# Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

Authors: Syed Tauseef Mohyud-Din; Farah Jabeen Awan; Jamshad Ahmad; Saleh M. Hassan
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.


---

## Body

## 1. Introduction

Nonlinear partial differential equations (NLPDEs) are mathematical models that are used to describe complex phenomena and dynamic processes arising in the world around us. The NLPDEs appear in many applications of science and engineering such as fluid dynamics, plasma physics, hydrodynamics, solid state physics, optical fibers, and acoustics, as well as other disciplines. Recently, lot of attention is paid to finding appropriate solutions of NLPDEs. In the similar context, various techniques including Adomian’s decomposition method (ADM) [1], Variational Iteration (VIM) [2], Homotopy Perturbation (HPM) [3], Homotopy Analysis (HAM) [4],F-Expansion [5], Exp-function [6], sine-cosine [7], differential transform method (DTM) [8–11], and reduced differential transform [9, 12–15] have been applied on wide range of linear and nonlinear problems of diversified physical nature. Inspired and motivated by ongoing research in this area, we apply reduced differential transform method (RDTM) [12–19] coupled with a complex transform to solve three important mathematical models [20–27], namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions. It is worth mentioning that derivatives are defined in Jumarie’s sense which is relatively a new approach and is easier to handle; however, other approaches like Caputo and Riemann Liouville may also be utilized. It is an established fact that models under discussion [20–23] are of extreme importance and hence appear frequently in various physical phenomena including nonlinear dispersive waves, shallow water waves, ion acoustic plasma waves, Lax pairs of a special form, four-reduction of KP hierarchy, Frenkel-Kontorova dislocation model; see [20–23] and the references therein. It is observed that the proposed technique is extremely simple and user friendly and has shown very useful results. It is to be highlighted that the suggested modified version may be extended to some other important nonlinear problems which have been solved by some other reliable methods; see [28–33]. It is to be highlighted that Ganji et al. have solved wide range of mathematical models [28–33] by making an appropriate use of some recently developed schemes and hence giving a new avenue of research.


## 2. Jumarie’s Fractional Derivative

Some useful formulas and results of Jumarie’s fractional derivative were summarized [24](1)
D
x
α
c
=
0
,
α
≥
0
,
c
=
constant
,
(2)
D
x
α
c
f
x
=
c
D
x
α
f
x
,
α
≥
0
,
c
=
constant
,
(3)
D
x
α
x
β
=
Γ
1
+
β
Γ
1
+
β
-
α
x
β
-
α
,
β
≥
α
≥
0
,
(4)
D
x
α
f
x
g
x
=
D
x
α
f
x
g
x
+
f
x
D
x
α
g
x
,
(5)
D
x
α
f
x
t
=
f
x
′
x
x
α
t
.


## 3. Fractional Complex Transform Method (FCTM)

The fractional complex transform was first proposed in [25] and is defined as (6)
T
=
p
t
α
Γ
α
+
1
,
X
=
q
x
β
Γ
β
+
1
,
Y
=
k
y
γ
Γ
1
+
γ
,
Z
=
l
z
λ
Γ
1
+
λ
,where p, q, k, and l are unknown constants: 0
<
α
≤
1, 0
<
β
≤
1, 0
<
γ
≤
1, and 0
<
λ
≤
1.


