Singular Integral Equations

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.

1 Reference Material.- 1.1 Introduction.- 1.2 Singular Integral Equations.- 1.3 Improper Integrals.- 1.3.1 The Gamma function.- 1.3.2 The Beta function.- 1.3.3 Another important improper integral.- 1.3.4 A few integral identities.- 1.4 The Lebesgue Integral.- 1.5 Cauchy Principal Value for Integrals.- 1.6 The Hadamard Finite Part.- 1.7 Spaces of Functions and Distributions.- 1.8 Integral Transform Methods.- 1.8.1 Fourier transform.- 1.8.2 Laplace transform.- 1.9 Bibliographical Notes.- 2 Abel’s and Related Integral Equations.- 2.1 Introduction.- 2.2 Abel’s Equation.- 2.3 Related Integral Equations.- 2.4 The equation $$\int_{0}^{s} {{{{(s - t)}}^{\beta }}g(t)dt = f(s), \Re e \beta > - 1}$$.- 2.5 Path of Integration in the Complex Plane.- 2.6 The Equation $$\int_{{{{C}_{{a\xi }}}}} {\frac{{g(z)dz}}{{{{{(z - \xi )}}^{\nu }}}}} + k\int_{{{{C}_{{\xi b}}}}} {\frac{{g(z)dz}}{{{{{(\xi - z)}}^{\nu }}}}} = f(\xi )$$.- 2.7 Equations on a Closed Curve.- 2.8 Examples.- 2.9 Bibliographical Notes.- 2.10 Problems.- 3 Cauchy Type Integral Equations.- 3.1 Introduction.- 3.2 Cauchy Type Equation of the First Kind.- 3.3 An Alternative Approach.- 3.4 Cauchy Type Equations of the Second Kind.- 3.5 Cauchy Type Equations on a Closed Contour.- 3.6 Analytic Representation of Functions.- 3.7 Sectionally Analytic Functions (z?a)n?v(z?b)m+v.- 3.8 Cauchy’s Integral Equation on an Open Contour.- 3.9 Disjoint Contours.- 3.10 Contours That Extend to Infinity.- 3.11 The Hilbert Kernel.- 3.12 The Hilbert Equation.- 3.13 Bibliographical Notes.- 3.14 Problems.- 4 Carleman Type Integral Equations.- 4.1 Introduction.- 4.2 Carleman Type Equation over a Real Interval.- 4.3 The Riemann-Hilbert Problem.- 4.4 Carleman Type Equations on a Closed Contour.- 4.5 Non-Normal Problems.- 4.6 A Factorization Procedure.- 4.7 An Operational Approach.- 4.8 Solution of a Related Integral Equation.- 4.9 Bibliographical Notes.- 4.10 Problems.- 5 Distributional Solutions of Singular Integral Equations.- 5.1 Introduction.- 5.2 Spaces of Generalized Functions.- 5.3 Generalized Solution of the Abel Equation.- 5.4 Integral Equations Related to Abel’s Equation.- 5.5 The Fractional Integration Operators ??.- 5.6 The Cauchy Integral Equation over a Finite Interval.- 5.7 Analytic Representation of Distributions of ?’[a, b].- 5.8 Boundary Problems in A[a,b].- 5.9 Disjoint Intervals.- 5.9.1 The problem [RjF]j =hj.- 5.9.2 The equation A1?1(0F) + A2?2(F) = G.- 5.10 Equations Involving Periodic Distributions.- 5.11 Bibliographical Notes.- 5.12 Problems.- 6 Distributional Equations on the Whole Line.- 6.1 Introduction.- 6.2 Preliminaries.- 6.3 The Hilbert Transform of Distributions.- 6.4 Analytic Representation.- 6.5 Asymptotic Estimates.- 6.6 Distributional Solutions of Integral Equations.- 6.7 Non-Normal Equations.- 6.8 Bibliographical Notes.- 6.9 Problems.- 7 Integral Equations with Logarithmic Kernels.- 7.1 Introduction.- 7.2 Expansion of the Kernel In |x-y|.- 7.3 The Equation $$\int_{a}^{b} {\ln } \left| {x - y} \right|g(y)dy = f(x)$$.- 7.4 Two Related Operators.- 7.5 Generalized Solutions of Equations with Logarithmic Kernels.- 7.6 The Operator $$\int_{a}^{b} {(P(x - y)\ln \left| {x - y} \right| + Q(x,y))g(y)dy}$$.- 7.7 Disjoint Intervals of Integration.- 7.8 An Equation Over a Semi-Infinite Interval.- 7.9 The Equation of the Second Kind Over a Semi-Infinite Interval.- 7.10 Asymptotic Behavior of Eigenvalues.- 7.11 Bibliographical Notes.- 7.12 Problems.- 8 Wiener-Hopf Integral Equations.- 8.1 Introduction.- 8.2 The Holomorphic Fourier Transform.- 8.3 The Mathematical Technique.- 8.4 The Distributional Wiener-Hopf Operators.- 8.5 Illustrations.- 8.6 Bibliographical Notes.- 8.7 Problems.- 9 Dual and Triple Integral Equations.- 9.1 Introduction.- 9.2 The Hankel Transform.- 9.3 Dual Equations with Trigonometric Kernels.- 9.4 Beltrami’s Dual Integral Equations.- 9.5 Some Triple Integral Equations.- 9.6 Erdélyi-Köber Operators.- 9.7 Dual Integral Equations of the Titchmarsh Type.- 9.8 Distributional Solutions of Dual Integral Equations.- 9.8.1 Fractional integration in H’?v.- 9.8.2 Solution of the distributional problem.- 9.8.3 Uniqueness.- 9.9 Bibliographical Notes.- 9.10 Problems.- References.