Exact non-classical symmetry solutions of Arrhenius reaction–diffusion

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 non-classical 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 also extends to heterogeneous diffusion wherein the nonlinear diffusivity factorizes 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.