## 4. Reduced Differential Transform Method (RDTM)

To illustrate the basic idea of the DTM, The differential transform ofkth derivative of a function u
x
,
t, which is analytic and differentiated continuously in the domain of interest, is defined as(7)
U
k
x
=
1
k
!
∂
k
u
x
,
t
∂
t
k
t
=
t
0
.The differential inverse transform of U
k
x is defined as follows:(8)
u
x
,
t
=
∑
k
=
0
∞
U
k
x
t
-
t
0
k
.Equation (8) is known as the Taylor series expansion of u
x
,
t, around t
=
t
0. Combining (7) and (8),(9)
u
x
,
t
=
∑
k
=
0
∞
1
k
!
∂
k
u
x
,
t
∂
t
k
t
=
t
0
t
-
t
0
k
,when t
0
=
0; the above equation reduces to(10)
u
x
,
t
=
∑
k
=
0
∞
1
k
!
∂
k
u
x
,
t
∂
t
k
t
=
t
0
t
k
,and (2) reduces to (11)
u
x
,
t
=
∑
k
=
0
∞
U
k
x
t
k
.Some properties of the reduced differential transform method are as follows.(1)
If the original function isu
x
,
t
=
w
x
,
t
+
v
x
,
t, then the transformed function is (12)
U
k
x
=
W
k
x
+
V
k
x
.
(2)
Ifu
x
,
t
=
α
w
x
,
t, then U
k
x
=
α
W
k
x.
(3)
Ifu
x
,
t
=
∂
m
w
x
,
t
/
∂
t
m, then U
k
x
=
(
k
+
m
!
/
k
!
)
W
k
x.
(4)
Ifu
x
,
t
=
∂
w
x
,
t
/
∂
x, then U
k
x
=
(
∂
/
∂
x
)
W
k
x.
(5)
Ifu
x
,
y
,
t
=
∂
w
x
,
y
,
t
/
∂
x, then U
k
x
,
y
=
(
∂
/
∂
x
)
W
k
x
,
y.
(6)
Ifu
x
,
y
,
z
,
t
=
∂
w
x
,
y
,
z
,
t
/
∂
x, then U
k
x
,
y
,
z
=
(
∂
/
∂
x
)
W
k
x
,
y
,
z.
(7)
Ifu
x
,
t
=
x
m
t
n
w
x
,
t, then U
k
x
=
x
m
W
k
-
n
x.
(8)
Ifu
x
,
t
=
w
2
x
,
t, then U
k
x
=
∑
r
=
0
k
W
r
x
W
k
-
r
x.


## 5. Numerical Applications

In this section, we apply the proposed fractional complex transform method coupled with reduced differential transform method to solve three important mathematical models. Numerical results are highly encouraging. For details about such equations, readers are referred to study [22, 23].


### 5.1. Fractional Kaup-Kupershmidt (FKK) Equation

It is an established fact that fractional Kaup-Kupershmidt (FKK) equation plays a major role in the study of nonlinear dispersive waves. Moreover, it describes a large number of important physical phenomena, such as shallow water waves and ion acoustic plasma waves.Consider the nonlinear KK equation [22, 23],(13)
