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The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

Fractional differential equations (FDEs) are viewed as alternative models to nonlinear differential equations. Varieties of them play important roles and serve as tools not only in mathematics but also in physics, biology, fluid flow, signal processing, control theory, systems identification, and fractional dynamics to create the mathematical modeling of many nonlinear phenomena. Besides, they are employed in social sciences such as food supplement, climate, finance, and economics. Oldham and Spanier first considered the fractional differential equations arising in diffusion problems [

In recent decades, some effective methods for fractional calculus appeared in open literature, such as the exp-function method [

The fractional complex transform [

This paper is organized as follows. In Section

In the last few decades, in order to improve the local behavior of fractional types, a few local versions of fractional derivatives have been proposed, that is, Caputo’s fractional derivative [

Assume that

Some useful formulas include

Let

We consider the following nonlinear FDE of the type

Li and He [

According to exp-function method, which was developed by He and Wu [

This equivalent formulation plays a significant and fundamental part for finding the exact solution of mathematical problems. To determine the values of

In the following sections, we present three examples to illustrate the applicability of the exp-function method and fractional complex transform to solve nonlinear fractional differential equations.

We consider a time fractional biological population model of the form [

For our goal, we present the following transformation:

Then by the use of (

Balancing the order of

For simplicity, we set

Substituting (

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

Consider

Consider

We consider the one-dimensional time fractional Burgers equation with the value problem [

For our purpose, we introduce the following transformations:

Substituting (

Integrating (

By the same procedure as illustrated in Section

For simplicity, we set

Substituting (

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

Consider

Consider

We consider the space-time fractional Cahn-Hilliard equation [

Firstly, we consider the following transformations:

Substituting (

Integrating (

Here take notice of nonlinear term in (

Substituting (

From (

In this paper, we have successfully developed fractional complex transform with the help of exp-function method to obtain exact solution of some fractional differential equations. The fractional complex transform and exp-function methods are extremely simple but effective and powerful for solving fractional differential equations. These methods are accessible to solve other similar nonlinear equations in fractional calculus. To our knowledge, these new solutions have not been reported in former literature; they may be of significant importance for the explanation of some special physical phenomena.

The authors declare that there is no conflict of interests in this paper.