Superconvergence in Iterated Solutions of Integral Equations

In this thesis, we investigate the superconvergence phenomenon of the iterated numerical solutions for the Fredholm integral equations of the second kind as well as a class of nonliner Hammerstein equations. The term superconvergence was first described in the early 70s in connection with the solution of two-point boundary value problems and other related partial differential equations. Superconvergence in this context was understood to mean that the order of convergence of the numerical solutions arising from the Galerkin as well as the collocation method is higher at the knots than we might expect from the numerical solutions that are obtained by applying a class of piecewise polynomials as approximating functions. The type of superconvergence that we investigate in this thesis is different. We are interested in finding out whether or not we obtain an enhancement in the global rate of convergence when the numerical solutions are iterated through integral operators. A general operator approximation scheme for the second kind linear equation is described that can be used to explain some of the existing superconvergence results. Moreover, a corollary to the general approximation scheme will be given which can be used to establish the superconvergence of the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for Hammerstein and weakly singular Hammerstein equations and its corresponding superconvergence phenomena for the iterated solutions. The type of regularities that the solution of weakly singular Hammerstein equations possess is investigated. Subsequently, we establish the singularity preserving Galerkin method for Hammerstein equations. Finally, the superconvergence results for the iterated solutions corresponding to this method will be described

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.

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

Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations

Due to the ability of function representation, hybrid functions and wavelets have a special position in research. In this thesis, we state elementary definitions, then we introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby- shev, sine-cosine and linear Legendre multi wavelets. The construction of most wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them. The properties of the Floor function are used to and a function which is one on the interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a; b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet, Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions. The main advantage of the wavelet technique for solving a problem is its ability to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential and Fredholm integral equations of the second kind. We also use collocation points and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. Illustrative examples are included to reveal the sufficiency of the technique. In linear integro-differential equations and Fredholm integral equations of the second kind cases, comparisons are done with CAS wavelets and differential transformation methods and it shows that the accuracy of these results are higher than them. Homotopy Analysis Method (HAM) is an analytic technique to solve the linear and nonlinear equations which can be used to obtain the numerical solution too. We extend the application of homotopy analysis method for solving Linear integro- differential equations and Fredholm and Volterra integral equations. We provide some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW, LLMW and CAS wavelets methods. For future studies, some problems are proposed at the end of this thesis

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