Superconvergence of Iterated Solutions for Linear and Nonlinear Integral Equations: Wavelet Applications

In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern of this dissertation is to address the issue of how this sparsity manifests itself in the setting of nonlinear equations, particularly for Hammerstein equations. We demonstrate that sparsity appears in the Jacobian matrix when one attempts to solve the system of nonlinear equations by the Newton\u27s iterative method. Overall, the dissertation generalizes the results of Alpert to nonlinear equations setting as well as the results of Chen and Xu, who discussed the Petrov-Galerkin method for Fredholm equation, to nonlinear equations setting

Existence and Uniqueness of Blow-up Solutions for a Parabolic Problem with a Localized Nonlinear Term via Semi-group Theory

International audienceHere, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classical solution of a semilinear parabolic problem with localized nonlinear term| a locally Lipschitz continuous function of the value of the solution at a point of a 1-dimensional domain. Our method, which uses Sobolev spaces and fractional power of operators, is in contrast with the classical ones (Green functions) which supply similar results in 1-dimensional settings