On a nonlinear reaction-diffusion boundary-value problem: application of a Lie-Bäcklund symmetry

Abstract

AbstractBy a systematic search for Lie-Bäetcklund symmetries, a class of linearisable reaction-diffusion equations is obtained that has, as a canonical form, ut = u2uxx + 2u2. One such nonlinear equation is θt = ∂x[a(b - θ)-2 θx] - ma(b -θ)-2 θx - q exp(-mx). This represents an extension of Fokas-Yortsos-Rosen equation (q = 0) to incorporate a reaction term. It is relevant to the modelling of unsaturated flow in a soil with a volumetric extraction mechanism, such as a web of plant roots. Here, a reciprocal transformation is used to solve a nonlinear boundary-value problem for transient flow into a finite layer of a soil subject to a constant flux boundary condition to compensate for such water extraction.