The generalized Cole–Hopf transformation for a generalized Burgers–Fisher equation with spatiotemporal variable coefficients

Abstract

In this paper, we investigate a generalized Burgers-Fisher equation with spatiotemporal variable coefficients(gvcBF), which can describe the nonlinear convection–diffusion phenomenon in chemical engineering and biology. It is shown that the gvc Burgers-Fisher equation can be exactly linearized as long as the variable coefficient functions satisfy some constraints conditions. We obtain a Bäcklund transformation between the gvc Burgers-Fisher equation and a linear parabolic equation with variable coefficients by the simplified homogeneous balance method. Furthermore, we derive out a generalized Cole-Hopf transformation between a class of the gvc Burgers-Fisher equation and the linear heat conduction equation with the help of a new unknown function transformation. Especially, we obtain the generalized Cole-Hopf transformation between the cylindrical and spherical Burgers-Fisher equations with algebraical decaying damping term and the heat conduction equation. Finally, we obtain a series of explicit exact solutions of the gvc Burgers-Fisher equation.