Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model

Abstract

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other not. A complete Lie symmetry classification, including a number of the cases characterised being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications tumour growth). Graphical representations of the solutions are provided and biological interpretation addressed briefly. The results are generalised on multi-dimensional case under assumption of radially symmetrical shape of the tumour.