An algorithm for solving the Burgers–Huxley equation using the Elzaki transform

Abstract

In this research work, we propose an algorithm which involves the coupling of a new integral transform called the Elzaki transform and the well-known homotopy perturbation method on the Burgers-Huxley equation which is a type of nonlinear advection-diffusion partial differential equation. The Burgers-Huxley equation which models reaction mechanisms, diffusion transports and nerve ion propagation which is applicable in traffic flows, acoustics, turbulence theory, hydrodynamics and generally mechanics is the fusion of the well-known Burgers equation and the Huxley equation. We proffer an analytical solution in the form of a Taylor multivariate series of displacement x and time t using the proposed Elzaki homotopy transformation perturbation method (EHTPM) to three cases of the Burgers-Huxley equation. This solution converges rapidly to a closed form which is the same as the exact solutions obtained using the normal analytical methods from the existing literatures. The exact results and that of our proposed EHTPM when compared via tables and 3D plots show an excellent agreement devoid of errors.

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

“…Most recently, Loyinmi Adedapo C. and Akinfe Timilehin K. (2020) implemented an algorithm using the Elzaki transform to provide exact solutions to the Burgers-Huxley equation of three distinct cases as a result of variation in the equation parameters [7] . Again in (2019), using a hybrid algorithm involving Elzaki transform and homotopy perturbation method (EHTPM), they proffered exact solution to the family of Fisher's reaction-diffusion equation which is well applicable in genetics, stochastic processes, nuclear reactor theory, and so on.…”

“…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.…”

Numerical Analysis of Fractional-Order Parabolic Equations via Elzaki Transform

“…It is important to note that there are many differential equations with variable coefficients that Sumudu and Laplace cannot accomplish transforms but can be conveniently done by using ET [27][28][29][30]. Many mathematicians have been solving differential equations with the aid of ET, such as Navier-Stokes equations [30], heat-like equations [31] and Burgers-Huxley equation [32].…”

A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems

“…So far, many modified techniques have been described to solve linear and nonlinear differential equations of integer and fractional orders. Those of which are Laplace homotopy perturbation method, Laplace Adomian decomposition method, and recently, Elzaki homotopy transformation perturbation method is used to solve a range of problems such as a family of Fisher's equation [21], spatial diffusion of Biological population [22], nonlinear oscillators [23], system of linear and nonlinear PDEs of fractional orders [24], time-fractional Navier-Stokes equations [25], and An algorithm for solving the Burgers-Huxley equation using the Elzaki transform [26] Elzaki integral transform is a modification of the Laplace and Sumudu transforms which was invented by Tariq [27], Elzaki transformation is an efficient and powerful technique that has found the exact solutions to several differential equations which cannot be solved by Sumudu transform [28]. Elzaki integral equation is a powerful and efficient technique that has been used to solve many differential equations of integer and fractional orders [25][29]- [33].…”