Lie symmetries, reduction and exact solutions of the (1+2)-dimensional nonlinear problem

TL;DR

This paper uses Lie symmetry methods to analyze a (1+2)-dimensional nonlinear tumor growth model, reducing it to simpler problems and finding exact solutions with biological relevance.

Contribution

It introduces a symmetry-based approach to reduce and solve a complex tumor growth PDE model, providing new exact solutions and insights.

Findings

Derived a nontrivial Lie symmetry for the model

Reduced the PDE to lower-dimensional problems

Obtained and analyzed exact solutions with biological implications

Abstract

The well known nonlinear model for describing the solid tumour growth [Byrne HM., et al. Appl Math Letters 2003;16:567-74] is under study using an approach based on Lie symmetries. It is shown that the model in the two-dimensional (in space) approximation forms a (1+2)-dimensional boundary value problem, which admits a highly nontrivial Lie symmetry. The special case involving the power-law nonlinearities is examined in details. The symmetries derived are applied for the reduction of the nonlinear boundary value problem in question to problems of lower dimensionality. Finally, the reduced problems with correctly-specified coefficients were exactly solved and the exact solutions derived were analysed, in particular, some plots were build in order to understand the time-space behaviour of these solutions and to discuss their biological interpretation.