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

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.

Prospective Analysis of Time-Fractional Emden–Fowler Model Using Elzaki Transform Homotopy Perturbation Method

“…The homotopy perturbation method (HPM) is one of the particularly prominent semianalytical approaches that combine homotopy and perturbation strategies because there are no strict constraints on the selection of its linear operator, and its series solution probably frequently fails to converge [17]. He and El-Dib [18] argued that their proposed approach depends on the traditional Taylor series, which does not need a higher-level approximation.…”

A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations

“…Many linear and nonlinear phenomena appear in several areas of scientific fields like physics, chemistry and biology can be modeled by different type of partial differential equation [1][2][3][4]. A broad class of analytical methods and numerical methods have been introduced such as (G′/G)-expansion method [5], Exp-function method [6], Homotopy perturbation method [7], Homotopy analysis method [8], Laplace transform [9], Residual power series [10], Quasi wavelet method [11], Fourier series [12], Chebyshev-Tau method [13], Haar wavelets method [14], trial equation method [15] and Two scale approach [16] to handle these linear and nonlinear PDEs but to reach exact solutions is not an easy way. In past few decades, The Klein-Gordon and sine-Gordon equations are a type of hyperbolic partial differential equation which are often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, i.e., physics, fluid dynamics, mathematical biology and quantum mechanics.…”

Analytical Approach for the Approximate Solution of Harry Dym Equation with Caputo Fractional Derivative

“…Several partial di erential equations (PDEs) have been demonstrated with the help of FC in the description of some physical phenomena such as uid dynamics, signal processing, astronomy, calculus, genetics, and kinetics [1][2][3][4]. Many researches have studied these PDEs with various analytical and numerical schemes such as Exp-function scheme [5], homotopy perturbation method [6], subequation technique [7], homotopy analysis method [8], rational function strategy [9], the rst integral approach [10], quasi-wavelet process [11], auxiliary equation procedure [12], fractional reduced di erential transform method [13], and fractional complex transform [14]. Kruskal and Moser [15] studied the Harry Dym equation for the rst time in 1974.…”

“…We construct the strategy of MHPTS in Section (4). We implement our suggested approach with a numerical example in Section (5), and finally, we give the detail of obtained results with discussion in Section (6), and the conclusion is in Section (7).…”