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Based on a fractional subequation and the properties of the modified Riemann-Liouville fractional derivative, we propose a new analytical method named extended fractional

The nonlinear fractional partial differential equations (FPDEs) have attracted much attention because of their potential applications in various fields of science, such as fluid mechanics (especially in viscoelastic flow and theory of viscoplasticity), medical (human tissue under mechanical load model), electrical engineering (ultrasound transmission), biochemistry (polymer and protein model), material diffusion (including normal diffusion and anomalous diffusion), signal processing, and control systems.

The exact solutions of FPDEs can facilitate illustrating the structural information about the complex physics phenomena and help better understand the physical interpretation. Thus, it is an important and significant task to find more exact solutions of different forms for the FPDEs. In recent decades, many mathematicians and physicists have made significant achievements and also presented some effective methods, for example, the fractional subequation method [

Searching for exact solutions of nonlinear ODEs plays an important role in the study of physical phenomena and gradually becomes one of the most important and significant tasks. In the past several decades, both mathematicians and physicists have made many significant works in this direction and presented some effective methods, such as the global error minimization method [

Recently, Feng [

In order to get as many results as possible, we propose a new analytical method named extended fractional

The organization of the paper is as follows. In Section

Jumarie’s modified Riemann-Liouville derivative of order

The property for the proposed modified Riemann-Liouville derivatives is listed in [

The above equations play an important role in fractional calculus in the following sections.

In this section, the main steps of the extended fractional

Suppose that a fractional partial differential equation in the variables

By using the traveling wave transformation,

We suppose the solution

Substituting (

Solving the equations system in Step

In order to obtain the general expressions for

Then by use of (

By the general expressions for

When

When

When

When

When

We consider a space-time fractional Fokas equation [

To solve (

Balancing

Substituting (

Substituting (

(

(

(

(

(

Substituting (

(

(

(

(

(

In particular, if

(

(

(

(

In this section, some typical wave figures are given as Figures

2D and 3D figures of solution

2D and 3D figures of solution

Finding exact traveling wave solutions of FPDEs is an important and difficult work. Based on Feng’s [

The authors declare that there are no conflicts of interest regarding the publication of this article.

This work was supported by the National Natural Science Foundation of China (11361023), the Natural Science Foundation of Yunnan Province (2013FZ117), and the Middle-Aged Academic Backbone of Honghe University (2014GG0105).