Reduction operators of variable coefficient semilinear diffusion equations with an exponential source

TL;DR

This paper investigates reduction operators, including nonclassical symmetries, of variable coefficient semilinear reaction-diffusion equations with exponential sources, using a mapping algorithm and applying findings to construct exact solutions.

Contribution

It introduces a systematic method for finding reduction operators of these equations and distinguishes inequivalent nonclassical symmetries from classical Lie symmetries.

Findings

Derived new reduction operators for the equations.

Constructed exact solutions using these operators.

Identified conditions under which reduction operators are inequivalent to Lie symmetries.

Abstract

Reduction operators (called also nonclassical or $Q$-conditional symmetries) of variable coefficient semilinear reaction-diffusion equations with exponential source $f(x)u_t=(g(x)u_x)_x+h(x)e^{mu}$ are investigated using the algorithm involving a mapping between classes of differential equations, which is generated by a family of point transformations. A special attention is paid for checking whether reduction operators are inequivalent to Lie symmetry operators. The derived reduction operators are applied to construction of exact solutions.