∂
α
u
∂
t
α
-
∂
5
u
∂
x
5
-
5
u
∂
3
u
∂
x
3
-
25
3
∂
u
∂
x
∂
2
u
∂
x
2
-
5
u
2
∂
u
∂
x
=
0
,with the initial condition(14)
u
x
,
0
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
,where k is an arbitrary constant.Applying the transformation [25], we get the following partial differential equation:(15)
∂
u
∂
T
-
∂
5
u
∂
x
5
-
5
u
∂
3
u
∂
x
3
-
25
3
∂
u
∂
x
∂
2
u
∂
x
2
-
5
u
2
∂
u
∂
x
=
0
.Applying the differential transform to (15) and (14), we obtain the following recursive formula:(16)
k
+
1
U
k
+
1
x
=
∂
5
U
k
x
∂
x
5
+
5
∑
r
=
0
k
U
k
-
r
x
∂
3
U
r
x
∂
x
3
+
25
3
∑
r
=
0
k
U
k
-
r
x
∂
2
U
r
x
∂
x
2
+
5
∑
r
=
0
k
∑
s
=
k
r
U
k
-
r
x
U
r
-
s
x
∂
U
s
x
∂
x
.Using the initial condition, we have(17)
U
0
x
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
.Now, substituting (17) into (16), and by straightforward iterative steps, yields(18)
U
1
x
=
-
264
k
7
e
k
x
-
1
+
e
k
x
1
+
e
k
x
3
,
U
2
x
=
-
1452
k
12
e
k
x
4
e
k
x
-
e
2
k
x
-
1
1
+
e
k
x
4
,
U
3
x
=
3524
k
17
e
k
x
-
11
e
k
x
+
11
e
2
k
x
-
e
3
k
x
+
1
1
+
e
k
x
5
,
⋮and so on.The series solution is given by(19)
u
x
,
T
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
-
264
k
7
e
k
x
-
1
+
e
k
x
1
+
e
k
x
3
T
-
1452
k
12
e
k
x
4
e
k
x
-
e
2
k
x
-
1
1
+
e
k
x
4
T
2
+
3524
k
17
e
k
x
-
11
e
k
x
+
11
e
2
k
x
-
e
3
k
x
+
1
1
+
e
k
x
5
T
3
+
⋯
.The inverse transformation will yield(20)
u
x
,
t
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
-
264
k
7
e
k
x
e
k
x
-
1
1
+
e
k
x
3
t
α
Γ
α
+
1
-
1452
k
12
e
k
x
4
e
k
x
-
e
2
k
x
-
1
1
+
e
k
x
4
t
2
α
Γ
2
α
+
1
+
3524
k
17
e
k
x
-
11
e
k
x
+
11
e
2
k
x
-
e
3
k
x
+
1
1
+
e
k
x
5
t
3
α
Γ
3
α
+
1
+
⋯
.This solution is convergent to the exact solution(21)
u
x
,
t
=
-
2
k
2
+
24
k
2
1
+
e
k
x
+
11
k
5
t
-
24
k
2
1
+
e
k
x
+
11
k
5
t
2
.In Figures 1(a) and 1(b), we have presented approximate solution at α
=
1 and exact solutions.Figure 1
(a) Approximate solution. (b) Exact solution.
(a)
(b)


