1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods

In this paper, we obtain the numerical solution of a 1-D generalised Burgers-Huxley equation under specified initial and boundary conditions, considered in three different regimes. The methods are Forward Time Central Space (FTCS) and a non-standard finite difference scheme (NSFD). We showed the schemes satisfy the generic requirements of the finite difference method in solving a particular problem. There are two proposed solutions for this problem and we show that one of the proposed solutions contains a minor error. We present results using FTCS, NSFD, and exact solution as well as show how the profiles differ when the two proposed solutions are used. In this problem, the boundary conditions are obtained from the proposed solutions. Error analysis and convergence tests are performed.

Numerical Approximation of Generalized Burger’s-Fisher and Generalized Burger’s-Huxley Equation by Compact Finite Difference Method

In this work, computational analysis of generalized Burger’s-Fisher and generalized Burger’s-Huxley equation is carried out using the sixth-order compact finite difference method. This technique deals with the nonstandard discretization of the spatial derivatives and optimized time integration using the strong stability-preserving Runge-Kutta method. This scheme inculcates four stages and third-order accuracy in the time domain. The stability analysis is discussed using eigenvalues of the coefficient matrix. Several examples are discussed for their approximate solution, and comparisons are made to show the efficiency and accuracy of CFDM6 with the results available in the literature. It is found that the present method is easy to implement with less computational effort and is highly accurate also.

The modified high-order Haar wavelet scheme with Runge–Kutta method in the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation

In this work, we focus on the modified high-order Haar wavelet numerical method, which introduces the third-order Runge–Kutta method in the time layer to improve the original numerical format. We apply the above scheme to two types of strong nonlinear solitary wave differential equations named as the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation. Numerical experiments verify the correctness of the scheme, which improves the speed of convergence while ensuring stability. We also compare the CPU time, and conclude that our scheme has high efficiency. Compared with the traditional wavelets method, the numerical results reflect the superiority of our format.