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

Cherniha, Roman; Davydovych, Vasyl; King, John R.


Roman Cherniha

Vasyl Davydovych

Professor of Theoretical Mechanics


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.


Cherniha, R., Davydovych, V., & King, J. R. (2018). Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model. Symmetry, 10(5), Article 171.

Journal Article Type Article
Acceptance Date May 14, 2018
Publication Date May 17, 2018
Deposit Date Jun 4, 2018
Publicly Available Date Jun 4, 2018
Journal Symmetry
Electronic ISSN 2073-8994
Publisher MDPI
Peer Reviewed Peer Reviewed
Volume 10
Issue 5
Article Number 171
Keywords Lie symmetry classification; Exact solution; Nonlinear reaction-diffusion system; Tumour growth model; Moving-boundary problem
Public URL
Publisher URL


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