Γ(β + α+ 1)tβ+α. (3)

The Caputo fractional derivative Dα of a function f(t) is defined as

Dαf(t) =1

Γ(n− α)

t∫0

f (n)(s)

(t− s)α−n+1ds, n− 1 < α 6 n, n ∈ N. (4)

The following are some proprieties of Caputo fractional derivatives

DαC = 0, (C is constant ), (5)

Dαtβ =

{0, β ∈ N ∪ {0} and β <

⌈α⌉

Γ(β+1)Γ(β−α+1) t

β−α, β ∈ N ∪ {0} and β >⌈α⌉

or β /∈ N. and β >⌊α⌋,

(6)

where⌈α⌉

denote the smallest integer greater than or equal to α and⌊α⌋

denote the largest integer less than orequal to α.

The Caputo fractional deferential operator is a linear operator, since,

Dα(λf(t) + µg(t)) = λDαf(t) + µDαg(t), (7)

where λ and µ are constants.

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104 Mohamed Elarabi Benattia et al.

3 Chebyshev Polynomial and Chebyshev Wavelets

3.1 Properties of shifted Chebyshev polynomials

The well known Chebyshev polynomials of degree m are defined on the interval[−1, 1

]and can be determined with

the aid of the following recurrence formula

Tm+1(t) = 2tTm(t)− Tm−1(t), m = 1, 2, ..,

where T0(t) = 1 and T1(t) = t. For one to use these polynomials on the interval [0, 1], we define the so called shiftedChebyshev polynomials by using the change of variable t = 2x−1. Let the shifted Chebyshev polynomials Tm(2x−1)be denoted by T ?m(x) Then T ?m(x) can be obtained as follows:

T ?m+1(x) = 2(2x− 1)T ?m(x)− T ?m−1(x), m = 1, 2, .., (8)

where T ?0 (x) = 1, T ?1 (x) = 2x− 1 and T ?2 (x) = 2(2x− 1)2 − 1 = 8x2 − 8x+ 1.The analytic form of the shifted Chebyshev polynomials T ?m(x) of degree m is given by

T ?m(x) = m

m∑k=0

(−1)m−k(m+ k − 1)!22k

(m− k)!(2k)!xk. (9)

The orthogonality condition is1∫0

T ?m(x)T ?n(x)√1− (2x− 1)2

=

{πγm

4 , m = n

0, m 6= n, (10)

where

γm =

{2, m = 0.

1, m > 1.

3.2 Wavelets and Chebyshev Wavelets

In recent years, wavelets have found their way into many different fields of science and engineering. Waveletsconstitute a family of functions constructed from dilation and translation of single function called the mother wavelet.When the dilation parameter a and the translation parameter b vary continuously, we have the following family ofcontinuous wavelets:

ψa,b(t) =∣∣a∣∣− 1

2ψ(t− ba

) a, b ∈ R, a 6= 0. (11)

Chebyshev wavelets ψnm(t) = ψ(k, n,m, t) have four arguments; n argument k can assume any positive integer, mis the order for Chebyshev polynomials and t is the normalized time. They are defined on the interval [0, 1) by

ψn,m(t) =

{2

k+12 Pm(2k+1t− 2n− 1),

0,

n2k 6 t < n+1

2k

otherwise, (12)

where

Pm(t) =

1√π√

2πTm(t)

, m = 0.

, m > 1.

3.3 Function Approximations

A function f(t) defined over L2 [0, 1] can be expanded in the terms of Chebyshev wavelets as

f(t) =

∞∑n=0

∞∑m=0

cnmψnm(t), (13)

where the coefficient cn,m is given bycn,m =

⟨f(t), ψn,m(t)

⟩,

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Numerical Solution for Solving Fractional Differential Equations 105

in which⟨., .⟩

denotes the inner product. If the infinite series in (13) is truncated, then it can be written as

f(t) =

2k−1∑n=0

M∑m=0

cnmψnm(t) = CTΨ(t). (14)

Where C and Ψ(t) are 2k(M + 1)× 1 matrices given by{C =

[c0,0 , c0,1 , ..., c0,M , c1,0 c1,1 , ...c1,M , ...., c(2k−1),0 , c(2k−1),1 , ...., c(2k−1),M

]Ψ =

[ψ0,0 , ψ0,1 , ..., ψ0,M , ψ1,0 , ψ1,1 , ...ψ1,M , ...., ψ(2k−1),0 , ψ(2k−1),1 , ...., ψ(2k−1),M

]T.