### 5.2. Generalized Fractional Drinfeld-Sokolov (GFDS) Equations [20, 21]

This system was introduced independently by Drinfeld and Sokolov [20, 21]. This coupled system was given as one of the numerous examples of nonlinear equations possessing Lax pairs of a special form. Also, the coupled system was found as a special case of the four-reduction of the KP hierarchy; see [20, 21] and the references therein.We consider the system of generalized fractional Drinfeld-Sokolov (GFDS) equations [20, 21](22)
∂
β
u
∂
t
β
+
∂
3
u
∂
x
3
-
6
u
∂
u
∂
x
-
6
∂
v
α
∂
x
=
0
,
∂
β
v
∂
t
β
-
2
∂
3
v
∂
x
3
+
6
u
∂
v
α
∂
x
=
0
,
0
<
x
,
t
<
π
,
0
<
β
≤
1
,with the initial conditions(23)
u
x
,
0
=
-
b
2
-
4
k
4
4
k
2
+
2
k
2
tanh
2
⁡
k
x
,
v
x
,
0
=
b
tanh
⁡
k
x
,where α is a constant.Applying the transformation [25], we get the following partial differential equations:(24)
∂
u
∂
T
+
∂
3
u
∂
x
3
-
6
u
∂
u
∂
x
-
6
∂
v
α
∂
x
=
0
,
∂
v
∂
T
-
2
∂
3
v
∂
x
3
+
6
u
∂
v
α
∂
x
=
0
.Applying the differential transform to (24) and (23), we obtain the following recursive formula:(25)
k
+
1
U
k
+
1
x
=
-
∂
3
U
k
x
∂
x
3
+
6
∑
r
=
0
k
U
k
-
r
x
∂
U
r
x
∂
x
+
6
∂
V
k
α
x
∂
x
,
k
+
1
V
k
+
1
x
=
2
∂
3
V
k
x
∂
x
3
-
6
∑
r
=
0
k
U
k
-
r
x
∂
V
r
x
∂
x
.Using the initial condition, we have(26)
U
0
x
=
-
b
2
-
4
k
4
4
k
2
+
2
k
2
tanh
2
⁡
k
x
,
V
0
x
=
b
tanh
⁡
k
x
.Now, substituting (26) into (25) when (α
=
2), and by straightforward iterative steps, yields(27)
U
1
x
=
2
k
4
k
2
+
3
b
2
sinh
⁡
k
x
cosh
⁡
k
x
3
,
V
1
x
=
1
2
b
4
k
2
+
3
b
2
cosh
⁡
k
x
2
k
,
U
2
x
=
-
1
2
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
2
cosh
⁡
k
x
4
,
V
2
x
=
-
1
4
b
4
k
2
+
3
b
2
2
sinh
⁡
k
x
cosh
⁡
k
x
3
k
2
,
U
3
x
=
1
3
sinh
⁡
k
x
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
cosh
⁡
k
x
5
,
V
3
x
=
1
24
b
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
3
cosh
⁡
k
x
4
,
⋮and so on.The series solution is given by(28)
u
x
,
T
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
+
2
k
4
k
2
+
3
b
2
sinh
⁡
k
x
cosh
⁡
k
x
3
T
-
1
2
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
2
cosh
⁡
k
x
4
T
2
+
1
3
sinh
⁡
k
x
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
cosh
⁡
k
x
5
T
3
+
⋯
,
v
x
,
T
=
b
tanh
⁡
k
x
+
1
2
b
4
k
2
+
3
b
2
cosh
⁡
k
x
2
k
T
-
1
4
b
4
k
2
+
3
b
2
2
sinh
⁡
k
x
cosh
⁡
k
x
3
k
2
T
2
+
1
24
b
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
3
cosh
⁡
k
x
4
T
3
+
⋯
.The inverse transformation will yields(29)
u
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
+
2
k
4
k
2
+
3
b
2
sinh
⁡
k
x
cosh
⁡
k
x
3
t
β
Γ
β
+
1
-
1
2
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
2
cosh
⁡
k
x
4
t
2
β
Γ
2
β
+
1
+
1
3
sinh
⁡
k
x
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
cosh
⁡
k
x
5
t
3
β
Γ
3
β
+
1
+
⋯
,
v
=
b
tanh
⁡
k
x
+
1
2
b
4
k
2
+
3
b
2
cosh
⁡
k
x
2
k
t
β
Γ
β
+
1
-
1
4
b
4
k
2
+
3
b
2
2
sinh
⁡
k
x
cosh
⁡
k
x
3
k
2
t
2
β
Γ
2
β
+
1
+
1
24
b
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
3
cosh
⁡
k
x
4
t
3
β
Γ
3
β
+
1
+
⋯
.This solution is convergent to the exact solution(30)
u
x
,
t
=
-
b
2
-
4
k
4
4
k
2
+
2
k
2
tanh
2
⁡
k
x
+
3
b
2
+
4
k
4
2
k
t
,
v
x
,
t
=
b
tanh
⁡
k
x
+
3
b
2
+
4
k
4
2
k
t
.In Figures 2(a)–2(d), we have presented approximate u
x
,
t and v
x
,
t at α
=
1 and exact solutions.Figure 2
(a) Approximate solution. (b) Exact solution. (c) Approximate solution. (d) Exact solution.
(a)
(b)
(c)
(d)


