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An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.

In recent years, nonlinear fractional differential equations (NFDEs) have been attracted great interest. It is caused by both the development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, engineering, and biology [

Recently, He [

The present paper is motivated by the desire to improve the work made in [

The rest of this paper is organized as follows. In Section

The Jumarie's modified Riemann-Liouville derivative is defined as

Some properties for the proposed modified Riemann-Liouville derivative are listed in [

In the above formulas (

We present the main steps of the extended fractional sub-equation method as follows.

For a given NFDEs with independent variables

By means of the traveling wave transformation,

We suppose that (

Here

Substituting (

Using the results obtained in the above steps and the solutions of (

As we know, the choice of an appropriate ansätz is very important when using the direct method to look for exact solutions. It can be easily found that the transformation (

We first consider the space-time fractional coupled Burgers' equations [

Using the traveling wave transformations

According to the method described in Section

Substituting (

Consider

Consider

Consider

From Case

Consider

Consider

Consider

Consider

From Cases

We next consider the following space-time fractional coupled MKdV equations [

Using the traveling wave transformations

According to the method described in Section

Substituting (

Consider

Consider

Consider

From Case

Consider

Consider

Consider

Consider

From Cases

It should be noted that when using ansätz (

It seems that the Exp-function method [

In this paper, based on a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions, we propose a new method called extended fractional sub-equation method to construct exact solutions of fractional differential equations. In order to illustrate the validity and advantages of the algorithm, we apply it to space-time fractional coupled Burgers' equations and coupled MKdV equations. As a result, many exact solutions are obtained. The results show that the extended fractional sub-equation method is direct, effective, and can be used for many other fractional differential equations in mathematical physics.

This work is supported by the NSF of China (nos. 10971166, 11171269, 61163027).