Linear integral equations.

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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