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2014 | Buch

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Linear Integral Equations

verfasst von: Rainer Kress

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter.

For this third edition in order to make the introduction to the basic functional analytic tools more complete the Hahn–Banach extension theorem and the Banach open mapping theorem are now included in the text. The treatment of boundary value problems in potential theory has been extended by a more complete discussion of integral equations of the first kind in the classical Holder space setting and of both integral equations of the first and second kind in the contemporary Sobolev space setting. In the numerical solution part of the book, the author included a new collocation method for two-dimensional hypersingular boundary integral equations and a collocation method for the three-dimensional Lippmann-Schwinger equation. The final chapter of the book on inverse boundary value problems for the Laplace equation has been largely rewritten with special attention to the trilogy of decomposition, iterative and sampling methods

Reviews of earlier editions:

"This book is an excellent introductory text for students, scientists, and engineers who want to learn the basic theory of linear integral equations and their numerical solution."

(Math. Reviews, 2000)

"This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a student. The presentation of the subject matter is lucid, clear and in the proper modern framework without being too abstract." (ZbMath, 1999)

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Basic Functional Analysis

Abstract

The topic of this book is linear integral equations of which

$$\displaystyle{ \int _{a}^{b}K(x,y)\varphi (y)\,dy = f(x),\quad x \in [a,b], }$$

(1.1)

and

$$\displaystyle{ \varphi (x) -\int _{a}^{b}K(x,y)\varphi (y)\,dy = f(x),\quad x \in [a,b], }$$

(1.2)

are typical examples.

Rainer Kress

Chapter 2. Bounded and Compact Operators

Abstract

In this chapter we briefly review the basic properties of bounded linear operators in normed spaces and then introduce the concept of compact operators that is of fundamental importance in the study of integral equations.

Rainer Kress

Chapter 3. Riesz Theory

Abstract

We now present the basic theory for an operator equation

$$\displaystyle{ \varphi -A\varphi = f }$$

(3.1)

of the second kind with a compact linear operator A: X → X on a normed space X.

Rainer Kress

Chapter 4. Dual Systems and Fredholm Alternative

Abstract

In the case when the homogeneous equation has nontrivial solutions, the Riesz theory, i.e., Theorem 3.4 gives no answer to the question of whether the inhomogeneous equation for a given inhomogeneity is solvable. This question is settled by the Fredholm alternative, which we shall develop in this chapter. Rather than presenting it in the context of the Riesz–Schauder theory with the adjoint operator in the dual space we will consider the Fredholm theory for compact adjoint operators in dual systems generated by non-degenerate bilinear or sesquilinear forms. This symmetric version is better suited for applications to integral equations and contains the Riesz–Schauder theory as a special case.

Rainer Kress

Chapter 5. Regularization in Dual Systems

Abstract

In this chapter we will consider equations that are singular in the sense that they are not of the second kind with a compact operator. We will demonstrate that it is still possible to obtain results on the solvability of singular equations provided that they can be regularized, i.e., they can be transformed into equations of the second kind with a compact operator.

Rainer Kress

Chapter 6. Potential Theory

Abstract

The solution of boundary value problems for partial differential equations is one of the most important fields of applications for integral equations. In the second half of the 19th century the systematic development of the theory of integral equations was initiated by the treatment of boundary value problems and there has been an ongoing fruitful interaction between these two areas of applied mathematics. It is the aim of this chapter to introduce the main ideas of this field by studying the basic boundary value problems of potential theory. For the sake of simplicity we shall confine our presentation to the case of two and three space dimensions. The extension to more than three dimensions is straightforward. As we shall see, the treatment of the boundary integral equations for the potential theoretic boundary value problems delivers an instructive example for the application of the Fredholm alternative, since both its cases occur in a natural way. This chapter covers the classical approach to boundary integral equations of the second kind in the space of continuous functions. The treatment of boundary integral equations of the first and of the second kind in Hölder spaces and in Sobolev spaces will be the topic of the two subsequent chapters.

