Linear integral equations.

Title page
PREFACE
CHAPTER I INTRODUCTORY
1. Linear Integral Equation of the First Kind
2. Abel's Problem
3. Linear Integral Equation of the Second Kind
4. Relation between Linear Differential Equation and Volterra's Integral Equation
5. Non-linear Equations
6. Singular Equations
7. Types of Solutions
Exercises
CHAPTER II SOLUTION OF INTEGRAL EQUATION OF SECOND KIND BY SUCCESSIVE SUBSTITUTIONS
8. Solution by Successive Substitutions
9. Volterra's Equation
10. Successive Approximations
11. Iterated Functions
12. Reciprocal Functions
13. Volterra's Solution of Fredhohn's Equation
14. Discontinuous Solutions
Exercises
CHAPTER III SOLUTION OF FREDHOLM'S EQUATION EXPRESSED AS RATIO OF TWO INTEGRAL SERIES IN λ
15. Fredholm's Equation as Limit of a Finite System of Linear Equations
16. Hadamard's Theorem
17. Convergence Proof
18. Fredholm's Two Fundamental Relations
19. Fredholm's Solution of the Integral Equation when D(λ) ≠ 0
20. Solution of the Homogeneous Equation when D(λ) = 0, D'(λ) ≠ 0
21. Solution of the Homogeneous Integral Equation when D(λ) = 0
22. Characteristic Constant. Fundamental Functions
23. The Associated Homogeneous Integral Equation
24. The Non-homogeneous Integral Equation when D(λ) = 0
25. Kernels of the Form ∑ a_i(x)b_i(y)
Exercises.
CHAPTER IV APPLICATIONS OF THE FREDHOLM THEORY
I. Free Vibrations of an Elastic String
26. The DifferentiaI Equations of the Problem
27. Reduction to a One-dimensional Boundary Problem
28. Solution of the Boundary Problem
29. Construction of Green's Function
30. Equivalence between the Boundary Problem and a Linear Integral Equation
II. Constrained Vibrations of an Elastic String
31. The Differential Equations of the Problem
32. Equivalence between the Boundary Problem and a Linear Integral Equation
33. Remarks on Solution of the Boundary Problem
III. Auxiliary Theorems on Harmonic Functions
34. Harmonic Functions
35. Definitions about Curves
36. Green's Theorem
37. The Analogue of Theorem IX for the Exterior Region
38. Generalization of the Preceding
IV. Logarithmic Potential of a Double Layer
39. Definition
40. Properties of w(x,y) at Points not on C
41. Behavior of w(x,y) on C
42. Behavior of ∂w/∂n on the Boundary C and at Infinity
43. Case where w_i or w_c Vanish along C
V. Fredholm's Solution of Dirichlet's Problem
44. Dirichlet's Problem
45. Reduction to an Integral Equation
46. Solution of the Integral Equation
47. Index of λ = 1 for K(s₀,s)
VI. Logarithmic Potentiel of a Simple Layer
48. Definition
49. Properties of v(x,y)
VII. Fredholm's Solution of Neumann's Problem
50. Neumann's Problem
51. Reduction to an Integral Equation
52. Solution of the Integral Equation
CHAPTER V HILBERT-SCHMIDT THEORY OF INTEGRAL EQUATIONS WITH SYMMETRIC KERNELS. SOLUTION EXPRESSED IN TERMS OF A SET OF FUNDAMENTAL FUNCTIONS
I. Existence of at Least One Characteristic Constant
53. Introductory Remarks
54. Power Series for D'(λ)/D(λ)
55. Plan of Kneser's Proof
56. Lemmas on Iterations of a Symmetric Kernel
57. Schwarz's Inequality
58. Application of Schwarz's Inequality
II. Orthogonality
59. Orthogonality Theorem
60. Rea1ity of the Characteristic Constants
61. Complete Normalized Orthogonal System of Fundamental Functions
III. Expansion of an Arbitrary Function Arcording to the Fundamental Functions of a Complete Normalized Orthogonal System
62. (a) Problem of the Vibrating String Resumed
(b) Determination of the Coefficients in the General Problem
IV. Expansion of the Kernel According to the Fundamental Functions of a Complete Normalized Orthogonal System
63. (a) Determination of the Coefficients
(b) The Bilinear Formula for the Case of a Finite Number of Fundamental Functions
(c) The Bilinear Formula for Kernels Having an Infinite Number of Characteristic Constants
64. The Complete Normalized Orthogonal System for the Iterated Kernel K_n(x,t)
V. Auxiliary Theorems
65. Bessel's Inequality
66. Proof of the Bilinear Formula for the Iterated Kernel K_n(x,t) for n >= 4
67. An Auxiliary Theorem of Schmidt
68. VI. Expansion of an Arbitrary Function According to the Complete Normalized Orthogonal System of Fundamental Functions of a Symmetric Kernel
VII. Solution of the Integral Equation
69. Schmidt's Solution of the Non-homogeneous Integral Equation when λ is not a Characteristic Constant
70. Schmidt's Solution of the Non-homogeneous Integral Equation when λ is a Characteristic Constant
71. Remarks on Obtaining a Solution
Exercises
CHAPTER VI APPLICATIONS OF THE HILBERT-SCHMIDT THEORY
I. Boundary Problems for Ordinary Linear Differential Equations
72. Introductory Remarks
73. Construction of Green's Function
74. Equivalence between the Boundary Problem and a Homogeneous Linear Integral Equation
75. Special Case g(x) == 1
76. Miscellaneous Remarks
II. Applications to some Problems of the Calculus of Variations
77. Some Auxiliary Theorems of the Calculus of Variations
78. Dirichlet's Problem
79. Applications to the Second Variation
80. Connection with Jacobi's Condition
III. Vibration Problems
81. Vibrating String
82. Vibrations of a Rope
83. The Rotating Rope
IV. Applications of the Hilbert-Schmidt Theory to the Flow of Heat in a Bar
84. The Partial Differential Equations of the Problem
85. Application to an Example
86. General Theory of the Exceptional Case
87. Flow of Heat in a Ring
88. Stationary Flow of Heat Procluced by an Interior Source
89. Direct Computation of the Charactcristic Constants and Fundamental Functions
INDEX