### 5.3. System of Coupled Fractional Sine-Gordon Equations [26, 27]

Coupled Sine-Gordon equations were introduced by Ray et al. [26, 27]. The coupled Sine-Gordon equations generalize the Frenkel-Kontorova dislocation model; see [26, 27] and the references therein.We now consider a system of coupled Sine-Gordon equations [26, 27]:(31)
∂
2
α
u
∂
t
2
α
-
∂
2
u
∂
x
2
=
-
a
2
sin
⁡
u
-
v
,
∂
2
α
v
∂
t
2
α
-
c
2
∂
2
v
∂
x
2
=
sin
⁡
u
-
v
,
0
<
x
,
t
<
π
,
0
<
α
≤
1
,with the initial conditions(32)
u
x
,
0
=
A
cos
⁡
k
x
,
u
t
x
,
0
=
0
,
v
x
,
0
=
0
,
v
t
x
,
0
=
0
,where c is the ratio of the acoustic velocities of the components u and v.Applying the transformation [25] to (31), we get the following partial differential equations:(33)
∂
2
u
∂
T
2
-
∂
2
u
∂
x
2
=
-
a
2
sin
⁡
u
-
v
,
∂
2
v
∂
T
2
-
c
2
∂
2
v
∂
x
2
=
sin
⁡
u
-
v
.Applying the differential transform to (33) and (32), we obtain the following recursive formula:(34)
k
+
2
!
k
!
U
k
+
2
x
=
∂
2
U
k
x
∂
x
2
-
a
2
N
k
x
,
k
+
2
!
k
!
V
k
+
2
x
=
c
2
∂
2
U
k
x
∂
x
2
+
N
k
x
.Using the initial condition, we have(35)
U
0
x
=
A
cosh
⁡
k
x
,
U
1
x
=
0
,
V
0
x
=
0
,
V
1
x
=
0
.Now, substituting (35) into (34), and by straightforward iterative steps, yields(36)
U
2
x
=
-
A
k
2
cosh
⁡
k
x
2
-
a
2
sin
⁡
A
cosh
⁡
k
x
2
,
V
2
x
=
sin
⁡
A
cosh
⁡
k
x
2
,
U
3
x
=
0
,
V
3
x
=
0
,
U
4
x
=
A
k
4
cosh
⁡
k
x
24
+
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
-
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cos
2
⁡
k
x
24
+
a
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cos
⁡
k
x
12
+
a
4
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
+
a
2
cos
⁡
A
cosh
⁡
k
x
s
i
n
A
cosh
⁡
k
x
24
,
V
4
x
=
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cosh
2
⁡
k
x
24
-
c
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
-
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
,
⋮and so on.The series solution is given by(37)
u
x
,
T
=
A
cosh
⁡
k
x
-
A
k
2
cosh
⁡
k
x
2
+
a
2
sin
⁡
A
cosh
⁡
k
x
2
T
2
+
A
k
4
cosh
⁡
k
x
24
+
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
-
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cos
2
⁡
k
x
24
+
a
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cos
⁡
k
x
12
+
a
4
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
+
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
T
4
+
⋯
,
v
x
,
T
=
sin
⁡
A
cosh
⁡
k
x
2
T
2
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cosh
2
⁡
k
x
24
-
c
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
-
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
T
3
+
⋯
.The inverse transformation will yield(38)
u
x
,
t
=
A
cosh
⁡
k
x
-
A
k
2
cosh
⁡
k
x
2
+
a
2
sin
⁡
A
cosh
⁡
k
x
2
t
2
α
Γ
2
α
+
1
+
A
k
4
cosh
⁡
k
x
24
+
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
-
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cos
2
⁡
k
x
24
+
a
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cos
⁡
k
x
12
+
a
4
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
+
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
t
4
α
Γ
4
α
+
1
+
⋯
,
v
x
,
t
=
sin
⁡
A
cosh
⁡
k
x
2
t
2
α
Γ
2
α
+
1
+
c
2
A
2
k
2
s
i
n
A
cosh
⁡
k
x
24
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cosh
⁡
2
⁡
k
x
24
-
1
+
c
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
1
+
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
t
4
α
Γ
4
α
+
1
+
⋯
.This solution is convergent to Adomian’s decomposition method solution [26, 27].In Figures3(a) and 3(b) we have presented approximate solutions u
x
,
t and v
x
,
t at α
=
1.Figure 3
(a) Approximate solutionu
x
,
t. (b) Approximate solution v
x
,
t.
(a)
(b)


