By introducing the fractional derivative in the sense of Caputo and combining the pretreatment technique to deal with long nonlinear items, the generalized two-dimensional differential transform method is proposed for solving the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. The presented method is a numerical method based on the generalized Taylor series expansion which constructs an analytical solution in the form of a polynomial. The numerical results show that the generalized two-dimensional differential transform method is very effective for the fractional coupled equations.

In the last past decade, the fractional differential equations have been widely used in various fields of physics and engineering. The analytical approximation of such problems has attracted great attention and became a considbased onerable interest in mathematical physics. Some powerful methods including the homotopy perturbation method [

The variational iteration method and the homotopy perturbation method were first proposed by Professor He in [

The differential transform method was used firstly by Zhou in 1986 to study electric circuits [

The aim of this paper is to directly extend the generalized two-dimensional differential transform method to obtain the approximate analytic solutions of a time-fractional Hirota-Satsuma coupled KdV equation,

The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem.

The fractional derivative of

For

The Caputo fractional derivative of the power function satisfies

Consider a function of two variables

The fundamental theorems of the generalized two-dimensional differential transform are as follows.

Suppose that

if

if

if

if

if

If

Some details of the aformentioned theorems can be found in [

Consider the following time-fractional Hirota-Satsuma coupled KdV equation:

The exact solutions of (

Suppose that the solutions

The generalized two-dimensional differential transforms of the initial conditions can be obtained as follows:

Utilizing the recurrence relations (

The closed forms of

Similarly, substituting all

The closed forms of

The closed forms of

The approximate solutions of (

In order to verify whether the approximate solutions of (

The surface shows the solution

In the following, we will construct an approximate solution of (

The generalized two-dimensional differential transforms of the initial conditions of (

Utilizing the recurrence relations in (

Consider the following time-fractional coupled MKdV equation:

The exact solutions of (

Equation system of (

According to (

Suppose that the solutions

The generalized two-dimensional differential transforms of the initial conditions of (

Utilizing the recurrence relations of (

The effectiveness and accuracy of the approximate solutions can be seen from the comparison figures.

Figures

The surface shows the solution

The surface shows the solution

The surface shows the exact solutions (

The surface shows the approximate solutions of (

In this paper, combining the Caputo fractional derivative, the GDTM was applied to derive approximate analytical solutions of the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation with initial conditions. The numerical solutions obtained from the GDTM are shown graphically. The obtained results demonstrate the reliability of the algorithm and its wider applicability to nonlinear fractional coupled partial differential equations.

In [

Setting

The project is supported by the NNSF of China (11061021), the NSF-IMU of China (2012MS0105, 2012MS0106), and the YSF-IMU of China (ND0811).