(15)

4 Chebyshev Wavelet Operational Matrix of Fractional Order Deriva-tive

In this section, we drive the Chebyshev wavelet operational matrix of fractional derivative by first transforming thewavelets to shifted Chebyshev polynomials, we then make use of the shifted Chebyshev operational matrix of thefractional derivative in , and finally we derive the Chebyshev wavelet operational matrix of the fractional derivative.

4.1 Transformation Matrix of the Chebyshev Wavelets to Chebyshev polynomials

An arbitrary function y(t) ∈ L2 [0, 1]can be expanded into shifted Chebyshev polynomials as

y(x) =

M∑m=0

pmT?m(x) = PΨ(x),

where the shifted Chebyshev coefficient vector P and the shifted Chebyshev vector Ψ′(x) are given by

P = [p0, p1, ......, pM ] , (16)

Ψ′(x) = [T ?0 (x), T ?1 (x), ...., T ?m(x)] . (17)

The Chebyshev wavelet may be expanded in to (M + 1) terms shifted Chebyshev polynomials as

Ψ2k(M+1)×1(t) = Φ2k(M+1)×(M+1)Ψ′(M+1)×1, (18)

where Φ is the transformation matrix of the Chebyshev wavelet to Chebyshev polynomials. Let M = 2 and k = 1,we have {

Ψ = [ψ0,0 , ψ0,1 , ψ0,2 , ψ1,0 ψ1,1 , ψ1,2]T

Ψ′(x) = [T ?0 (x), T ?1 (x), T ?2 (x)],

where

ψ0,0 = 2P0(4t− 1) = 2√πT ?0 (t)

ψ0,1 = 2P1(4t− 1) = 2√

2πT1(4t− 1) = 2

√2π (4t− 1) = 2

√2π (2(2t− 1) + 1)

= 4√

2πT

?1 (t) + 2

√2πT

?0 (t)

ψ0,2 = 2P2(4t− 1) = 2√

2πT2(4t− 1) = 2

√2π

[2(4t− 1)T1(4t− 1)− 1

]= 2√

[2(4t− 1)(4t− 1)− 1

]= 2√

[8(2t− 1)2 + 8(2t− 1) + 1

]= 8√

2πT

?2 (t) + 16

√2πT

?1 (t) + 10

√2πT

?0 (t)

, 0 6 t < 12

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106 Mohamed Elarabi Benattia et al.

ψ1,0 = 2P0(4t− 3) = 2√πT ?0 (t)

ψ1,1 = 2P1(4t− 3) = 2√

2πT1(4t− 3) = 2

√2π (4t− 3) = 2

√2π (2(2t− 1)− 1)

= 4√

2πT

?1 (t)− 2

√2πT

?0 (t)

ψ1,2 = 2P2(4t− 3) = 2√

2πT2(4t− 3) = 2

√2π

[2(4t− 3)T1(4t− 3)− 1

]= 2√

[2(4t− 3)(4t− 3)− 1

]= 2√

[8(2t− 1)2 − 8(2t− 1) + 1

]= 8√

2πT

?2 (t)− 16

√2πT

?1 (t) + 10

√2πT

?0 (t)

, 12 6 t < 1 .

