Integral Equations

Name: Integral Equations
Brand: Dover Publications
Price: 17.49 EUR
Availability: OnlineOnly

B. L. Moiseiwitsch(Autor*in)

Dover Publications (Verlag)

Erschienen am 30. November 2011

311 Seiten

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978-0-486-15212-7 (ISBN)

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Inhalt

Cover
Title Page
Copyright Page
Preface to the Dover Edition
Preface
Contents
Errata
1: Classification of integral equations
1.1 Historical introduction
1.2 Linear integral equations
1.3 Special types of kernel
1.3.1 Symmetric kernels
1.3.2 Kernels producing convolution integrals
1.3.3 Separable kernels
1.4 Square integrable functions and kernels
1.5 Singular integral equations
1.6 Non-linear equations
Problems
2: Connection with differential equations
2.1 Linear differential equations
2.2 Green's function
2.3 Influence function
Problems
3: Integral equations of the convolution type
3.1 Integral transforms
3.2 Fredholm equation of the second kind
3.3 Volterra equation of the second kind
3.4 Fredholm equation of the first kind
3.4.1 Stieltjes integral equation
3.5 Volterra equation of the first kind
3.5.1 Abel's integral equation
3.6 Fox's integral equation
Problems
4: Method of successive approximations
4.1 Neumann series
4.2 Iterates and the resolvent kernel
Problems
5: Integral equations with singular kernels
5.1 Generalization to higher dimensions
5.2 Green's functions in two and three dimensions
5.3 Dirichlet's problem
5.3.1 Poisson's formula for the unit disc
5.3.2 Poisson's formula for the half plane
5.3.3 Hilbert kernel
5.3.4 Hilbert transforms
5.4 Singular integral equation of Hilbert type
Problems
6: Hilbert space
6.1 Euclidean space
6.2 Hilbert space of sequences
6.3 Function space
6.3.1 Orthonormal system of functions
6.3.2 Gram-Schmidt orthogonalization
6.3.3 Mean square convergence
6.3.4 Riesz-Fischer theorem
6.4 Abstract Hilbert space
6.4.1 Dimension of Hilbert space
6.4.2 Complete orthonormal system
Problems
7: Linear operators in Hilbert space
7.1 Linear integral operators
7.1.1 Norm of an integral operator
7.1.2 Hermitian adjoint
7.2 Bounded linear operators
7.2.1 Matrix representation
7.3 Completely continuous operators
7.3.1 Integral operator with square integrable kernel
Problems
8: The resolvent
8.1 Resolvent equation
8.2 Uniqueness theorem
8.3 Characteristic values and functions
8.4 Neumann series
8.4.1 Volterra integral equation of the second kind
8.4.2 Bocher's example
8.5 Fredholm equation in abstract Hilbert space
Problems
9: Fredholm theory
9.1 Degenerate kernels
9.2 Approximation by degenerate kernels
9.3 Fredholm theorems
9.3.1 Fredholm theorems for completely continuous operators
9.4 Fredholm formulae for continuous kernels
Problems
10: Hilbert-Schmidt theory
10.1 Hermitian kernels
10.2 Spectrum of a Hilbert-Schmidt kernel
10.3 Expansion theorems
10.3.1 Hilbert-Schmidt theorem
10.3.2 Hubert's formula
10.3.3 Expansion theorem for iterated kernels
10.4 Solution of Fredholm equation of second kind
10.5 Bounds on characteristic values
10.6 Positive kernels
10.7 Mercer's theorem
10.8 Variational principles
10.8.1 Rayleigh-Ritz variational method
Problems
Bibliography
Index
Answers to problems

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