Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

Haji Gul,Kamal Shah,Saowaluck Chasreechai,Sajjad Ali,Shakoor Muhammad,Thanin Sitthiwirattham

doi:10.3390/sym13112215

Abstract

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

Highlights

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.In recent years, fractional calculus has made a great contribution to the fields of science and engineering due to its many applications in the fields of damping visco elasticity, biology, electronics, genetic algorithms, signal processing, robotic technology, traffic systems, telecommunication, chemistry, physics, and economics and finance
We introduce and apply the asymptotic homotopy perturbation method (AHPM) to obtain the approximate solution of fractional order Helmholtz Equations (1) and (2)
All the figures show that the symmetry or asymmetry of the solutions of the original problems is invariant by using the homotopy deformation Equation (10)

Summary

Introduction

Fractional calculus has made a great contribution to the fields of science and engineering due to its many applications in the fields of damping visco elasticity, biology, electronics, genetic algorithms, signal processing, robotic technology, traffic systems, telecommunication, chemistry, physics, and economics and finance. This has all been possible due to such mathematicians as Riemann, Liouville, Leibniz, Euler, Bernoulli, Wallis, and L’ Hospital, who played an important role in the development of fractional calculus. Our research focuses on the following fractional order equations of Helmholtz, which are important in fractional calculus.

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Methods

Conclusion