## 5.1. Fractional Kaup-Kupershmidt (FKK) Equation

It is an established fact that fractional Kaup-Kupershmidt (FKK) equation plays a major role in the study of nonlinear dispersive waves. Moreover, it describes a large number of important physical phenomena, such as shallow water waves and ion acoustic plasma waves.Consider the nonlinear KK equation [22, 23],(13)
∂
α
u
∂
t
α
-
∂
5
u
∂
x
5
-
5
u
∂
3
u
∂
x
3
-
25
3
∂
u
∂
x
∂
2
u
∂
x
2
-
5
u
2
∂
u
∂
x
=
0
,with the initial condition(14)
u
x
,
0
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
,where k is an arbitrary constant.Applying the transformation [25], we get the following partial differential equation:(15)
∂
u
∂
T
-
∂
5
u
∂
x
5
-
5
u
∂
3
u
∂
x
3
-
25
3
∂
u
∂
x
∂
2
u
∂
x
2
-
5
u
2
∂
u
∂
x
=
0
.Applying the differential transform to (15) and (14), we obtain the following recursive formula:(16)
k
+
1
U
k
+
1
x
=
∂
5
U
k
x
∂
x
5
+
5
∑
r
=
0
k
U
k
-
r
x
∂
3
U
r
x
∂
x
3
+
25
3
∑
r
=
0
k
U
k
-
r
x
∂
2
U
r
x
∂
x
2
+
5
∑
r
=
0
k
∑
s
=
k
r
U
k
-
r
x
U
r
-
s
x
∂
U
s
x
∂
x
.Using the initial condition, we have(17)
U
0
x
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
.Now, substituting (17) into (16), and by straightforward iterative steps, yields(18)
U
1
x
=
-
264
k
7
e
k
x
-
1
+
e
k
x
1
+
e
k
x
3
,
U
2
x
=
-
1452
k
12
e
k
x
4
e
k
x
-
e
2
k
x
-
1
1
+
e
k
x
4
,
U
3
x
=
3524
k
17
e
k
x
-
11
e
k
x
+
11
e
2
k
x
-
e
3
k
x
+
1
1
+
e
k
x
5
,
⋮and so on.The series solution is given by(19)
u
x
,
T
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
-
264
k
7
e
k
x
-
1
+
e
k
x
1
+
e
k
x
3
T
-
1452
k
12
e
k
x
4
e
k
x
-
e
2
k
x
-
1
1
+
e
k
x
4
T
2
+
3524
k
17
e
k
x
-
11
e
k
x
+
11
e
2
k
x
-
e
3
k
x
+
1
1
+
e
k
x
5
T
3
+
⋯
.The inverse transformation will yield(20)
u
x
,
t
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
-
264
k
7
e
k
x
e
k
x
-
1
1
+
e
k
x
3
t
α
Γ
α
+
1
-
1452
k
12
e
k
x
4
e
k
x
-
e
2
k
x
-
1
1
+
e
k
x
4
t
2
α
Γ
2
α
+
1
+
3524
k
17
e
k
x
-
11
e
k
x
+
11
e
2
k
x
-
e
3
k
x
+
1
1
+
e
k
x
5
t
3
α
Γ
3
α
+
1
+
⋯
.This solution is convergent to the exact solution(21)
u
x
,
t
=
-
2
k
2
+
24
k
2
1
+
e
k
x
+
11
k
5
t
-
24
k
2
1
+
e
k
x
+
11
k
5
t
2
.In Figures 1(a) and 1(b), we have presented approximate solution at α
=
1 and exact solutions.Figure 1
(a) Approximate solution. (b) Exact solution.
(a)
(b)


