A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

Shabir Ahmad,Manuel De La Sen,Ali Akgül,Aman Ullah,Alexander Meskhi

doi:10.1155/2021/8770488

Abstract

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

Highlights

Introduction and MotivationNoninteger calculus is a popular field that is aimed at explaining real-world phenomena that are modeled with operators of fractional order
A general procedure for solving nonlinear PDEs described by the CF derivative has been developed
We have demonstrated the estimated solution’s convergence and provided a result for the absolute error calculation

Summary

Introduction and Motivation

Noninteger calculus is a popular field that is aimed at explaining real-world phenomena that are modeled with operators of fractional order. Caputo and Fabrizio (CF) [12] suggested a new fractional operator with an exponential kernel in a recent attempt around the middle of the last decade This derivative’s kernel is nonsingular, so the results are more reasonable than the classical one. Differential equations of fractional order are notoriously difficult to solve, and finding an exact solution is even more difficult. Elzaki introduced new integral transform “Elzaki Transform” and used heavily in solving partial differential equations [18]. The main purpose of this article is to apply newly introduced integral transform called “Yang Transform” discovered by Yang [24] with HPM to solve nonlinear fractional order PDEs. We solve two popular nonlinear PDEs through the proposed method. There is no requirement for a method like discretizing the problem and no linearization for the nonlinear problem, and just a few iterations can lead to a solution that can be estimated using these techniques

Preliminaries

Main Work

F χðvÞ: ð11Þ v

Algorithm of the Proposed Method

Conclusion