Differential Quadrature Solutions of the Generalized Burgers–Fisher Equation with a Strong Stability Preserving High-Order Time Integration

Abstract

Numerical solutions of the generalized Burgers-Fisher equation are presented based on a polynomial-based differential quadrature method with minimal computational effort. To achieve this, a combination of a polynomial-based differential quadrature method in space and a third-order strong stability preserving Runge-Kutta scheme in time have been used. The proposed technique successfully worked to give reliable results in the form of numerical approximation converging very rapidly. The computed results have been compared with the exact solution to show the required accuracy of the method. The approximate solutions to the nonlinear equations were obtained. The approach is seen to be a very reliable alternative to the rival techniques for realistic problems.

A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation

“…In order to test the accuracy and efficiency of the proposed scheme, comparisons of the obtained results are made with the above exact solution and traditional methods such as [26, (44) 32,35,36]. MATLAB 8.1 has been utilized in this work for simulations.…”

“…Ismail et al [24] applied Adomian decomposition method (ADM), Javidi [25] employed modified pseudospectral method, Rashidi et al [26] used homotopy perturbation method (HPM), Khattak [27] employed collocation-based radial basis functions method (CBRBF) and Xu and Xian [28] applied Exp-function method to find the analytic as well as numerical solutions of the generalized Burgers-Fisher equation. Also many other authors used different methods to obtain the analytical and numerical solution of the generalized Burgers-Fisher equation; for example, Zhu and Kang [29] used the B-spline quasi-interpolation method, Zhang and Yan [30] used a lattice Boltzmann model, Sari et al [31] used the compact finite difference method, Sari et al [32] developed the polynomial-based differential quadrature method, Zhang et al [33] used the local discontinuous Galerkin (LDG) methods and Nawaz et al [34] employed optimal homotopy asymptotic method (OHAM).…”

Implementation of Wavelet Based Transform for numerical solutions of Partial differential equations

“…, 2 can be successively calculated from equation (29). This process is started with the initial condition (12). These coefficients are then substituted in equations (25)-(27) to obtain the approximate solutions at different time levels.…”

“…The Burgers-Fisher equation which describes the interaction between the reaction mechanism, convection effect, and diffusion transport [6] is considered in this paper. Many numerical schemes have been proposed for obtaining approximate solutions of the Burger-fisher equation [7][8][9][10][11][12].…”

An Improved Second-Order Numerical Method for the Generalized Burgers–fisher Equation

“…Many researchers have studied it theoretically and numerically. Because of its strong nonlinearity, it is often used as a model problem to test various numerical methods, where, among others, finite-difference methods [16,23,24], the Adomian decomposition method [17,19] and differential quadrature [22] have been used. Powerful mathematical methods such as the tanh [11,28], extended tanh [10], tanh-coth [29,30], exp-function [31], variational iteration [21], homotopy analysis [2], factorization [12] and spectral collocation [14,18] methods have also been used for this equation.…”