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1999 | Buch | 2. Auflage

Normed Spaces Kapitel lesen Erstes Kapitel lesen

Linear Integral Equations

verfasst von: Rainer Kress

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

In the ten years since the first edition of this book appeared, integral equations and integral operators have revealed more of their mathematical beauty and power to me. Therefore, I am pleased to have the opportunity to share some of these new insights with the readers of this book. As in the first edition, the main motivation is to present the fundamental theory of integral equations, some of their main applications, and the basic concepts of their numerical solution in a single volume. This is done from my own perspective of integral equations; I have made no attempt to include all of the recent developments. In addition to making corrections and adjustments throughout the text and updating the references, the following topics have been added: In Sec­ tion 4.3 the presentation of the Fredholm alternative in dual systems has been slightly simplified and in Section 5.3 the short presentation on the index of operators has been extended. The treatment of boundary value problems in potential theory now includes proofs of the jump relations for single-and double-layer potentials in Section 6.3 and the solution of the Dirichlet problem for the exterior of an arc in two dimensions (Section 7.6). The numerical analysis of the boundary integral equations in Sobolev space settings has been extended for both integral equations of the first kind in Section 13.4 and integral equations of the second kind in Section 12.4.

Inhaltsverzeichnis

Frontmatter
1. Normed Spaces
Abstract
The topic of this book is linear integral equations of which
$$ \int_a^b {K(x,y)\rho (y)dy = f(x),x \in \left[ {a,b} \right]} , $$
and
$$ rho (x) - \int_a^b {K(x,y)\rho (y)dy = f(x),x \in \left[ {a,b} \right]} , $$
are typical examples. In these equations the function ϕ is the unknown, and the so-called kernel K and the right-hand side f are given functions. The above equations are called Fredholm integral equationsof the firstand second kind,respectively. We will regard them as operator equations of the firstand second kindin appropriate normed function spaces. The symbol A: X→Ywill mean a single-valued mapping whose domain of definition is a set Xand whose range is contained in a set Y,i.e., for every ϕ ∈ Xthe mapping Aassigns a unique element Aϕ ∈ Y. The range A(X)is the set A(X):= Aϕ: ϕ ∈ X of all image elements. We will use the terms mapping, function,and operatorsynonymously.
Rainer Kress
2. Bounded and Compact Operators
Abstract
In this chapter we briefly explain the basic properties of bounded linear operators and then introduce the concept of compact operators that is of fundamental importance in the study of integral equations.
Rainer Kress
3. Riesz Theory
Abstract
We now present the basic theory for an operator equation
$$ \rho - A\rho = f $$
of the second kind with a compact linear operator A: X → Xon a normed space X.This theory was developed by Riesz [153] and initiated through Fredholm’ s [42] work on integral equations of the second kind.
Rainer Kress
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 for the adjoint operator in the dual space we will consider the Fredholm theory for compact adjoint operators in dual systems generated by nondegenerate bilinear or sesquilinear forms. This symmetric version is more elementary and better suited for applications to integral equations.
Rainer Kress
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
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. About a century ago 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.
Rainer Kress
7. Singular 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. 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. For our introduction to integral equations they will provide an application of the general idea of regularizing singular operators as described in Chapter 5. We assume the reader is acquainted with basic complex analysis.
Rainer Kress
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
9. The Heat Equation
Abstract
The temperature distribution uin a homogeneous and isotropic heat conducting medium with conductivity k,heat capacity c, and mass density psatisfies the partial differential
$$ \frac{{\partial u}}{{\partial t}} = k\Delta u $$
equation where K= k/cp.This is called the equation of heat conductionor, shortly, the heat equation;it was first derived by Fourier. Simultaneously, the heat equation also occurs in the description of diffusion processes. The heat equation is the standard example for a parabolicdifferential equation. In this chapter we want to indicate the application of Volterra-type integral equations of the second kind for the solution of initial boundary value problems for the heat equation. Without loss of generality we assume the constant K=1.For a more comprehensive study of integral equations of the second kind for the heat equation we refer to Cannon [20], Friedman [45], and Pogorzelski [145].
Rainer Kress
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
