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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 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]
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.
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)
⟩,
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√
2π
[2(4t− 1)(4t− 1)− 1
]= 2√
2π
[8(2t− 1)2 + 8(2t− 1) + 1
]= 8√
2πT
?2 (t) + 16
√2πT
?1 (t) + 10
√2πT
?0 (t)
, 0 6 t < 12
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√
2π
[2(4t− 3)(4t− 3)− 1
]= 2√
2π
[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)
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)
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)
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) =√
2π
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) =
√
2π
2√π
(2x− 1)2√π
(8x2 − 8x+ 1)
=⇒ Ψ(0) =
√
2π
− 2√π
2√π
. (45)
y(0) = CTΨ(0) = d0 = 0⇐⇒ (c00, c01, c02)
√
2π
− 2√π
2√π
= 0 which implies
c00 −√
2c01 +√
2c02 = 0. (46)
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
3π
, 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
3π
, 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
) √
2π
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.
Numerical Solution for Solving Fractional Differential Equations 111
References
[1] A. Hanyga, Fractional-order relaxation laws in non-linear viscoelasticity, Continuum Mechanics and Thermo-dynamics, 19 (2007), 25-36.
[2] A. H. Bhrawy, A. S. Alofi. The operational matrix of fractional integration for shifted Chebyshev polynomials.AML 26(2013)25-31.
[3] A. Isah, P. Chang. Legendre Wavelet Operational Matrix of Fractional Derivative Through Wavelet-PolynomialTransformation and its Application in Solving Fractional Order Differential Equations. IJPAM. 105(2015), 97-114.
[4] A. Saadatmandia, M. Dehghanb. Anew operational matrix for solving fractional-order differential equa-tions.CMA 59(2010) 1326-1336.
[5] E. Keshavarz, Y. Ordokhani, M. Razzaghi. Bernoulli wavelet operational matrix of fractional order integrationand its applications in solving the fractional order differential equations. Applied Mathematical Modelling 38(2014) 6038–6051.
[6] F. Mohammadi. Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix offractional derivative. IJAAMM. 2(1) (2014) 83-91
[7] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, 1994.
[8] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equations,Journal of Computational and Applied Mathematics 181 (2005) 245 – 251.
[9] Mountassir HamdiI Cherif, Kacem Belghaba And Djelloul Ziane, Homotopy Perturbation Method For SolvingThe Fractional Fisher’s Equation, International Journal of Analysis and Applications, ISSN 2291-8639, Volume10, Number 1 (2016), 9-16..
[10] Podlubny I. . Fractional differential equations. Academic press, New York; 1999
[11] J. Biazar, E. Babolian, R. Islam, Solution of the system of ordinary differential equations by Adomian decom-position method, Appl. Math. Comput. 147 (3) (2004) 713–719 Comput. 147 (3) (2004) 713–719 .
[12] Ji-Huan He, Variational iteration method — a kind of non-linear analytical technique: some examples, Interna-tional Journal of Non-Linear Mechanics 34 (1999) 699—708.
[13] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordonand Breach, Yverdon, 1993
[14] Shaher Momani, Zaid Odibat, Homotopy perturbation method for nonlinear partial differential equations offractional order, Physics Letters A, Volume 365, Issues 5–6, 11 June 2007, Pages 345-350.
[15] S. Gh. Hosseini, F. Mohammadi. A New Operational Matrix of Derivative for Chebyshev Wavelets and itsApplications in Solving Ordinary Differential Equations with Non analytic Solution. AMS, 5 (51) (2011) 2537-2548.
[16] V. E. Tarasov, Fractional integro-di erential equations for electromagnetic waves in dielectric media, Theoreticaland Math. Phys, 158 (2009), 355-359.
[17] W. A. Khan, F. A. Ansari, European Option Pricing of Fractional Black-Scholes Model Using Sumudu Transformand its Derivatives, General Letters in Mathematics, Vol 1 (3) 2016 pp 74-80.