Exact nonclassical symmetry solutions of Arrhenius reaction-diffusion

TL;DR

This paper derives exact nonclassical symmetry solutions for multidimensional nonlinear Arrhenius reaction-diffusion equations, revealing exponential-in-time heat flux solutions and extending to heterogeneous diffusion with practical applications.

Contribution

It introduces a novel nonclassical Lie symmetry approach to solve complex reaction-diffusion equations with exponential heat flux solutions, including heterogeneous cases.

Findings

Solutions exhibit exponential growth or decay in heat flux.

Extension to heterogeneous diffusion with temperature and position dependence.

Applications to heat conduction and soil-water flow problems.

Abstract

Exact solutions for nonlinear Arrhenius reaction-diffusion are constructed in $n$ dimensions. A single relationship between nonlinear diffusivity and the nonlinear reaction term leads to a nonclassical Lie symmetry whose invariant solutions have a heat flux that is exponential in time (either growth or decay), and satisfying a linear Helmholtz equation in space. This construction extends also to heterogeneous diffusion wherein the nonlinear diffusivity factorises to the product of a function of temperature and a function of position. Example solutions are given with applications to heat conduction in conjunction with either exothermic or endothermic reactions, and to soil-water flow in conjunction with water extraction by a web of plant roots.