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 xalone by functions of yalone. In particular, we will describe the construction of appropriate degenerate kernels by interpolation of the given kernel and by orthonormal expansions. The corresponding error analysis will be settled by our results in Section 10.1. We also include a discussion of some basic facts on piecewise linear interpolation and trigonometric interpolation, which will be used in this and subsequent chapters.
Rainer Kress
12. Quadrature Methods
Abstract
In this chapter we shall describe the quadratureor Nyström methodfor the approximate solution of integral equations of the second kind with continuous or weakly singular kernels. As we have pointed out in Chapter 11, the implementation of the degenerate kernel method, in general, requires some use of numerical quadrature. Therefore it is natural to try the application of numerical integration in a more direct approach to approximate integral operators by numerical integration operators. This will lead to a straightforward but widely applicable method for approximately solving equations of the second kind. The reason we placed the description of the quadrature method after the degenerate kernel method is only because its error analysis is more involved.
Rainer Kress
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 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
14. Iterative Solution and Stability
Abstract
The approximation methods for integral equations described in Chapters 11-13 lead to full linear systems. Only if the number of unknowns is reasonably small may these equations be solved by direct methods like Gaussian elimination. But, in general, a satisfying accuracy of the approximate solution to the integral equation will require a comparatively large number of unknowns, in particular for integral equations in more than one dimension. Therefore iterative methods for the resulting linear systems will be preferable. For this, in principle, in the case of positive definite symmetric matrices the classical conjugate gradient method (see Problem 13.2) can be used. In the general case, when the matrix is not symmetric more general Krylov subspace iterations may be used among which a method called generalized minimum residual method (GMRES)due to Saad and Schultz [158] is widely used. Since there is a large literature on these and other general iteration methods for large linear systems (see Freud, Golub, and Nachtigal [43], Golub and van Loan [52], Greenbaum [53], Saad [157], and Trefethen and Bau [175], among others), we do not intend to present them in this book. At the end of this chapter we will only briefly describe the main idea of the panel clustering methodsand the fast multipole methodsbased on iterative methods and on a speed-up of matrix-vector multiplications for the matrices arising from the discretization of integral equations.
Rainer Kress
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 socalled ill-posed problems.In this chapter we introduce regularization methods for the stable solution of equations of the first kind in a Hilbert space setting. For a more comprehensive study of ill-posed problems, we refer to Baumeister [13], Engl, Hanke, and Neubauer [38], Groetsch [58], Kirsch [87], Louis [116], Morozov [129], and Tikhonov and Arsenin [174].
Rainer Kress
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. We shall explain the concepts of quasi-solutions and minimum norm solutions as strategies for the selection of the regularization parameter. The final section of this chapter is devoted to discussion of the classical regularization of integral equations of the first kind as introduced by Tikhonov [172] and Phillips [142].
Rainer Kress
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. For further studies of regularization through discretization, we refer to Baumeister [13], Kirsch [87], Louis [116], and Natterer [136].
Rainer Kress
18. Inverse Boundary Value Problems
Abstract
To end this book we shall briefly indicate the application of ill-posed integral equations of the first kind and regularization techniques to inverse boundary value problems. In the ten years since the first edition of this book was written, the monograph [25] on inverse acoustic and electromagnetic scattering has appeared. Therefore, instead of considering an inverse obstacle scattering problem as in the first edition, in order to introduce the reader to current research in inverse boundary value problems we shall consider an inverse Dirichlet problem for the Laplace equation as a model problem. Of course, in a single chapter it is impossible to give a complete account of inverse boundary value problems. Hence we shall content ourselves with developing some of the main principles. For a detailed study of inverse boundary value problems, we refer to Colton and Kress [25] and Isakov [77].
Rainer Kress
Backmatter
Metadaten
Titel
Linear Integral Equations
verfasst von
Rainer Kress
Copyright-Jahr
1999
Verlag
Springer New York
Electronic ISBN
978-1-4612-0559-3
Print ISBN
978-1-4612-6817-8
DOI
https://doi.org/10.1007/978-1-4612-0559-3