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A model of malignant gliomas throug symmetry reductions

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  • 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.
    Mathematics Subject Classification: Primary: 76M60, 92D25; Secondary: 35Q91.


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    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.


    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.


    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.


    E. Yombaa, Exact Solitary Waves of the Fisher Equation, IMA Preprint Series, (2005), 2061.

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