Thus, in this case

Φ =

{Φ1 =

[ai,j]6× 3, 0 6 t < 1

2

Φ2 =[bi,j]6× 3, 1

2 6 t < 1

where

Φ1 =

2√π

0 0

2√

2π 4

√2π 0

10√

2π 16

√2π 8

√2π

0 0 00 0 00 0 0

, Φ2 =

0 0 00 0 00 0 02√π

0 0

−2√

2π 4

√2π 0

10√

2π −16

√2π 8

√2π

.

4.2 Chebyshev operational matrix to fractional calculus

The fractional derivative of order α of the vector Ψ′(t) can be expressed by

DαΨ′(t) = L(α)Ψ′(t), (19)

where L(α) is the (m+ 1)× (m+ 1) operational matrix of fractional derivative of order α.In the following theorem we generalize the operational matrix of derivative of shifted Chebyshev polynomials forfractional derivative. The L(α) operational matrix of fractional derivative of order α defined as:

L(α) =

0 0 · · · 0...

... · · ·...

0 0 · · · 0dαe∑k=dαe

ϑdαe,0,kdαe∑k=dαe

ϑdαe,1,k · · ·dαe∑k=dαe

ϑdαe,m,k

...... · · ·

...i∑

k=dαeϑi,0,k

i∑k=dαe

ϑi,1,k · · ·i∑

k=dαeϑi,m,k

...... · · ·

...m∑

k=dαeϑm,0,k

m∑k=dαe

ϑm,1,k · · ·m∑

k=dαeϑm,m,k

where ϑi,j,k is given by:

ϑi,j,k =4ij

π

j∑l=0

(−1)i+j−k−l(j + l − 1)!(i+ k − 1)!k!22l+2k

(j − l)!(i− k)!(2l)!(2k)!(k + l − α+ 1)Γ(k − α+ 1), j > 1.

ϑi,j,k = (−1)i−k2i

π(k − α+ 1)

(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1), j = 0.

(20)

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Numerical Solution for Solving Fractional Differential Equations 107

Proof. The analytic form of the shifted Chebyshev polynomial T ?i (t) of degree i given by

T ?i (x) = i

i∑k=0

(−1)i−k(i+ k − 1)!22k

(i− k)!(2k)!xk. (21)

Note that T ?i (0) = (−1)i and T ?i (1) = 1. Using equations (6), (7) and (21) we haveDαT ?i (x) = i

i∑k=0

(−1)i−k(i+ k − 1)!22k

(i− k)!(2k)!Dα(xk)

= ii∑

k=dαe(−1)i−k

(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1)xk−α

, i = dαe , ...,m. (22)

Now, approximate xk−α by (m+ 1) terms of shifted Chebyshev series, we have

xk−α ≈m∑j=0

bkjT?j (x), (23)

where

bk0 =2

π

1∫0

xk−αT ?0 (x)dx =2

π

∫ 1

0

xk−α =2

π(k − α+ 1), (24)

and

bk,j =4

π

1∫0

xk−αT ?j (x)dx

=4

π

1∫0

xk−αjj∑l=0

(−1)j−l (j+l−1)!22l

(j−l)!(2l)! xldx

=4j

π

1∫0

j∑l=0

(−1)j−l (j+l−1)!22l

(j−l)!(2l)! xk−α+ldx

=4j

π

j∑l=0

(−1)j−l (j+l−1)!22l

(j−l)!(2l)!(k−α+l+1) ,

, j > 1 ,

so

bk,j =4j

π

j∑l=0

(−1)j−l(j + l − 1)!22l

(j − l)!(2l)!(k − α+ l + 1), j > 1. (25)

Employing the equations (22) and (25), we get

DαT ?i (x) ' ii∑

k=dαe(−1)i−k

(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1)

m∑j=0

bkjT?j (x).

