On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

Muhammad Sinan,Zareen A Khan,Qasem Al-Mdallal,Fathalla Rihan,Kamal Shah,Ndolane Sene

doi:10.1155/2021/6045722

Abstract

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

Highlights

PDEs have important applications in physics, engineering, and other applied sciences. ey can describe different phenomena and processes of real-world problems
E Kawahara partial differential equation (KPDE) was first suggested by Kawahara [5] in 1972. Since these nonlinear equations need to be solved by using some approximate methods, researchers have solved several nonlinear problems by using homotopy perturbation method (HPM). is method was first proposed by He [6] and has been applied in [7] for the solution of differential equations and integral equations in both linear and nonlinear cases. e said method is a combination of topological homotopy and traditional perturbation methods. e advantage of this method is to provide an analytic approximate solution in applied sciences Journal of Mathematics with a capacious range, and in this method, a small parameter is not necessary for an equation. is method is applied to the system of the nonlinear system of equations as in [8] for the analytic approximate solution for the model of rabies transmission dynamics
Upon the use of the homotopy perturbation method (HPM), we have investigated the Kawahara fractional-order partial differential equation of fifth-order under fractional order

Summary

Introduction

PDEs have important applications in physics, engineering, and other applied sciences. ey can describe different phenomena and processes of real-world problems. We apply a semianalytic HPM to solve the fifth-order KPDEs. As in the last several decades’ investigation, traveling-waves solutions for nonlinear equations played an important role in the study of the nonlinear physical phenomenon [4]. E KPDE was first suggested by Kawahara [5] in 1972 Since these nonlinear equations need to be solved by using some approximate methods, researchers have solved several nonlinear problems by using HPM. Is method is applied to the system of the nonlinear system of equations as in [8] for the analytic approximate solution for the model of rabies transmission dynamics. E mentioned derivative extends order from integer to any real or complex number which provides a detailed explanation to physical problems. We present the solutions graphically and at the end, provide conclusion and discussion

Preliminaries and Notations

A General Algorithm about HPM

Approximate Solution to Considered Problem

Fractional Spatial Numerical Solution

Results and Discussion

Conclusion