## 5.2. Generalized Fractional Drinfeld-Sokolov (GFDS) Equations [20, 21]

This system was introduced independently by Drinfeld and Sokolov [20, 21]. This coupled system was given as one of the numerous examples of nonlinear equations possessing Lax pairs of a special form. Also, the coupled system was found as a special case of the four-reduction of the KP hierarchy; see [20, 21] and the references therein.We consider the system of generalized fractional Drinfeld-Sokolov (GFDS) equations [20, 21](22)
∂
β
u
∂
t
β
+
∂
3
u
∂
x
3
-
6
u
∂
u
∂
x
-
6
∂
v
α
∂
x
=
0
,
∂
β
v
∂
t
β
-
2
∂
3
v
∂
x
3
+
6
u
∂
v
α
∂
x
=
0
,
0
<
x
,
t
<
π
,
0
<
β
≤
1
,with the initial conditions(23)
u
x
,
0
=
-
b
2
-
4
k
4
4
k
2
+
2
k
2
tanh
2
⁡
k
x
,
v
x
,
0
=
b
tanh
⁡
k
x
,where α is a constant.Applying the transformation [25], we get the following partial differential equations:(24)
∂
u
∂
T
+
∂
3
u
∂
x
3
-
6
u
∂
u
∂
x
-
6
∂
v
α
∂
x
=
0
,
∂
v
∂
T
-
2
∂
3
v
∂
x
3
+
6
u
∂
v
α
∂
x
=
0
.Applying the differential transform to (24) and (23), we obtain the following recursive formula:(25)
k
+
1
U
k
+
1
x
=
-
∂
3
U
k
x
∂
x
3
+
6
∑
r
=
0
k
U
k
-
r
x
∂
U
r
x
∂
x
+
6
∂
V
k
α
x
∂
x
,
k
+
1
V
k
+
1
x
=
2
∂
3
V
k
x
∂
x
3
-
6
∑
r
=
0
k
U
k
-
r
x
∂
V
r
x
∂
x
.Using the initial condition, we have(26)
U
0
x
=
-
b
2
-
4
k
4
4
k
2
+
2
k
2
tanh
2
⁡
k
x
,
V
0
x
=
b
tanh
⁡
k
x
.Now, substituting (26) into (25) when (α
=
2), and by straightforward iterative steps, yields(27)
U
1
x
=
2
k
4
k
2
+
3
b
2
sinh
⁡
k
x
cosh
⁡
k
x
3
,
V
1
x
=
1
2
b
4
k
2
+
3
b
2
cosh
⁡
k
x
2
k
,
U
2
x
=
-
1
2
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
2
cosh
⁡
k
x
4
,
V
2
x
=
-
1
4
b
4
k
2
+
3
b
2
2
sinh
⁡
k
x
cosh
⁡
k
x
3
k
2
,
U
3
x
=
1
3
sinh
⁡
k
x
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
cosh
⁡
k
x
5
,
V
3
x
=
1
24
b
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
3
cosh
⁡
k
x
4
,
⋮and so on.The series solution is given by(28)
u
x
,
T
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
+
2
k
4
k
2
+
3
b
2
sinh
⁡
k
x
cosh
⁡
k
x
3
T
-
1
2
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
2
cosh
⁡
k
x
4
T
2
+
1
3
sinh
⁡
k
x
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
cosh
⁡
k
x
5
T
3
+
⋯
,
v
x
,
T
=
b
tanh
⁡
k
x
+
1
2
b
4
k
2
+
3
b
2
cosh
⁡
k
x
2
k
T
-
1
4
b
4
k
2
+
3
b
2
2
sinh
⁡
k
x
cosh
⁡
k
x
3
k
2
T
2
+
1
24
b
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
3
cosh
⁡
k
x
4
T
3
+
⋯
.The inverse transformation will yields(29)
u
=
-
2
k
2
+
24
k
2
1
+
e
k
x
-
24
k
2
1
+
e
k
x
2
+
2
k
4
k
2
+
3
b
2
sinh
⁡
k
x
cosh
⁡
k
x
3
t
β
Γ
β
+
1
-
1
2
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
2
cosh
⁡
k
x
4
t
2
β
Γ
2
β
+
1
+
1
3
sinh
⁡
k
x
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
cosh
⁡
k
x
5
t
3
β
Γ
3
β
+
1
+
⋯
,
v
=
b
tanh
⁡
k
x
+
1
2
b
4
k
2
+
3
b
2
cosh
⁡
k
x
2
k
t
β
Γ
β
+
1
-
1
4
b
4
k
2
+
3
b
2
2
sinh
⁡
k
x
cosh
⁡
k
x
3
k
2
t
2
β
Γ
2
β
+
1
+
1
24
b
2
cosh
⁡
k
x
2
-
3
4
k
2
+
3
b
2
3
k
3
cosh
⁡
k
x
4
t
3
β
Γ
3
β
+
1
+
⋯
.This solution is convergent to the exact solution(30)
u
x
,
t
=
-
b
2
-
4
k
4
4
k
2
+
2
k
2
tanh
2
⁡
k
x
+
3
b
2
+
4
k
4
2
k
t
,
v
x
,
t
=
b
tanh
⁡
k
x
+
3
b
2
+
4
k
4
2
k
t
.In Figures 2(a)–2(d), we have presented approximate u
x
,
t and v
x
,
t at α
=
1 and exact solutions.Figure 2
(a) Approximate solution. (b) Exact solution. (c) Approximate solution. (d) Exact solution.
(a)
(b)
(c)
(d)


