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In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

Fractional differential equations are the generalizations of classical differential equations with integer order derivatives. It is well known that fractional partial differential equations have proved to be very useful in many research fields such as physics, mathematical biology, engineering, fluid mechanics, plasma physics, optical fibers, neural physics, solid state physics, viscoelasticity, electromagnetism, electrochemistry, signal processing, control theory, chaos, finance, and fractal dynamics. In the last few decades, research on various aspects for fractional partial differential equations has received more and more attention by many authors, such as the oscillation [

In this paper, we develop an auxiliary equation method for solving fractional partial differential equations, where the fractional derivative is defined in the sense of the conformable fractional derivative. Our aim is to seek exact solutions with variable coefficient function forms for some certain fractional partial differential equations.

The conformable fractional derivative is defined by [

The paper is organized as follows. In Section

In this section, we give the description of the auxiliary equation method for solving fractional partial differential equations.

Suppose that a fractional partial differential equation in the independent variables

For those variables involving fractional derivatives, fulfil corresponding fractional transformations so that the fractional partial derivatives can be converted into integer order partial derivatives with respect to new variables.

Taking the expressions

Suppose that the solution of (

Substituting (

Solving the equations yielded in Step

In the present method, the expression of the transformation denoted by

As the partial differential equations yielded in Step

We consider the time fractional two-dimensional Boussinesq equation with the following form:

The fractional Boussinesq equation is used in the analysis of long waves in shallow water and also used in the analysis of many other physical applications, such as the percolation of water in a porous subsurface of a horizontal layer of material. Tasbozan et al. [

Now we use the proposed auxiliary equation method to solve (

Next we will discuss two cases, in which

Substituting (

It is obvious that if we let

On the other hand, it is well known that the solution of (

By a combination of the results denoted in Families 1–5 and (

Similarly, by a combination of the results in Families 1–5 and (

In Figure

The solitary wave solution

Substituting (

For the solutions of (

Substituting the results denoted by Families 1–4 into (

If we set

By a combination of the results in Families 1–4 and (

We consider the space-time fractional (2+1)-dimensional breaking soliton equation [

Now we solve (

Let

If

By a combination of (

By a combination of (

If

By a combination of the results above and (

Similarly, by a combination of (

Based on the properties of conformable fractional calculus, we have proposed an auxiliary equation method for seeking exact solutions with variable coefficient function forms for fractional partial differential equations and applied it to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for them have been successfully found. These solutions are new exact solutions so far in the literature to the best of our knowledge. We note that, by a combination of other auxiliary equations different from the two equations used here, more exact solutions can be found subsequently.

The data used to support the findings of this study are included within the article.

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

This work was partially supported by Natural Science Foundation of China (11671227) and the development supporting plan for young teachers in Shandong University of Technology.