Abstract

The objective of this present paper is to utilize an auxiliary equation method for constructing exact solutions associated with variable coefficient function forms for certain nonlinear partial differential equations (NPDEs) in the sense of the conformable derivative. Utilizing the specific fractional transformations, the conformable derivatives appearing in the original equation can be converted into integer order derivatives with respect to new variables. As for applications of the method, we particularly obtain variable coefficient exact solutions for the conformable time (2 + 1)-dimensional Kadomtsev–Petviashvili equation and the conformable space-time (2 + 1)-dimensional Boussinesq equation. As a result, the obtained exact solutions for the equations are solitary wave solutions including a soliton solitary wave solution and a bell-shaped solitary wave solution. The advantage of the used method beyond other existing methods is that it provides variable coefficient exact solutions covering constant coefficient ones. In consequence, the auxiliary equation method based on setting all coefficients of an exact solution as variable function forms can be more extensively used, straightforward and trustworthy for solving the conformable NPDEs.

Highlights

  • Over the last few decades, many researchers have thoroughly investigated many methods for finding exact solutions of nonlinear partial differential equations (NPDEs). This is because the exact solutions play a crucial role in exactly describing physical phenomena such as nonlinear wave spreads in incompressible fluid, electromagnetic field, shallow water waves, epidemic diseases and water pollutant modeled by certain NPDEs [1,2,3,4,5]

  • This paper addresses the use of an auxiliary equation method used to generate variable coefficient exact solutions for a particular type of NPDEs

  • The brief algorithm of an auxiliary equation method for solving a nonlinear conformable PDE for an exact solution, in which its coefficients are in the form of variable functions, comprises the following steps

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Summary

Introduction

Over the last few decades, many researchers have thoroughly investigated many methods for finding exact solutions of nonlinear partial differential equations (NPDEs). This is because the exact solutions play a crucial role in exactly describing physical phenomena such as nonlinear wave spreads in incompressible fluid, electromagnetic field, shallow water waves, epidemic diseases and water pollutant modeled by certain NPDEs [1,2,3,4,5]. Symbolic computer software packages (e.g., Mathematica and Maple) have been continuously and progressively developed for several years According to these reasons, it brings a vital challenge for solving NPDEs to scholars’ attention

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