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The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.

In recent years, fractional differential equations (FDEs) have become one of the most exciting and extremely active areas of research because of their potential applications in physics and engineering. These include electromagnetic, fluid flow, dynamical process in self-similar and porous structures, electrical networks, probability and statistics control theory of dynamical systems, chemical physics, optics, acoustic, viscoelasticity, electrochemistry, and material science [

There are different kinds of fractional integration and differentiation operators. The most famous one is the Riemann-Liouville definition [

In the literature, there are many effective methods to treat FDEs such as the Adomian decomposition method, the variational iteration method, the homotopy perturbation method, the differential transform method, the finite difference method, the finite element method, the exponential function method [

The moving boundary problem is a nonlinear initial-boundary value problem that requires extra boundary conditions to determine the motion of the boundary [

Burger’s equation is a classical nonlinear differential equation which was firstly introduced by Burger in 1948. It is used as a model for many nonlinear physical phenomena such as acoustics, continuous stochastic processes, dispersive water waves, gas dynamics, heat conduction, longitudinal elastic waves in an isotropic solid, number theory, shock waves, and turbulence [

The rest of this paper is organized as follows: the description of the fractional Riccati expansion method with variable coefficients is presented in Section

In this section, we present the fractional Riccati expansion method with variable coefficients to find exact analytical solutions of nonlinear FDEs. The fractional derivatives are described in sense of the modified Riemann-Liouville derivative defined by Jumarie [

The above properties play an important role in the fractional Riccati expansion method with variable coefficients.

Consider a nonlinear FDE in two variables

Suppose that

If

If

If

If

If

If

If

If

If

Determining the integer

It can be easily found that if

To solve the space-time fractional Burger’s equation (

This transformation maps equation (

In particular, when

Substituting (

From the solutions of (

The remaining solutions can be obtained in a similar manner. When

Thus, the solutions (

Meanwhile, from solutions (

Firstly, we study the features of the solution (

Taking

Note that in the static case when

Evolutional behavior of

In the special case of a linear moving boundary, the solution of the Burger’s equation can be found directly from (

By taking the dependent variable to be

Secondly, we discuss the features of the kink soliton solution (

The initial-boundary conditions have the form

Figure

Evolutional behavior of

When

Considering the transformation

In this paper, the fractional Riccati expansion method with variable coefficients has been successfully applied to the forced space-time fractional Burger’s equation. A number of new analytical solutions have been obtained. Figures

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

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