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Ⅰ．Volterra Equations 1

1．1．A Mechanical Problem Leading to an Integral Equation 1

1．2．Integral Equations and Algebraic Systems of Linear Equations 3

1．3．Volterra Equations 5

1．4．L2-Kernels and Functions 8

1．5．Solution of Volterra Integral Equations of the Second Kind 10

1．6．Volterra Equations of the First Kind 15

1．7．An Example 17

1．8．Volterra Integral Equations and Linear Differential Equations 18

1．9．Equations of the Faltung Type(Closed Cycle Type) 22

1．10．Transverse Oscillations of a Bar 26

1．11．Application to the Bessol Functions 32

1．12．Some Generalizations of the Theory of Volterra Equations 38

1．13．Non-Linear Volterra Equations 42

Ⅱ．Fredholm Equations 49

2．1．Solution by the Method of Successive Approximations:Neumann＇s Series 49

2．2．An Example 53

2．3．Fredholm＇s Equations with Pincherle-Goursat Kernels 55

2．4．The Fredholm Theorem for General Kernels 64

2．5．The Formulae of Fredholm 66

2．6．Numerical Solution of Integral Equations 75

2．7．The Fredholm Solution of the Dirichlet Problem 76

Ⅲ．Symmetric Kernels and Orthogonal Systems of Functions 81

3．1．Introductory Remarks and a Process of Orthogonalization 81

3．2．Approximation and Convergence in the Mean 83

3．3．The Riesz-Fischer Theorem 88

3．4．Completeness and Closure 90

3．5．Completeness of the Trigonometric System and of the Polynomials 95

3．6．Approximation of a General L2-Kernel by Means of PG-Kernels 98

3．7．Enskog＇s Method 100

3．8．The Spectrum of a Symmetric Kernel 102

3．9．The Bilinear Formula 106

3．10．The Hilbert-Schmidt Theorem and Its Applications 110

3．11．Extremal Properties and Bounds for Eigenvalues 118

3．12．Positive Kernels—Mercer＇s Theorem 124

3．13．Connection with the Theory of Linear Differential Equations 127

3．14．Critical Velocities of a Rotating Shaft and Transverse Oscillations of a Beam 136

3．15．Symmetric Fredholm Equations of the First Kind 143

3．16．Reduction of a Fredholm Equation to a Similar One with a Symmetric Kernel 145

3．17．Some Generalizations 150

3．18．Vibrations of a Membrane 154

Ⅳ．Some Types of Singular or Non-Linear Integral Equations 161

4．1．Orientation and Examples 161

4．2．Equations with Cauchy＇s Principal Value of an Integral and Hilbert＇s Transformation 166

4．3．The Finite Hilbert Transformation and the Airfoil Equation 173

4．4．Singular Equations of the Carleman Type 185

4．5．General Remarks About Non-Linear Integral Equations 197

4．6．Non-Linear Equations of the Hammerstein Type 202

4．7．Forced Oscillations of Finite Amplitude 213

Appendix Ⅰ．Algebraic Systems of Linear Equations 219

Appendix Ⅱ．Hadamard＇s Theorem 223

Exercises 227

References 231

Index 235