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## Über dieses Buch

Many physical problems that are usually solved by differential equation techniques can be solved more effectively by integral equation methods. This work focuses exclusively on singular integral equations and on the distributional solutions of these equations. A large number of beautiful mathematical concepts are required to find such solutions, which in tum, can be applied to a wide variety of scientific fields - potential theory, me­ chanics, fluid dynamics, scattering of acoustic, electromagnetic and earth­ quake waves, statistics, and population dynamics, to cite just several. An integral equation is said to be singular if the kernel is singular within the range of integration, or if one or both limits of integration are infinite. The singular integral equations that we have studied extensively in this book are of the following type. In these equations f (x) is a given function and g(y) is the unknown function. 1. The Abel equation x x) = l g (y) d 0 < a < 1. ( / Ct y, ( ) a X - Y 2. The Cauchy type integral equation b g (y) g(x)=/(x)+).. l--dy, a y-x where).. is a parameter. x Preface 3. The extension b g (y) a (x) g (x) = J (x) +).. l--dy , a y-x of the Cauchy equation. This is called the Carle man equation.

## Inhaltsverzeichnis

### 1. Reference Material

Abstract
In this chapter we give a brief summary of several topics and results that are used throughout the book.

### 2. Abel’s and Related Integral Equations

Abstract
In this chapter we give the solutions of Abel’s and some related integral equations. In the first section we present two methods for the solution of Abel’s equation and by using similar techniques solve some integral equations that can be reduced to Abel’s equation in the next section. The range of the parameter α appearing in Abel’s equation is extended in the third section, while in the fourth, we generalize our analysis to equations over a simple contour in the complex plane. Equations involving combinations of both the Abel integral operator and its adjoint over open and closed contour are considered in Sections 2.5 and 2.6. Several examples are solved in the last section.

### 3. Cauchy Type Integral Equations

Abstract
In this chapter we study various singular integral equations with kernels of the Cauchy type, starting from the most basic Cauchy type integral equation.

### 4. Carleman Type Integral Equations

Abstract
Many problems in physics and engineering which can be reduced to the integral equation
$$\alpha (\xi )g(\xi ) - \lambda \beta (\xi ){\text{p}}{\text{.v}}{\text{.}}\int\limits_C {\frac{{\gamma (\omega )g(\omega )}}{{\omega - \xi }}} d\omega = f(\xi )$$
(4.1)
, where α(ξ),β(ξ), γ(ξ) and f (ξ) are prescribed functions of a real or complex variable ξ. The range of integration C can be an interval of the real line, a closed or an open contour in the complex plane C. An explicit solution of this equation was first given by Carleman [10] for a real interval and, therefore, the equation bears his name. It has been recognized for several decades that this equation plays a pivotal role in the theory of singular integral equations. When β(ξ) and β(ξ) are constants, equation (4.1) reduces to the Cauchy type integral equation. Subsequent to the analysis of Carleman, many more results have been found and have occurred extensively in the literature [56,62, 70,86,103].

### 5. Distributional Solutions of Singular Integral Equations

Abstract
In this chapter we study the distributional solutions of singular integral equations. The distributional framework is rather convenient for the study of several singular operators, including the singular integrals that we have studied in the previous chapters. Therefore, the study of singular integral equations in spaces of distributions is a natural and important subject. Interestingly, however, the interpretation of singular integrals at the end-points is not straightforward, and so the study of singular integral equations in spaces of distributions over finite intervals turns out to be somewhat difficult.

### 6. Distributional Equations on the Whole Line

Abstract
In this chapter we continue the study of the distributional solutions of integral equations. Our aim is to present the solution of singular integral equations of the type where
$$a(x)g(x) + b(x)H\{ g(t);x\} = f(x), - \infty < x < \infty$$
(6.1)
, where H is the Hilbert transform
$$H(g) = - \frac{1}{\pi }{{x}^{{ - 1}}}*g(x) = \frac{1}{\pi }p.v.\int_{{ - \infty }}^{\infty } {\frac{{g(t)}}{{t - x}}dt}$$
(6.2)
. In the previous chapter we considered the distributional solution of integral equations over finite intervals; here we conduct the analysis of the distributional Cauchy and Carleman equations over the whole real line.

### 7. Integral Equations with Logarithmic Kernels

Abstract
Singular integral equations with logarithmic kernels arise in analysis and in many two-dimensional problems in mathematical physics, mechanics and engineering such as potential and scattering theories.

### 8. Wiener-Hopf Integral Equations

Abstract
The purpose of this chapter is to study the distributional solution of the integral equations of the type
$$g(x) + \lambda \int_{0}^{\infty } {k(x - y)g(y)dy = f(x), x \geqslant 0}$$
(8.1)
, as well as the corresponding equations of the first kind, the so-called Wiener-Hopf integral equations. Observe that the kernel k(x–y) is a difference kernel and that the interval of integration is [0, ∞).