The equivalence group and invariant solutions of a tumour growth model. (English) Zbl 1032.92017

Summary: Recently, several mathematical models appeared in the literature for describing spread of malignant tumours. These models are formulated as systems of nonlinear partial differential equations containing, in general, several unknown functions of dependent variables. Determination of these unknown functions (called in group analysis arbitrary elements) is a complicated problem that challenges researchers. Our aim is to calculate the generators of the equivalence group for one of the known models and, using the equivalence generators, specify arbitrary elements, find additional symmetries and calculate group invariant solutions.


92C50 Medical applications (general)
54H15 Transformation groups and semigroups (topological aspects)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI


[1] Perumpanini, A.J.; Sherratt, J.A.; Norbury, J.; Byrne, H.M., A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Physica D, 126, 145-159, (1999) · Zbl 1001.92523
[2] Aznavoorian, S.; Stracke, M.L.; Krutzsch, H.; Schiffman, E.; Liotta, L.A., Signal transduction for chemotaxis and hapotaxis by matrix molecules in tumour cells, J. cell. biol., 110, 1427-1438, (1990)
[3] Stewart, J.M.; Broadbridge, P.; Goard, J.M., Symmetry analysis and numerical modelling of invasion by malignant tissue, Nonlinear dynam., 28, 2, 175-193, (2002) · Zbl 1005.92017
[4] Ovsiannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002
[5] Akhatov, I.Sh.; Gazizov, R.K.; Ibragimov, N.H., Nonlocal symmetries: heuristic approach, Itogi nauki i tekhniki (VINITI, Moscow), J. sov. math., 55, 1401-1450, (1991), English translation in · Zbl 0760.35002
[6] Ibragimov, N.H.; Torrisi, M.; Valenti, A., Preliminary group classification of equations vtt=f(x,vx)vxx+g(x,vx), J. math. phys., 32, 11, 2988-2995, (1991) · Zbl 0737.35099
[7] Ibragimov, N.H.; Torrisi, M., A simple method for group analysis and its application to a model of detonation, J. math. phys., 33, 11, 3931-3937, (1992) · Zbl 0761.35104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.