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Symmetry 2018, 10(5), 171; https://doi.org/10.3390/sym10050171

Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model

1
Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs’ka Street, 01601 Kyiv, Ukraine
2
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
Author to whom correspondence should be addressed.
Received: 25 April 2018 / Revised: 13 May 2018 / Accepted: 14 May 2018 / Published: 17 May 2018
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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 is not. A complete Lie symmetry classification, including a number of the cases characterised as 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 a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour. View Full-Text
Keywords: lie symmetry classification; exact solution; nonlinear reaction-diffusion system; tumour growth model; moving-boundary problem lie symmetry classification; exact solution; nonlinear reaction-diffusion system; tumour growth model; moving-boundary problem
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Cherniha, R.; Davydovych, V.; King, J.R. Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model. Symmetry 2018, 10, 171.

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