Thesis Summary:

The main objective of this work is the mathematical and numerical analysis of partial differential equations, specifically systems of nonlinear parabolic equations and in particular systems for models with application in Biology. Mathematical analysis involves proving the existence and uniqueness of a weak solution, that is, solving the continuous problem. Numerical analysis refers to the implementation, development and analysis of numerical methods for the specific models studied. Initially, we demonstrate the existence and non-negativity of weak solution of a reaction-diffusion system that models the growth of an early tumor, using the Faedo-Galerkin method, a priori estimates and compactness results. We construct a finite volume scheme based on the classical methods of the Eymard - Gallouët - Herbin Handbook, for this scheme we demonstrate the existence of solution and also the convergence of this discrete solution to a weak solution of the continuous problem. Finally, some numerical results are obtained that highlight the formation of patterns. The reaction-diffusion system with which it was worked was obtained from the model given by Marciniak-Czochra and Kimmel, that is to say, it is an extension. Subsequently, a reaction-diffusion-convection system is addressed in a contaminated environment, that is, a system of parabolic partial differential equations that model the interaction between two species, coupled with a system of ordinary differential equations that model the concentration of the Pollutant in the environment and organisms of the two species. This model is an extension of the model studied by Yang et al. The existence of a weak solution of this system is demonstrated, for which the Schauder Fixed Point Theorem is used. On the other hand, in the numerical aspect a numerical scheme is constructed which combines non-conforming finite elements and finite volumes based on the method described in the Vohralik thesis. It proves the existence of a discrete solution of the scheme and also the convergence of this solution to a weak solution of the continuous problem. Numerical tests were performed. Finally, a reaction-diffusion system with non-local diffusion and non-linear cross-diffusion is considered. This model describes the interaction between three species. The system studied is based on the model of the food chain of Hastings and Powell. A finite volume scheme is constructed based on the classical methods of the Eymard - Gallouët - Herbin Handbook, for which the existence and uniqueness of a discrete solution is demonstrated, and the convergence of that solution to a weak solution of the continuous problem is proved. Some numerical results are presented.

ISI Publications from the Thesis

Veronica ANAYA, Mostafa BENDAHMANE, Mauricio SEPúLVEDA: Mathematical and numerical analysis for predator-prey system in a polluted environment. Networks and Heterogeneous Media, vol. 5, 4, pp. 813-847, (2010)

Veronica ANAYA, Mostafa BENDAHMANE, Mauricio SEPúLVEDA: A numerical analysis of a reaction-diffusion system modelling the dynamics of growth tumors. Mathematical Models and Methods in Applied Sciences, vol. 20, 5, pp. 731-756, (2010)

Veronica ANAYA, Mostafa BENDAHMANE, Mauricio SEPúLVEDA: Mathematical and numerical analysis for reaction-diffusion systems modeling the spread of early tumors. Boletin de la Sociedad Espagnola de Matematica Aplicada, vol. 47, pp. 55-62. (2009)