Abstract
We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.
Highlights
During the last few decades, nonlinear diffusion equation (1)ut + αuux − uxx = f (u, x, t) (1)has played an important role in nonlinear physics
A nonstandard finite difference scheme for the Burgers-Fisher equation was given by Mickens and Gumel [7]
We obtain the exact finite difference schemes based on the solitary wave solutions of two equations
Summary
Has played an important role in nonlinear physics. Recently, it began to become important in various other fields of science, for example, biology, chemistry, and economics [1,2,3]. A nonstandard finite difference scheme for the Burgers-Fisher equation was given by Mickens and Gumel [7]. Mickens et al [17] considered a second-order, linear equation (d2x/dt2) + a(t)(dx/dt) + b(t)x = f(t) with constant coefficients and gave an exact finite difference scheme of the equation. Roeger and Mickens [19] gave NSFD schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed points. They constructed a nonexact NSFD scheme preserving the critical properties of the original differential equation. Roeger [20] studied a two-dimensional linear system with constant coefficients and constructed exact finite-difference scheme for the system. The first objective is to consider the Burgers and Burgers-Fisher equations
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