Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

Abstract

In this research work, we propose an unprecedented hybrid algorithm which involves the coupling of a new integral transform, namely, the Elzaki transform and the well‐known homotopy perturbation method called the Elzaki homotopy transformation perturbation method (EHTPM) to solve for the exact solution of three distinct types of Fisher's equation, namely, the Fisher's equation of two cases, the sixth‐order Fisher's equation and the nonlinear diffusion equation of the Fisher type. These equations are prominent in mathematical biology and highly applicable in genetic propagation, population dynamics, stochastic processes, combustion theory, as well as a prototype model for a spreading flame, and so on. The efficacy and authenticity of this method was established via convergence and error analysis, and shows that the solutions obtained from EHTPM are unique and convergent. The results of EHTPM when compared with the exact results, homotopy results, and results from the other existing literature via table of comparison, 3D plots, and the convergence plots validates that EHTPM is an efficient and reliable tool of providing exact solutions to a wider class of nonlinear partial differential equations in a simple and straightforward manner, with no discretization, linearization, computation of Adomian polynomials, and devoid of errors.

Qualitative Analysis and Dynamical Behavior of a Lassa Haemorrhagic Fever Model with Exposed Rodents and Saturated Incidence Rate

“…The above positivity and boundedness analysis which makes us verify equations (22) and (27) is the proof to the following theorem:…”

A solitary wave solution to the generalized Burgers-Fisher's equation using an improved differential transform method: A hybrid scheme approach

“…Numerous researchers have studied the nonlinear ordinary and partial differential equations over the years [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , but consistent findings still yield highly pertinent results and recommendations, as there is no best method or algorithm in providing exact solutions to an equation. Nonlinear PDEs can be classified as the integrable and non-integrable [6] , [7] depending the nature of the equation in question.…”

“…Several methods have been established in the literature over the years for solving nonlinear partial differential equations including the Burgers-Fisher's equation viz: the Taylor collocation method [21] , [22] , wavelet collocation method [23] , [24] , cubic B-spline method [25] , [26] , [27] , differential quadrature method [28] , homotopy analysis method (HAM) [29] , [30] , [31] , reduced differential transform method (RDTM) [32] , [33] , new iterative method (NIM) [34] , [35] , Sumudu decomposition method (SDM) [36] , Elzaki decomposition method (EDM) [37] , [38] , perturbation iteration method (PIM) [39] , variational iteration method (VIM) [40] , [41] , Adomian decomposition method (ADM) [18] , [42] , [43] , Elzaki homotopy transformation perturbation method (EHTPM) [6] , [7] , homotopy perturbation transformation method (HPTM) [44] , Elzaki differential transform [45] , Elzaki projected differential transform method (EPDTM) [46] , new iterative transform method [47] , Laplace Adomian decomposition method (LADM) [48] , Laplace variational iteration method [49] , homotopy analysis transform method [50] , and so on.…”

An efficient numerical approach for fractional multidimensional diffusion equations with exponential memory

“…Recently, Elzaki [31] proposed a new integral transform known as Elzaki transform (ET). Several authors used this transform to solve differential equations of noninteger order such as Khalid et al [32] applied ET on fractional differential equation and find useful results, Mohamed and Elzaki [33] studied ET to solve linear and nonlinear partial differential equations of fractional order, Jena and Chakraverty [34] discussed time-fractional Navier-Stokes equations by employing homotopy perturbation ET, Loyinmi and Akinfe [35] demonstrated on Elzaki homotopy transformation perturbation method for finding the exact solutions of Fisher's reaction-diffusion equation, and so on.…”