Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

Supaporn Kaewta,Sekson Sirisubtawee,Surattana Sungnul,Sanoe Koonprasert

doi:10.3390/fractalfract5030088

Abstract

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

Highlights

Nonlinear evolution equations (NLEEs), which are interpreted as the differential law of the development in time of a system and typically expressed in terms of nonlinear partial differential equations (NPDEs), can be utilized to describe many interesting and sophisticated phenomena in physics, mathematical physics, engineering, and other various scientific fields such as fluid mechanics [1,2], plasma physics [3], quantum mechanics [4], biology [5], nonlinear wave theory [6], and fiber optics [7]
The modified Korteweg de Vries (KdV)–Zakharov–Kuznetsov (mKdV–ZK) equation plays a significant role in explaining dynamics of many branches of physics such as plasma physics, nonlinear optics, fluid dynamics, shallow water waves in oceanography, quantum mechanics and mathematical physics so that fundamental properties of nonlinear propagation for such various physical phenomena are analyzed [41,43,44,45]
All of the exact solutions obtained in this paper were substituted back into their corresponding equations with the help of the Maple package program and their satisfactions confirm the validity of the solutions expressed in the current article

Summary

Introduction

Nonlinear evolution equations (NLEEs), which are interpreted as the differential law of the development in time of a system and typically expressed in terms of nonlinear partial differential equations (NPDEs), can be utilized to describe many interesting and sophisticated phenomena in physics, mathematical physics, engineering, and other various scientific fields such as fluid mechanics [1,2], plasma physics [3], quantum mechanics [4], biology [5], nonlinear wave theory [6], and fiber optics [7]. In 2020, the ( G 0 /G2 )-expansion approach was employed to construct some novel exact traveling wave solutions of the (2 + 1)-dimensional Boiti–Leon–Pempinelli system [28]. The mKdV–ZK equation plays a significant role in explaining dynamics of many branches of physics such as plasma physics, nonlinear optics, fluid dynamics, shallow water waves in oceanography, quantum mechanics and mathematical physics so that fundamental properties of nonlinear propagation for such various physical phenomena are analyzed [41,43,44,45]. Some scientists have devoted substantial efforts to finding exact solutions of the (3 + 1)-dimensional mKdV–ZK equation in the sense of the classical partial, conformable, and Jumarie’s modified Riemann–Liouville derivatives using different and reliable approaches.

Conformable Derivative and Its Properties

Result

Graphical Representations of the Selected Solutions

Conclusions and Future Work