Applied Mathematical Sciences, Vol. 8, 2014, no. 44, 2195 - 2210

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.4285

Solution of Nonlinear Fractional Differential

Equations Using the Homotopy Perturbation

Sumudu Transform Method

Eltayeb A. Yousif

Department of Applied Mathematics, Faculty of Mathematical Sciences,

University of Khartoum, Khartoum 11111, Sudan

Department of Mathematics, Faculty of Science, Northern Border University,

Arar 91431, Saudi Arabia

Sara H. M. Hamed

Department of Mathematics, Faculty of Mathematical Sciences and Statistics,

Alneelain University, Khartoum 11121, Sudan

Copyright © 2014 Eltayeb A. Yousif

and Sara H. M. Hamed. This is an open access article

distributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we obtain exact analytical solutions of nonlinear fractional

differential equations using a combined form of the Homotopy perturbation

method with the Sumudu transform. The solutions are given in closed forms in

terms of Mittage-Leffler functions. The fractional derivatives are considered in

Caputo sense. The method is illustrated through a number of test examples.

Keywords: Homotopy Perturbation Method, Sumudu Transform, Nonlinear

Fractional Differential Equations, Mittage-Leffler functions

2196 Eltayeb A. Yousif and Sara H. M. Hamed

1. Introduction

Fractional calculus is a generalization of differentiation and integration to

non-integer orders. Many problems in physics and engineering are modulated in

terms of fractional differential and integral equations, such as acoustics, diffusion,

signal processing, electrochemistry, and may other physical phenomena [14,26].

During last decades, a great deal of interest appears in fractional differential

equations. The solutions of fractional equations are investigated by many authors

using powerful methods in obtaining exact and approximate solutions

[1,25,30-32,35,36].

The Homotopy perturbation method (HPM) is proposed by He in 1999 [17]. This

method is a coupling of traditional perturbation method and homotopy in topology.

Later on He himself drawn many modifications and developments of the method

[17-22]. In recent years Homotopy perturbation method has been extensively

introduced by numerous authors, and implemented to obtain exact and

approximate analytical solutions to a wide range of both linear and nonlinear

problems in science and engineering [4,6,15,16,20,22,23,30,32].

Watugala in 1993 [10] introduced a new integral transform and named it as

Sumudu transform, used it in obtaining the solution of ordinary differential

equations in control engineering problems. Asiru [28] implemented the Sumudu

transform for solving integral equations of convolution type. Belgacem et al

[18,19] presented the fundamental properties of Sumudu transform. Kilicman and

Eltayeb [2,3,12] investigated various types of problems via Sumudu transform,

including ordinary and partial differential equations. Gupta and Bhavna [37] used

Sumudu transform in determining the solution of reaction-diffusion equation.

Rana et al [29] applied He's homotopy perturbation method to compute Sumudu

transform. Several authors [1,31,34-36] have discussed many fractional partial

differential equations using Sumudu transform. The Homotopy perturbation

Sumudu transform method [7,11,24,25,29,33] is applied to solve many problems,

for example, nonlinear equations, heat and wave-like equations.

Solution of nonlinear fractional differential equations 2197

In this paper the authors implemented the Homotopy perturbation Sumudu

transform method (HPSTM) to evaluate the exact analytical solution of nonlinear

fractional partial differential equations.

This work is organized as follows: In section 2 we provide some preliminaries.

Section 3 introduces the concept of Homotopy perturbation method, while section 4

gives the Sumudu transform. The Homotopy perturbation Sumudu transform

method (HPSTM) is analyzed in section 5. Numerical examples are provided in

section 6. The conclusions are given in section 7.

2. Preliminaries

Definition (2.1): The Caputo fractional derivative of order of a function

( ) is defined by [14,26]

( )

{

( )∫( ) ( )( )

( )

( )

Where is called the Caputo derivative operator.

Definition (2.2): The Mittag-Leffler function with two parameters, is defined by

[13,14,26,27]

( ) ∑

( )

( )

The following results are obtained directly from definition ( ):

( ) ( ) ( ) (

) ( )

( ) ( )

( )

( ) (

)

[ ( )]

} ( )

Note (2.3): The special type of Mittag-Leffler function ( √ ) is given by

[13]

( √ )

( √ ) ( )

By using (4), we have to drive

2198 Eltayeb A. Yousif and Sara H. M. Hamed

( √ )

( √ )

√ ( )

where is a complementary error function. These two functions are used

further in this paper. The derivation of formula (5) based on definition of

Mittage-Leffler function and formula (4), we have

( √ ) ∑

( √ )

( )

( )

Replacing with in the RHS of ( ), we get

( √ ) ∑

( √ )

( )

(∑( √ )

( )

√

( ))

( ( √ )

√ )

Substituting ( √ )

( √ ) , then we get the result.

3. Homotopy Perturbation Method

To illustrate the concept of Homotopy perturbation method, we consider the

nonlinear differential equation:

( ) ( ) ( ) ( )

with the boundary conditions:

(

) ( )

Where is a linear operator, is nonlinear operator, is boundary operator,

is the boundary of the domain and ( ) is a known analytic function.

