Post-Processing Techniques and Wavelet Applications for Hammerstein Integral Equations

This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation. In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a \u27linearization\u27 technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation

ANALYSIS OF ITERATIVE METHODS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH APPLICATIONS TO THE HELMHOLTZ PROBLEM

This thesis is concerned with the numerical solution of boundary integral equations and the numerical analysis of iterative methods. In the first part, we assume the boundary to be smooth in order to work with compact operators; while in the second part we investigate the problem arising from allowing piecewise smooth boundaries. Although in principle most results of the thesis apply to general problems of reformulating boundary value problems as boundary integral equations and their subsequent numerical solutions, we consider the Helmholtz equation arising from acoustic problems as the main model problem. In Chapter 1, we present the background material of reformulation of Helmhoitz boundary value problems into boundary integral equations by either the indirect potential method or the direct method using integral formulae. The problem of ensuring unique solutions of integral equations for exterior problems is specifically discussed. In Chapter 2, we discuss the useful numerical techniques for solving second kind integral equations. In particular, we highlight the superconvergence properties of iterated projection methods and the important procedure of Nystrom interpolation. In Chapter 3, the multigrid type methods as applied to smooth boundary integral equations are studied. Using the residual correction principle, we are able to propose some robust iterative variants modifying the existing methods to seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease, faster convergence of multigrid type methods and fixed step convergence of the conjugate gradient method. In the case of non-smooth integral boundaries, we first derive the singular forms of the solution of boundary integral solutions for Dirichlet problems and then discuss the numerical solution in Chapter 5. Iterative methods such as two grid methods and the conjugate gradient method are successfully implemented in Chapter 6 to solve the non-smooth integral equations. The study of two grid methods in a general setting and also much of the results on the conjugate gradient method are new. Chapters 3, 4 and 5 are partially based on publications [4], [5] and [35] respectively.Department of Mathematics and Statistics, Polytechnic South Wes

A Personal Perspective on the History of the Numerical Analysis of Fredholm Integral Equations of the Second Kind

This is a personal perspective on the development of numerical methods for solving Fredholm integral equations of the second kind, discussing work being done principally during the 1950s and 1960s. The principal types of numerical methods being studied were projection methods (Galerkin, collocation) and Nyström methods. During the 1950s and 1960s, functional analysis became the framework for the analysis of numerical methods for solving integral equations, and this in‡uenced the questions being asked. This paper looks at the history of the analyses being done at that time.

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

Singularity Preserving Numerical Methods for Boundary Integral Equations

In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad