Integral equations 积分方程

Geared toward upper-level undergraduate students, this text begins with a straightforward account, accompanied by simple examples of a variety of integral equations and the methods of their solution. The treatment becomes gradually more abstract, with discussions of Hilbert space and linear operators, the resolvent, Fredholm theory, and more. 1977 edition.

Preface to the Dover Edition
Preface
Errata
1:Classification of integral epuations
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 dtthrential 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
……
9：Fredholm theory
10:Hilbert-Schmidt theory
Bibliography
Index
Answers to problems