Integral Equations of First Kind

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Linear Equation in Finite-Dimensional Spaces

• Linear Equations in Space l₂

• Fredholm Integral Equations of the Second Kind. The Fredholm Theorems

• One–dimensional Linear Singular Integral Equations. The Noether Theorems

• Normally Solvable Linear Equations in Linear Metric Spaces. The Hausdorff Theorems.

The following sections are included:

• Integral transformations as integral equations of the first kind

• On the representation of the solution of the Dirichlet problem for harmonic functions as a potential of a simple layer

• A tangential derivative of the potential of a simple layer

• On the representation of the solution of Neumann's problem in the form of a potential of a double layer

• An integral equation of the contact problem in the elasticity theory

• A stream problem for the flow around a finite straight segment

• Integral equations of the first kind in the theory of boundary value problems for elliptic systems

• The tautochrone problem

• Some general remarks on integral equations the book is concerned with

The following sections are included:

• Preliminary remarks

• Spectral problem for one system of the associated homogeneous integral equations

• Source–like functions. The Hilbert–Schmidt theorem

• The Fischer–Riesz theorem

• The Picard theorem

• The case of a symmetric kernel

• Some remarks concerning to the Picard theorem

• The spread of the Picard theorem to the equations of the first kind in the Hilbert space

The following sections are included:

• The concept of the Schwarz kernel and its explicit expressions for some domains

• Integral Equations of the First Kind with the Hilbert Kernel

• The Hardy Inversion Formulae

• The Inversion of the System of the Singular Integral Equations of the First Kind with the Kernels generated by the Schwarz Kernel in a Strip

• Integral Equations of the First Kind with the Kernels Generated by the Schwarz Kernel for an annulus

The following sections are included:

• The Concept of the Poisson Kernel

• The Concept of the Neumann Kernel

• The Neumann Kernel for the Unit Ball

• A Many–Dimensional Analogue of the Hilbert Inversion Formulae

• Many–Dimensional Analogues of the Hardy Inversion Formulae

• The Neumann Generalized Problem

The following sections are included:

• The Integral In The Sense Of A Cauchy Principle Value

• The Behaviour of the Derivative of the Cauchy Type Integral Near the Line of Integration

• The Behaviour of the Cauchy Type Integral (6.1) Near the Segment γ_δ

• The Behaviour of the Cauchy Type Integral (6.1) Near the Ends of γ

• The Construction of Solutions of a Class of Integral Equations with the Cauchy Principle Value Integral in a Finite Interval

• The Integral Equation of the First Kind with a Logarithmic Kernel

• The Dirichlet Problem for a Plane with a Gap

• The Foepl Equation

• The Study of the Integral Equation (2.10)

• The Study of the Equation (2.17)

• The Basis Property of the System of Functions {cos nυ, sin nυ}

The following sections are included:

• The Volterra Integral Equation of the First Kind

• The Volterra Equations of the First Kind with Singular Kernels

• The Abel Integral Equation

• The Abel Integral Equation with Constant Limits of Integration

• Reduction of Some Classes of Integral Equations of the First Kind with Constant Limits of Integration to the Fredholm Equations of the Second Kind

• The Origin of the Kernel of the Integral Equation (7.44)

• The Construction of the Solution of Equation (7.44)

• Some Other Conclusions from the Main Result of the Previous Section

• The Study of the Integral Equation (2.31)

The following sections are included:

• The Hilbert Inversion Formulae in the Case of Smooth Closed Contours on the Plane

• The Concept of a Two–Dimensional Holomorphic Vector

• The Inversion of the Integral Equation (8.1) in Vector Notations

• The Formulae (8.11) and (8.12) in the Cases of a Circumference and a Straight Line

• Domains with a Smooth and Piecewise Smooth Boundaries. Some Integral Identities

• A Three–dimensional Analogue of the Cauchy–Riemann System

• A Two–Dimensional Analogue of the Cauchy Principle Value Integral

• A Two–Dimensional Analogue of the Poincare–Bertrand Formula

• A Two–Dimensional Analogue of the Inversion Formulae (8.11) and (8.12)

• Formulae (8.61) and (8.69) in the Cases of the Plane and Sphere

• The Inversion of One System of Two–Dimensional Singular Integral Equations

• The Dirichlet Problem for a Space with a Crack

• A Two–Dimensional Analogue of the Integral Equation (6.60)

The following sections are included:

• REFERENCES