' ii∑

k=dαe(−1)i−k

(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1)

m∑j=0

bkjT?j (x)

' ii∑

k=dαe

m∑j=0

(−1)i−k(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1)bkjT

?j (x)

=m∑j=0

(i∑

k=dαeϑi,j,k)T ?j (x)

, (26)

where

ϑi,j,k =4ij

π

j∑l=0

(−1)i+j−k−l(j + l − 1)!(i+ k − 1)!k!22l+2k

(j − l)!(i− k)!(2l)!(2k)!(k + l − α+ 1)Γ(k − α+ 1), i = dαe , ..,m, j > 1. (27)

Employing the equations (24)and (25), we getϑi,0,k = i(−1)i−k

(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1)bk,0

= (−1)i−k2i

π(k − α+ 1)

(i+ k − 1)!22kk!

(i− k)!(2k)!Γ(k − α+ 1)

, j = 0 . (28)

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108 Mohamed Elarabi Benattia et al.

Where ϑi,j,k is given in equation (20). Rewrite equation (26) as a vector form we have

DαT ?i (x) = (

i∑k=dαe

ϑi,0,k,

i∑k=dαe

ϑi,1,k, ......,

i∑k=dαe

ϑi,m,k)Ψ′(x), i = dαe , ..,m. (29)

4.3 Chebyshev Wavelet Operational Matrix of Fractional Order Derivative

Now, we derive Chebyshev wavelet operational matrix of fractional order derivative. Let

DαΨ(t) = H(α)Ψ(t), (30)

where H(α) is the Chebyshev wavelet operational matrix of fractional order derivative. Using (18) and (19) we get

DαΨ(t) = DαΦΨ′(t) = ΦDαΨ′(t) = ΦL(α)Ψ′(t), (31)

from equation (30) and (31), we have

H(α)Ψ(t) = H(α)ΦΨ′(t) = ΦL(α)Ψ′(t). (32)

Thus, the Chebyshev wavelet operational matrix of fractional derivative H(α) is given by

H(α) = ΦL(α)Φ−1 (33)

5 Applications of the operational matrix of fractional derivative

In this section, in order to show the high importance of operational matrix of fractional derivative, we apply it tosolve multi-order fractional differential equation.

5.1 Linear multi-order fractional differential equation

Consider the linear multi-order fractional differential equation

Dαy(x) = a1Dµ1y(x) + a2D

µ2y(x) + ...+ asDµsy(x) + as+1y(x) + as+2g(x), (34)

with initial conditionsy(i)(x) = di, i = 1, .., n. (35)

Where aj , for j = 1, ......, s+ 2 are real constant coefficients and also n < α 6 n+ 1,0 < µ1 < µ2 < ..... < µs < α. Dα denotes the Caputo fractional derivative of order α.

To solve the problem (34) and (35), we approximate y(x) and g(x) by the Chebyshev wavelets as,

y(x) ≈2k−1∑n=0

M∑m=0

cn,mψn,m = CTΨ(t). (36)

g(x) ≈2k−1∑n=0

M∑m=0

zn,mψn,m = GTΨ(t), (37)

whereG =

[z0,0 , z0,1 , ..., z0,M , z1,0 , z1,1 , ...z1,M , ..., z(2k−1),0 , z(2k−1),1 , ...., z(2k−1),M

],

is know but C as defined in (15) is the unknown vector.Now, using (30) and (36) we get

Dαy(x) ≈ CTDαΨ(x) ≈ CTH(α)Ψ(x); (38)

Dµjy(x) ≈ CTDµjΨ(x) ≈ CTH(µj)Ψ(x). (39)

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Numerical Solution for Solving Fractional Differential Equations 109

Using (36), (39) the residual R(x) for equation (34) can be written as

R(x) ≈ (CTH(α)Ψ(x)− a1CTH(µ1)Ψ(x)− ....− akCTH(µk)Ψ(x)− ak+1C

TΨ(x)− ak+2GTΨ(x)). (40)

As in typical tau method we generate 2k(M + 1)− n linear equations by applying

⟨R(x), Ψ(x)

⟩=

1∫0

R(x)Ψ(x)dx, j = 1, ..., 2k(M + 1)− n. (41)

Also by substituting initial conditions (35) in to (36) and (39) we havey(0) ≈ CTΨ(0) = d0

y′(0) ≈ CTH(1)Ψ(0) = d1

...

y(n)(0) ≈ CTH(n)Ψ(0) = dn

. (42)

Equations (41) and (42) generate 2k(M+1) set of linear equations. These linear equations can be solved for unknowncoefficients of the vector C.

