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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.

MSC:
92C50Medical applications of mathematical biology
54H15Transformation groups and semigroups of topological spaces
35Q80Applications of PDE in areas other than physics (MSC2000)
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[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, No. 2, 175-193 (2002) · Zbl 1005.92017
[4] Ovsiannikov, L. V.: Group analysis of differential equations. (1982) · Zbl 0485.58002
[5] Akhatov, I. Sh.; Gazizov, R. K.; Ibragimov, N. H.: Nonlocal symmetries: heuristic approach. Itogi nauki i tekhniki (VINITI, Moscow) 34, 3-83 (1989) · Zbl 0722.35004
[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, No. 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, No. 11, 3931-3937 (1992) · Zbl 0761.35104