Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

TL;DR

This paper develops a new symmetry-based approach to classify and solve variable coefficient semilinear reaction-diffusion equations, leading to new exact solutions and a comprehensive understanding of their transformational properties.

Contribution

It introduces a novel method using mappings between classes of differential equations for symmetry analysis and classification of variable coefficient reaction-diffusion equations.

Findings

Performed group classifications for the entire class and singular subclass.

Identified admissible transformations for the imaged class.

Constructed new exact solutions via Lie reductions and transformations.

Abstract

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).