6 Illustrative Examples

In this section, we demonstrate the effectiveness of the proposed Chebyshev wavelet method with numerical examples.we consider the following initial value problem,

D2y(x) +D12 y(x) + y(x) = g(x), (43)

with g(x) = x2 + 2 +8

3√πx

32 .

The exact solution of this problem is y(x) = x2. By applying the technique described in the section (??) withM = 2 and k = 0 we may write the approximate solution as{

y(x) = CTΨ(x)⇐⇒ y(0) = CTΨ(0) = d0 = 0.

y′(x) = CTH(1)Ψ(x)⇐⇒ y′(0) = CTH(1)Ψ(0) = d1 = 0.

Now, we calculate Ψ(x) =(ψ00(x), ψ01(x), ψ02(x)

)T, where

ψ00(x) =√

2P0(x) =√

ψ01(x) =√

2P1(2x− 1) =√

2√

2πT1(2x− 1) = 2√

π(2x− 1)

ψ01(x) =√

2P2(2x− 1) =√

2√

2πT2(2x− 1) = 2√

π

[2(2x− 1)2 − 1

]= 2√

π(8x2 − 8x+ 1)

. (44)

So

Ψ(x) =

2√π

(2x− 1)2√π

(8x2 − 8x+ 1)

=⇒ Ψ(0) =

− 2√π

2√π

. (45)

y(0) = CTΨ(0) = d0 = 0⇐⇒ (c00, c01, c02)

− 2√π

2√π

= 0 which implies

c00 −√

2c01 +√

2c02 = 0. (46)

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110 Mohamed Elarabi Benattia et al.

Now, we calculate the matrices L(1)

, L(2) and L( 12 ). From the equation (20) we have

L(1) =

0 0 04π 0 −8

3π0 16

3π 0

, L(2) =

0 0 00 0 032π 0 −64

, L( 12 ) =

0 0 04

3π√π

245π√π

−247π√π

−415π√π

247π√π

824135π

, (47)

using the equation (33) and

Φ =

2π 0 0

0 2√π

0

0 0 2√π

we get the matrices H(1), H(2) and H( 1

2 ) as follow:

H(1) =

0 0 04√

2π 0 −8

3π0 16

3π 0

, H(2) =

0 0 00 0 0

32√

2π 0 −64

, H( 12 ) =

0 0 04√

2π3π

24√π

5π−24√π

7π−4√

2π15π

24√π

7π824√π

135π

. (48)

Then, applying the equations (42) we obtain

c01 − 4c02 = 0. (49)

We can expand the function g(x) of the problem by Chebyshev wavelets as

g(x) =(2.513562154 1.1221745532 0.2077870929

)Ψ(x).

And by using the equation (41) we get

4√

2π3π c01 − 4

√2π

15π c02 + 32√

2π c02 + c00 = 2.513562154. (50)

Solving the following system by the Gauss eliminationc00 −

√2c01 +

√2c02 = 0

c01 − 4c02 = 04√

2π3π c01 − 4

√2π

15π c02 + 32√

2π c02 + c00 = 2.513562154

, (51)

we obtainc00 = 0.4699864272, c01 = 0.4431074542, c02 = 0.1107768636.

Finally,

y(x) =(0.4699864272 0.4431074542 0.1107768636

) √

2√π

(2x− 1)2√π

(8x2 − 8x+ 1)

≈ x2

7 Conclusion

In this article, a general formulation foe deriving the Chebyshev wavelet operational matrix of fractional derivativeshas been derived, and as an important application, we describe how to solve numerically the FDEs. Maple softwareis used to obtain the approximate solution.

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Numerical Solution for Solving Fractional Differential Equations 111

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