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.

Codimension two Lie invariant solutions of the modified Khokhlov–Zabolotskaya–Kuznetsov equation

“…Theorem 3.1. An optimal list of two-dimensional subalgebras for the Lie algebra spanned by (22) reads as (28), where a 3 and a 5 are arbitrary constants.…”

“…Galas 21 refined Ovsiannikov's method by removing equivalent subalgebras for the solvable algebra and also discussed the problem of nonsolvable algebra. Over the years, many researchers focused on classification of lower dimensional subalgebras 24‐28 . Bihlo 29‐31 introduced a new notion of finding two‐dimensional optimal system without using one‐dimensional optimal classification.…”

“…Over the years, many researchers focused on classification of lower dimensional subalgebras. [24][25][26][27][28] Bihlo [29][30][31] introduced a new notion of finding two-dimensional optimal system without using one-dimensional optimal classification. He introduced a coefficient matrix for general element of two-dimensional Lie algebra.…”

Group classification and analytical solutions of a radially symmetric avascular cancer model

“…[27][28][29], where symmetries were used to explore problems related to dengue; Refs. 30,31, where the authors considered and examined generalizations of the Fisher equation; Ref. 32, where several disease models were delved; Refs.…”