Rainer Kress

Chapter 7. Singular Boundary Integral Equations

Abstract

In this chapter we will consider one-dimensional singular integral equations involving Cauchy principal values that arise from boundary value problems for holomorphic functions in the classical Hölder space setting. The investigations of these integral equations with Cauchy kernels by Gakhov, Muskhelishvili, Vekua, and others have had a great impact on the further development of the general theory of singular integral equations.

Rainer Kress

Chapter 8. Sobolev Spaces

Abstract

In this chapter we study the concept of weak solutions to boundary value problems for harmonic functions. We shall extend the classical theory of boundary integral equations as described in the two previous chapters from the spaces of continuous or Hölder continuous functions to appropriate Sobolev spaces. For the sake of brevity we will confine ourselves to interior boundary value problems in two dimensions.

Rainer Kress

Chapter 9. The Heat Equation

Abstract

The temperature distribution u in a homogeneous and isotropic heat conducting medium with conductivity k, heat capacity c, and mass density ρ satisfies the partial differential equation

$${ \partial u \over \partial t} =\kappa \varDelta u$$

where κ = k∕c ρ.

Rainer Kress

Chapter 10. Operator Approximations

Abstract

In subsequent chapters we will study the numerical solution of integral equations. It is our intention to provide the basic tools for the investigation of approximate solution methods and their error analysis. We do not aim at a complete review of all the various numerical methods that have been developed in the literature. However, we will develop some of the principal ideas and illustrate them with a few instructive examples.

Rainer Kress

Chapter 11. Degenerate Kernel Approximation

Abstract

In this chapter we will consider the approximate solution of integral equations of the second kind by replacing the kernels by degenerate kernels, i.e., by approximating a given kernel K(x, y) through a sum of a finite number of products of functions of x alone by functions of y alone.

Rainer Kress

Chapter 12. Quadrature Methods

Abstract

In this chapter we shall describe the quadrature or Nyström method for the approximate solution of integral equations of the second kind with continuous or weakly singular kernels.

Rainer Kress

Chapter 13. Projection Methods

Abstract

The application of the quadrature method, in principle, is confined to equations of the second kind. To develop numerical methods that can also be used for equations of the first kind we will describe projection methods as a general tool for approximately solving linear operator equations. After introducing into the principal ideas of projection methods and their convergence and error analysis we shall consider collocation and Galerkin methods as special cases. We do not intend to give a complete account of the numerous implementations of collocation and Galerkin methods for integral equations that have been developed in the literature. Our presentation is meant as an introduction to these methods by studying their basic concepts and describing their numerical performance through a few typical examples.

Rainer Kress

Chapter 14. Iterative Solution and Stability

Abstract

The approximation methods for integral equations described in Chapters 11–13 lead to full linear systems.

Rainer Kress

Chapter 15. Equations of the First Kind

Abstract

Compact operators cannot have a bounded inverse. Therefore, equations of the first kind with a compact operator provide a typical example for so-called ill-posed problems.

Rainer Kress

Chapter 16. Tikhonov Regularization

Abstract

This chapter will continue the study of Tikhonov regularization and will be based on its classical interpretation as a penalized residual minimization. For this we will consider the more general case of merely bounded linear operators. In particular, we shall explain the concepts of quasi-solutions and minimum norm solutions as strategies for the selection of the regularization parameter.

Rainer Kress

Chapter 17. Regularization by Discretization

Abstract

We briefly return to the study of projection methods and will consider their application to ill-posed equations of the first kind. In particular we will present an exposition of the moment discretization method.

Rainer Kress

Chapter 18. Inverse Boundary Value Problems

Abstract

In this book, so far, we have considered only so-called direct boundary value problems where, given a differential equation, its domain, and a boundary condition, we want to determine its solution.

Rainer Kress

Backmatter

Titel: Linear Integral Equations
verfasst von: Rainer Kress
Verlag: Springer New York
Electronic ISBN: 978-1-4614-9593-2
Print ISBN: 978-1-4614-9592-5
DOI: https://doi.org/10.1007/978-1-4614-9593-2