Symmetry analysis for a Fisher equation with exponential diffusion

Abstract

In this paper, we consider a generalized Fisher equation with exponential diffusion from the point of view of the theory of symmetry reductions in partial differential equations. The generalized Fisher‐type equation arises in the theory of population dynamics. These types of equations have appeared in many fields of study such as in the reaction‐diffusion equations, in heat transfer problems, in biology, and in chemical kinetics. By using the symmetry classification, simplified by equivalence transformations, for a special family of Fisher equations, all the reductions are derived from the optimal system of subalgebras and symmetry reductions are used to obtain exact solutions.