Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.
Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It 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 the Hilbert-Schmidt theory of linear operators in Hilbert space. This new edition of 'Integral Equations' offers the additional benefit of solutions to selected problems.
Author(s): B. L. Moiseiwitsch
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 2005
Language: English
Pages: 176
Tags: Математика;Интегральные уравнения;
Preface to the Dover Edition iii
Preface v
1: Classification of integral equations
1.1 Historical introduction 1
1.2 Linear integral equations 3
1.3 Special types of kernel 4
1.3.1 Symmetric kernels 4
1.3.2 Kernels producing convolution integrals 5
1.3.3 Separable kernels 6
1.4 Square integrable functions and kernels 8
1.5 Singular integral equations 9
1.6 Non-linear equations 11
Problems 12
2: Connection witlt difterential equations
2.1 Linear differential equations 14
2.2 Green's function 18
2.3 Influence function 20
Problems 22
3: Integral equations of the convolution type
3.1 Integral transforms 24
3.2 Fredholm equation of the second kind 26
3.3 Volterra equation of the second kind 31
3.4 Fredholm equation of the first kind 34
3.4.1 Stieltjes integral equation 34
3.5 Volterra equation of the first kind 36
3.5.1 Abel's integral equation 37
3.6 Fox's integral equation 39
Problems 40
4: Method of successive approximations
4.1 Neumann series 43
4.2 Iterates and the resolvent kernel 46
Problems 51
5: Integral equations witlt sinplar kernels
5.1 Generalization to higher dimensions 53
5.2 Green's functions in two and three dimensions 54
5.3 Dirichlet's problem 55
5.3.1 Poisson's formula for the unit disc 59
5.3.2 Poisson's formula for the half plane 60
5.3.3 Hilbert kernel 61
5.3.4 Hilbert transforms 63
5.4 Singular integral equation of Hilbert type 65
Problems 67
6: Hilbert space
6.1 Euclidean space 69
6.2 Hilbert space of sequences 71
6.3 Function space 74
6.3.1 Orthonormal system of functions 75
6.3.2 Gram-Schmidt orthogonalization 76
6.3.3 Mean square convergence 77
6.3.4 Riesz-Fischer theorem 79
6.4 Abstract Hilbert space 80
6.4.1 Dimension of Hilbert space 82
6.4.2 Complete orthonormal system 82
Problems 83
7: Linear operators in Hilbert space
7.1 Linear integral operators 85
7.1.1 Norm of an integral operator 87
7.1.2 Hermitian adjoint 88
7.2 Bounded linear operators 89
7.2.1 Matrix representation 91
7.3 Completely continuous operators 92
7.3.1 Integral operator with square integrable kernel 93
Problems 95
8: The resolvent
8.1 Resolvent equation 98
8.2 Uniqueness theorem 99
8.3 Characteristic values and functions 101
8.4 Neumann series 102
8.4.1 Volterra integral equation of the second kind 105
8.4.2 Bacher's example 109
8.5 Fredholm equation in abstract Hilbert space 109
Problems 111
9: Fredholm theory
9.1 Degenerate kernels 114
9.2 Approximation by degenerate kernels 120
9.3 Fredholm theorems 121
9.3.1 Fredholm theorems for completely continuous operators 125
9.4 Fredholm formulae for continuous kernels 126
Problems 135
10: Hilbert-Schmidt theory
10.1 Hermitian kernels 136
10.2 Spectrum of a Hilbert-Schmidt kernel 136
10.3 Expansion theorems 139
10.3.1 Hilbert-Schmidt theorem 141
10.3.2 Hilbert's formula 143
10.3.3 Expansion theorem for iterated kernels 143
10.4 Solution of Fredholm equation of second kind 144
10.5 Bounds on characteristic values 146
10.6 Positive kernels 147
10.7 Mercer's theorem 148
10.8 Variational principles 150
10.8.1 Rayleigh-Ritz variational method 152
Problems 154
Bibliography 157
Index 158
Answers to problems 162
