1.
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Volterra Equations |
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1.1 |
A Mechanical Problem Leading to an Integral Equation |
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1.2 |
Integral Equations and Algebraic Systems of Linear Equations |
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1.3 |
Volterra Equations |
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1.4 |
L subscript 2-Kernels and Functions |
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1.5 |
Solution of Volterra Integral Equations of the Second Kind |
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1.6 |
Volterra Equations of the First Kind |
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1.7 |
An Example |
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1.8 |
Volterra Integral Equations and Linear Differential Equations |
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1.9 |
Equations of the Faltung type (Closed Cycle Type) |
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1.10 |
Transverse Oscillations of a Bar |
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1.11 |
Application to the Bessel Functions |
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1.12 |
Some Generalizations of the Theory of Volterra Equations |
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1.13 |
Non-Linear Volterra Equations |
2. |
Fredholm Equations |
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2.1 |
Solution by the Method of Successive Approximations: Neumann's Series |
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2.2 |
An Example |
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2.3 |
Fredholm's Equations with Pincherle-Goursat Kernels |
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2.4 |
The Fredholm Theorem for General Kernels |
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2.5 |
The Formulae of Fredholm |
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2.6 |
Numerical Solution of Integral Equations |
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2.7 |
The Fredholm Solution of the Dirichlet Problem |
3. |
Symmetric Kernels and Orthogonal Systems of Functions |
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3.1 |
Introductory Remarks and a Process of Orthogonalization |
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3.2 |
Approximation and Convergence in the Mean |
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3.3 |
The Riesz-Fischer Theorem |
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3.4 |
Completeness and Closure |
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3.5 |
Completeness of the Trigonometric System and of the Polynomials |
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3.6 |
Approximation of a General L subscript 2-Kernel by Means of PG-Kernels |
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3.7 |
Enskog's Method |
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3.8 |
The Spectrum of a Symmetric Kernel |
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3.9 |
The Bilinear Formula |
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3.10 |
The Hilbert-Schmidt Theorem and Its Applications |
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3.11 |
Extremal Properties and Bounds for Eigenvalues |
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3.12 |
Positive Kernels--Mercer's Theorem |
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3.13 |
Connection with the Theory of Linear Differential Equations |
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3.14 |
Critical Velocities of a Rotating Shaft and Transverse Oscillations of a Beam |
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3.15 |
Symmetric Fredholm Equations of the First Kind |
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3.16 |
Reduction of a Fredholm Equation to a Similar One with a Symmetric Kernel |
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3.17 |
Some Generalizations |
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3.18 |
Vibrations of a Membrane |
4. |
Some Types of Singular or Non-Linear Integral Equations |
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4.1 |
Orientation and Examples |
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4.2 |
Equations with Cauchy's Principal Value of an Integral and Hilbert's Transformation |
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4.3 |
The Finite Hilbert Transformation and the Airfoil Equation |
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4.4 |
Singular Equations of the Carleman Type |
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4.5 |
General Remarks About Non-Linear Integral Equations |
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4.6 |
Non-Linear Equations of the Hammerstein Type |
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4.7 |
Forced Oscillations of Finite Amplitude |
Appendix I. |
Algebraic Systems of Linear Equations |
Appendix II. |
Hadamard's Theorem |
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Exercises; References; Index |
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