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We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

Fractional differential equations (FDEs) are generalizations of classical differential equations of integer order. Recently, fractional differential equations have gained much attention as they are widely used to describe various complex phenomena in various applications such as the fluid flow, signal processing, control theory, systems identification, finance and fractional dynamics, and physics. The fractional differential equations have been investigated by many researchers [

He and Wu [

The present paper investigates for the first time the applicability and effectiveness of the Exp-function method on fractional nonlinear partial differential equations.

Jumarie proposed a modified Riemann-Liouville derivative. With this kind of fractional derivative and some useful formulas, we can convert fractional differential equations into integer-order differential equations by variable transformation in [

In this section, we firstly give some properties and definitions of the modified Riemann-Liouville derivative which are used further in this paper.

Assume that

A few properties of the fractional modified Riemann-Liouville derivative were summarized and three famous formulas of them are

Secondly, let us consider the time fractional differential equation with independent variables

Using the fractional variable transformation

Next, using the fractional variable transformation

The fractional differential equation (

We consider the general nonlinear ordinary differential equation in (

This equivalent formulation plays an important and fundamental part for finding the analytic solution of problems. To determine the value of

We suppose that the solution in (

Suppose that

To show the efficiency of the method described in the previous part, we present some FDEs examples.

We consider the nonlinear fractional Sharma-Tasso-Olver equation [

For our purpose, we introduce the following transformations:

Substituting (

Integrating (

Here take notice of the nonlinear term in (

For simplicity, we set

Substituting (

Solving this system of algebraic equations by using symbolic computation, we obtain the following results.

We have

If we set

If we set

We have

Comparing our results with the results [

We consider the space fractional Burgers equation [

For our purpose, we introduce the following transformations:

Substituting (

Integrating (

Here take notice of the nonlinear term in (

For simplicity, we set

Substituting (

Solving this system of algebraic equations by using symbolic computation, we obtain the following results.

We have

We have

We have

The obtained solutions for the space fractional Burgers equation are new to our best knowledge.

We consider the following fractional time fractional fmKdV equation [

For our purpose, we introduce the following transformations

Substituting (

By using the ansatz (

Substituting (

We have

We have

If we set

We have

We have

If we set

We have

If we take

The established solutions have been checked by putting them back into the original equation (

In this paper, we use the Exp-function method to calculate the exact solutions for the time and space fractional nonlinear partial differential equations. When the parameters take certain values, the solitary wave solutions are derived from the exponential form. Since this method is very efficient, reliable, simple, and powerful in finding the exact solutions for the nonlinear fractional differential equations, the proposed method can be extended to solve many systems of nonlinear fractional partial differential equations. We hope that the present solutions may be useful in further numerical analysis and these results are going to be very useful in further future research.

The authors declare that there is no conflict of interests regarding the publication of this paper.