American Institute of Mathematical Sciences

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

A model of malignant gliomas throug symmetry reductions

María Rosa 1, , María S. Bruzón 1, and  M. L. Gandarias 1,

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.
Keywords: Lie symmetries, partial differential equations, Fisher-equation, ordinary differential equation., malignant gliomas.
Mathematics Subject Classification: Primary: 76M60, 92D25; Secondary: 35Q9.
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).

[2]

N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations Chaos. Solitons and Fractals, 24 (2005), 1217-1231.

[3]

J. D. Murray, Mathematical Biology, Third Edition, Springer-Verlag, New York Berlin Heidelberg, 2002.

[4]

P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1993.

[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.

[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.

[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.

[8]

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

show all references

References:
[1]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations Applied Mathematical Sciences, 154 Springer (2002).

[2]

N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations Chaos. Solitons and Fractals, 24 (2005), 1217-1231.

[3]

J. D. Murray, Mathematical Biology, Third Edition, Springer-Verlag, New York Berlin Heidelberg, 2002.

[4]

P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1993.

[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.

[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.

[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.

[8]

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

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