In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.

Made available in DSpace on 2016-06-02T20:28:27Z (GMT). No. of bitstreams: 1 4698.pdf: 604113 bytes, checksum: 033e879a6c06b9c3ad1884a3611729fe (MD5) Previous issue date: 2012-08-10 / Universidade Federal de Sao Carlos / We study the behavior, as t ∞, of the energy for the solutions of the Cauchy problem for some strictly hyperbolic linear second order equations with coeficients very rapidly oscillating. / Nós estudamos o comportamento da energia, para t ∞, das soluções do problema de Cauchy para algumas equações estritamente hiperbólicas de segunda ordem com coeficientes que oscilam rapidamente.