American Institute of Mathematical Sciences

2015, 2015(special): 974-980. doi: 10.3934/proc.2015.0974

A model of malignant gliomas throug symmetry reductions

 1 Dpto. de Matemáticas, Universidad de Cádiz, Polígono del Río San Pedro s/n 11510 Puerto Real, Cádiz, Spain, Spain, Spain

Received  September 2014 Revised  May 2015 Published  November 2015

A glioma is a kind of tumor that starts in the brain or spine. The most common site of gliomas is in the brain. Most of the mathematical models in use for malignant gliomas are based on a simple reaction-diffusion equation: the Fisher equation [3]. A nonlinear wave model describing the fundamental features of these tumors has been introduced in [5], by V.M. Pérez and collaborators. In this work, we study this model from the point of view of the theory of symmetry reductions in partial differential equations. We obtain the classical symmetries admitted by the system, then, we use the transformations groups to reduce the equations to ordinary differential equations. Some exact solutions are derived from the solutions of a simple non-linear ordinary differential equation.
Citation: María Rosa, María S. Bruzón, M. L. Gandarias. A model of malignant gliomas throug symmetry reductions. Conference Publications, 2015, 2015 (special) : 974-980. doi: 10.3934/proc.2015.0974
References:
 [1] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations Applied Mathematical Sciences, 154 Springer (2002). Google Scholar [2] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations Chaos. Solitons and Fractals, 24 (2005), 1217-1231. Google Scholar [3] J. D. Murray, Mathematical Biology, Third Edition, Springer-Verlag, New York Berlin Heidelberg, 2002. Google Scholar [4] P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1993. Google Scholar [5] V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego, and L. Pérez-Romasanta, Bright solitary waves in malignant gliomas, Physical Review E., 84 (2011), 021921. Google Scholar [6] K. R. Swanson, C. Bridge, J. D. Murray, and E. C. Alvord, Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion, Journal of the Neurological Sciences, 216 (2003), 1-10. Google Scholar [7] N. K. Vitanov, Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1176-1185. Google Scholar [8] E. Yombaa, Exact Solitary Waves of the Fisher Equation, IMA Preprint Series, (2005), 2061. Google Scholar

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References:
 [1] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations Applied Mathematical Sciences, 154 Springer (2002). Google Scholar [2] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations Chaos. Solitons and Fractals, 24 (2005), 1217-1231. Google Scholar [3] J. D. Murray, Mathematical Biology, Third Edition, Springer-Verlag, New York Berlin Heidelberg, 2002. Google Scholar [4] P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1993. Google Scholar [5] V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego, and L. Pérez-Romasanta, Bright solitary waves in malignant gliomas, Physical Review E., 84 (2011), 021921. Google Scholar [6] K. R. Swanson, C. Bridge, J. D. Murray, and E. C. Alvord, Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion, Journal of the Neurological Sciences, 216 (2003), 1-10. Google Scholar [7] N. K. Vitanov, Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1176-1185. Google Scholar [8] E. Yombaa, Exact Solitary Waves of the Fisher Equation, IMA Preprint Series, (2005), 2061. Google Scholar
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