Exact non-classical symmetry solutions of Arrhenius reaction–diffusion

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.

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

“…By virtue of (13), this is identically satisfied given that a W exists satisfying (19), which in turn gives that a V exists satisfying (17), thus proving our claim.…”

“…The nonclassical method, first introduced by Bluman and Cole [8] (see, for example, [2] or [3]), seeks invariance of a given partial differential equation (PDE) augmented with the invariant surface condition. As the determining equations for these nonclassical symmetries are nonlinear, there seemed to be little hope for this new method; however, with the development of computer algebra systems, the nineties saw a huge explosion of interest as several authors took interest in the nonclassical method and continues today to be an active area of interest (e.g., [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and references within).…”

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

“…Finally, we point out that examples of exact solutions of multidimensional RD equations with exponential nonlinearities are presented in [11,43,44].…”

“…The Lie symmetry of Equation (4) is the three-dimensional Lie algebra [10]. Notably, the well-known RD equation with the Arrhenius reaction term e −A 0 /u (A 0 is a positive constant), which is widely used in applications, can be reduced to the form (4) (with e u → exp(A * 0 u − u 0 ), A * 0 > 0, u 0 > 0) via a corresponding approximation (see, for details, [11]). …”

Nonclassical Symmetry Solutions for 4th Order Phase Field Reaction-Diffusion

“…The derivation of (38) structure of the problem, one may look for solutions to (38) as the critical points of (39) and vice-versa.…”