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Published online by Cambridge University Press: 05 July 2015
Fredholm equations arise in many areas of science and engineering. Consequently, they occupy a central topic in applied mathematics. Traditional numerical methods developed during the period prior to the mid-1980s include mainly quadrature, collocation and Galerkin methods. Unfortunately, all of these approaches suffer from the fact that the resulting discretization matrices are dense. That is, they have a large number of nonzero entries. This bottleneck leads to significant computational costs for the solution of the corresponding integral equations.
The recent appearance of wavelets as a new computational tool in applied mathematics has given a new direction to the area of the numerical solution of Fredholm integral equations. Shortly after their introduction it was discovered that using a wavelet basis for a singular integral equation led to numerically sparse matrix discretization. This observation, combined with a truncation strategy, then led to a fast numerical solution of this class of integral equations.
Approximately 20 years ago the authors of this book began a systematic study of the construction of wavelet bases suitable for solving Fredholm integral equations and explored their usefulness for developing fast multi scale Galerkin, Petrov–Galerkin and collocation methods. The purpose of this book is to provide a self-contained account of these ideas as well as some traditional material on Fredholm equations to make this book accessible to as large an audience as possible.
The goal of this book is twofold. It can be used as a reference text for practitioners who need to solve integral equations numerically and wish to use the new techniques presented here. At the same time, portions of this book can be used as a modern text treating the subject of the numerical solution of integral equations, which is suitable for upper-level undergraduate students as well as graduate students. Specifically, the first five chapters of this book are designed for a one-semester course, which provides students with a solid background in integral equations and fast multi scale methods for their numerical solutions.
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