Numerical Investigation of Fractional-Order Kawahara and Modified Kawahara Equations by a Semianalytical Method

Weaam Alhejaili,Joshua Kiddy Kwasi Asamoah,Aatif Ali,Laiq Zada,Sharifah E Alhazmi,Rashid Nawaz,Anwar Saeed

doi:10.1155/2022/1985572

Abstract

In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractional-order Kawahara and modified Kawahara equations. The method convergence is controlled by a flexible function known as the auxiliary function. The values of the unknown arbitrary constants in the auxiliary function are computed using the Caputo derivative fractional-order and the well-known approach of least squares. Fractional-order derivatives are taken in the Caputo sense with numerical values in the closed interval 0 , 1 . The suggested method is directly applied to fractional-order Kawahara and modified Kawahara equations, with no need for small or large parameter assumptions. The numerical results obtained by the proposed method are compared to the new iterative method (NIM). Results reveal that the proposed method converges faster to the exact solution than other methods in the literature.

Highlights

Fractional computation was established as an important subject of mathematics in 1695
We assume that the time-fractional Kawahara equation is given in [35]: Table 4: Comparison absolute errors of 2nd-order optimal homotopy asymptotic method (OHAM) solution with 3rd-order new iterative method (NIM) solution for time-fractional Kawahara equations for different values of α
Figure 1: 3D surface obtained by OHAM solution for fractional Kawahara equation at α = 0:5

Summary

Introduction

Fractional computation was established as an important subject of mathematics in 1695. Very few researchers have drawn on the successful use of fractional systems in these fields to examine their mathematical approximation methods, since diagnostic frameworks are usually difficult to obtain. A variety of real-world problems can be modeled using fractional-order differential equations. These equations have many applications in fluid mechanics, electromagnetic theory, electric grids, diffuse transport, groundwater problems, biological sciences, etc. The exact solution for nonlinear problems is very hard to obtain, and an alternative way is to find the approximate solution. We extend the well-known optimal homotopy asymptotic method (OHAM) to fractional-order Kawahara and modified Kawahara equations

Methods

Results

Conclusion