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

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.

[1]  Aly R. Seadawy,et al.  The Solutions of the Boussinesq and Generalized Fifth-Order KdV Equations by Using the Direct Algebraic Method , 2012 .

[2]  K. Schmidt,et al.  On the homogenization of the Helmholtz problem with thin perforated walls of finite length , 2016, 1611.06001.

[3]  An efficient approach for solution of fractional-order Helmholtz equations , 2021 .

[4]  N. Herisanu,et al.  Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer , 2008 .

[5]  N. Herisanu,et al.  Optimal Homotopy Perturbation Method for a Non-Conservative Dynamical System of a Rotating Electrical Machine , 2012 .

[6]  I. Hashim,et al.  Analytical treatment of two-dimensional fractional Helmholtz equations , 2019, Journal of King Saud University - Science.

[7]  Hammad Khalil,et al.  Analytical Solutions of Fractional Order Diffusion Equations by Natural Transform Method , 2018 .

[8]  Ishak Hashim,et al.  Solving the generalized Burgers-Huxley equation using the Adomian decomposition method , 2006, Math. Comput. Model..

[9]  M. Arif,et al.  Numerical treatment of fractional order Cauchy reaction diffusion equations , 2017 .

[10]  Faranak Rabiei,et al.  A Semianalytical Approach to the Solution of Time-Fractional Navier-Stokes Equation , 2021, Advances in Mathematical Physics.

[11]  D. Lu,et al.  Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov–Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma , 2016 .

[12]  K. Shah,et al.  Computation of solution to fractional order partial reaction diffusion equations , 2020, Journal of advanced research.

[13]  Computation of iterative solutions along with stability analysis to a coupled system of fractional order differential equations , 2019, Advances in Difference Equations.

[14]  M. Arif,et al.  Approximate solutions to nonlinear factional order partial differential equations arising in ion-acoustic waves , 2019, AIMS Mathematics.

[15]  Vasile Marinca,et al.  On the flow of a Walters-type B’ viscoelastic fluid in a vertical channel with porous wall , 2014 .

[16]  A. Atangana,et al.  A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions , 2013 .

[17]  Nicolae Herisanu,et al.  Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust , 2019, Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems.

[18]  K. Shah,et al.  On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations , 2018, Mathematical Methods in the Applied Sciences.

[19]  Samad Noeiaghdam,et al.  A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection , 2021, Mathematics.

[20]  M. Arif,et al.  Stable monotone iterative solutions to a class of boundary value problems of nonlinear fractional order differential equations , 2019, Journal of Nonlinear Sciences and Applications.