Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equation

Zhao Li,Chun Huang,Tianyong Han

doi:10.1063/5.0029159

Abstract

In this paper, the dynamical behavior of a time-space fractional Phi-4 equation is investigated by utilizing the bifurcation method of a planar dynamical system. Under the given parameter conditions, phase portraits and bifurcations are obtained with the help of the mathematical software Maple. Moreover, some new exact traveling wave solutions are obtained, such as Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, kink solitary wave solutions, and periodic wave solutions.

Highlights

It is well known that the fractional partial differential equation (FPDE) is broadly used to describe various complex phenomena in different scientific research areas and engineering applications, especially in the field of physics
Some literature studies1–9 have reported that the exact traveling wave solutions10–12 can help us well understand natural phenomena in the real world and accurately indicate the dynamical processes modeled by the FPDE
II, we give the definition of a conformable fractional derivative and introduce an effective method of constructing the exact traveling wave solution to the time-space FPDE

Summary

INTRODUCTION

It is well known that the fractional partial differential equation (FPDE) is broadly used to describe various complex phenomena in different scientific research areas and engineering applications, especially in the field of physics. Many effective methods to construct exact traveling wave solutions of the FPDE have been established and developed. In Ref. 26, Sirisubtawee et al employed the (G′/G, 1/G)-expansion method to obtain the exact solutions of the space–time fractional Phi-4 equation in the sense of the conformable fractional derivative. In Ref. 28, Tariq and Akram developed the tanh method to establish exact solutions of the time fractional Phi-4 equation by means of Jumarie’s modified Riemann–Liouville derivative. This article is arranged as follows: In Sec. II, we give the definition of a conformable fractional derivative and introduce an effective method of constructing the exact traveling wave solution to the time-space FPDE.

Brief introductions of the bifurcation method for the time-space FPDE

APPLICATIONS

CONCLUSION