The He’s homotopy perturbation technique [8-10] defines the Homotopy

( ) [ ] which satisfies:

( ) ( )[ ( ) ( )] [ ( ) ( ) ( )] (9)

or

( ) ( ) ( ) ( ) [ ( ) ( )] ( )

Solution of nonlinear fractional differential equations 2199

Where and [ ] is an impending parameter, is an initial

approximation which satisfies the boundary condition. The basic assumption is that

the solution of equation (9) and equation (10) can be expressed as power series in

as follows:

( )

The approximate solution of equation (7) is given by

( )

4. Sumudu Transform

Consider functions in the set A, that defined by:

{ ( ) | ( )| | | ( ) [ )}

where is a constant must be finite, need not simultaneously exist,

and each may be finite. The Sumudu transform is defined by [10]

( ) ( ( )) ∫ ( )

( ) ( )

Definition (4.1): The Sumudu transform of fractional order derivative, is defined

by [25,34]

[ ( )]

[ ( )] ∑

[ ( )( )]

( )

5. Analysis of the method

In this section we need to illustrate the concept and construction of Homotopy

perturbation Sumudu transforms method (HPSTM) for fractional equations, that by

considering the general nonlinear nonhomogenous time-fractional partial

differential equation with the initial conditions to by:

( ) ( ) ( ) ( )

( )

( )

} ( )

2200 Eltayeb A. Yousif and Sara H. M. Hamed

Applying Sumudu transform on both sides of (15) with respect to t, we get

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

( ( )) ∑

( )

( ( ) ( )) ( ( ))

( ( )) ∑

( ) ( ( ) ( ) ( ))

Taking the inverse Sumudu transform to the above result, we have

( ) ∑ ( )

( )

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

Application of the Homotopy perturbation method to (16), yields

( )( ( ) ( ))

( ( ) ∑ ( )

( )

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

( ) ( ) ∑ ( )

( )

( ( ( ( ))

( ( ) ( )))) ( )

Let:

( ) ∑ ( )

( ) ( ) ( ) ∑ ( ( ))

( )

Solution of nonlinear fractional differential equations 2201

( ( ))

( ∑

)

Substituting (18) into equation (17), we get

∑ ( )

( ) ∑ ( )

( )

( ( ( ( ))

( ∑ ( )

∑ ( ( ))

))) ( )

( ) ( )

( ) ∑ ( )

( )

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

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

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

The solution of equation (15) is given by

∑

( )

6. Numerical examples

Example (1): Consider the nonlinear nonhomogenous time-fractional invicid

Burgers equation

( ) ( ) ( )

( ) } ( )

Solution: By applying (HPSTM) to equation (21), then from (19), we have

2202 Eltayeb A. Yousif and Sara H. M. Hamed

∑ ( )

( ( ( ) (∑ ( ( ))

)))

(

( )( )

( )

( ( (∑ ( ( ))

)))

)

( )

( )

( )( )

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

( )( )

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

( )

( )

( ) ( ( ( ( ))))

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

( ) (

( )

( ))

( ) ( ( ( ( ))))

( ( ( ) ( ) ( ) ( )

( ) ( )))

( )

( )

( )

Solution of nonlinear fractional differential equations 2203

( )

( )

( )

( )

( )

( )

( )

(

( )

( )

( ) )

(

( )

( )

( ) )

∑( )

( )

∑( )

( )

( ) (

)

Remark (1): As special case if we take

then from (4) and (5), we have

( ) (

) (

)

(√ ) √

.

Remark (2): If then, ( ) ( ) ( ) .

This agrees with the solution obtained by Wazwaz [5].

Example (2): Consider the following nonlinear time-fractional equation

( ) ( ) ( )

( ) ( ) } ( )

Solution: By applying (19) in (22), we get

∑ ( )

( ( ( ∑ ( )

∑ ( ( ))

)))

( )

2204 Eltayeb A. Yousif and Sara H. M. Hamed

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

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

( )

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

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

( )

( )

( )

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

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

( ) ( )))

( )

( )

( )

( )

( )

( )

( )

( )

( )

(

( )

( )

( ) )

(

( )

( ) )

∑

( )

∑

( )

( ) (

)

Remark (3): If then:

( ) ( ) (

) .

Example (3): Consider the time-fractional fifth order KdV equation

Solution of nonlinear fractional differential equations 2205

( )

( ) } ( )

Solution: Application of (19) into (23), yields

∑ ( )

( ( (∑ ( )

∑ ( ( ))

)))

( )

( ) ( ( (

( )

( ( )))))

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

( ( ( ( ))))

( )

( ) ( ( (

( )

( ( )))))

( ( (

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

( )

( ) ( ( (

( )

( ( )))))

2206 Eltayeb A. Yousif and Sara H. M. Hamed

( ( (

( ) ( ( ) ( ))

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

( )

( )

( )

( )

( ) (

)

Remark (4): If

, then from ( ), we have:

( ) ( √ ) (√ )

Remark (5): If then we get the solution of the classical equation as

( ) .

7. Conclusion

In the present paper, we applied the Homotopy perturbation Sumudu transform

method (HPSTM) for solving fractional nonlinear partial differential equations.

The time derivatives are considered in Caputo sense. Solutions are determined in

a compact form in terms of Mittag-Leffler functions. The terms are obtained in a

simplified way and straightforward. The method was tested on three different

problems. This method is powerful, reliable and effective, easy to implement.

Thus, this technique can be applied to solve many nonlinear problems in applied

science.

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Received: February 11, 2014