## 5.3. System of Coupled Fractional Sine-Gordon Equations [26, 27]

Coupled Sine-Gordon equations were introduced by Ray et al. [26, 27]. The coupled Sine-Gordon equations generalize the Frenkel-Kontorova dislocation model; see [26, 27] and the references therein.We now consider a system of coupled Sine-Gordon equations [26, 27]:(31)
∂
2
α
u
∂
t
2
α
-
∂
2
u
∂
x
2
=
-
a
2
sin
⁡
u
-
v
,
∂
2
α
v
∂
t
2
α
-
c
2
∂
2
v
∂
x
2
=
sin
⁡
u
-
v
,
0
<
x
,
t
<
π
,
0
<
α
≤
1
,with the initial conditions(32)
u
x
,
0
=
A
cos
⁡
k
x
,
u
t
x
,
0
=
0
,
v
x
,
0
=
0
,
v
t
x
,
0
=
0
,where c is the ratio of the acoustic velocities of the components u and v.Applying the transformation [25] to (31), we get the following partial differential equations:(33)
∂
2
u
∂
T
2
-
∂
2
u
∂
x
2
=
-
a
2
sin
⁡
u
-
v
,
∂
2
v
∂
T
2
-
c
2
∂
2
v
∂
x
2
=
sin
⁡
u
-
v
.Applying the differential transform to (33) and (32), we obtain the following recursive formula:(34)
k
+
2
!
k
!
U
k
+
2
x
=
∂
2
U
k
x
∂
x
2
-
a
2
N
k
x
,
k
+
2
!
k
!
V
k
+
2
x
=
c
2
∂
2
U
k
x
∂
x
2
+
N
k
x
.Using the initial condition, we have(35)
U
0
x
=
A
cosh
⁡
k
x
,
U
1
x
=
0
,
V
0
x
=
0
,
V
1
x
=
0
.Now, substituting (35) into (34), and by straightforward iterative steps, yields(36)
U
2
x
=
-
A
k
2
cosh
⁡
k
x
2
-
a
2
sin
⁡
A
cosh
⁡
k
x
2
,
V
2
x
=
sin
⁡
A
cosh
⁡
k
x
2
,
U
3
x
=
0
,
V
3
x
=
0
,
U
4
x
=
A
k
4
cosh
⁡
k
x
24
+
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
-
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cos
2
⁡
k
x
24
+
a
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cos
⁡
k
x
12
+
a
4
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
+
a
2
cos
⁡
A
cosh
⁡
k
x
s
i
n
A
cosh
⁡
k
x
24
,
V
4
x
=
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cosh
2
⁡
k
x
24
-
c
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
-
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
,
⋮and so on.The series solution is given by(37)
u
x
,
T
=
A
cosh
⁡
k
x
-
A
k
2
cosh
⁡
k
x
2
+
a
2
sin
⁡
A
cosh
⁡
k
x
2
T
2
+
A
k
4
cosh
⁡
k
x
24
+
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
-
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cos
2
⁡
k
x
24
+
a
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cos
⁡
k
x
12
+
a
4
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
+
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
T
4
+
⋯
,
v
x
,
T
=
sin
⁡
A
cosh
⁡
k
x
2
T
2
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cosh
2
⁡
k
x
24
-
c
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
-
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
T
3
+
⋯
.The inverse transformation will yield(38)
u
x
,
t
=
A
cosh
⁡
k
x
-
A
k
2
cosh
⁡
k
x
2
+
a
2
sin
⁡
A
cosh
⁡
k
x
2
t
2
α
Γ
2
α
+
1
+
A
k
4
cosh
⁡
k
x
24
+
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
24
-
a
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cos
2
⁡
k
x
24
+
a
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cos
⁡
k
x
12
+
a
4
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
+
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
A
cosh
⁡
k
x
24
t
4
α
Γ
4
α
+
1
+
⋯
,
v
x
,
t
=
sin
⁡
A
cosh
⁡
k
x
2
t
2
α
Γ
2
α
+
1
+
c
2
A
2
k
2
s
i
n
A
cosh
⁡
k
x
24
+
c
2
A
2
k
2
sin
⁡
A
cosh
⁡
k
x
cosh
⁡
2
⁡
k
x
24
-
1
+
c
2
A
k
2
cos
⁡
A
cosh
⁡
k
x
cosh
⁡
k
x
24
-
1
+
a
2
cos
⁡
A
cosh
⁡
k
x
sin
⁡
cosh
⁡
k
x
24
t
4
α
Γ
4
α
+
1
+
⋯
.This solution is convergent to Adomian’s decomposition method solution [26, 27].In Figures3(a) and 3(b) we have presented approximate solutions u
x
,
t and v
x
,
t at α
=
1.Figure 3
(a) Approximate solutionu
x
,
t. (b) Approximate solution v
x
,
t.
(a)
(b)


## 6. Conclusion

In this research, we present new applications of the fractional complex transform method with coupling reduced differential transform method (RDTM) by handling three nonlinear physical fractional dynamical models. This coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique, which does not require linearization, discretization or perturbation, gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool for solving nonlinear equationsand hence may be extended to other nonlinear problems also.



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*Source: 364853-2015-12-06.xml*