Integral Equations
And Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology
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Integral Equations: And their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, Second Revised Edition contains an account of the general theory of Fredholm and Hilbert-Schmidt. This edition discusses methods of approximate solution of Fredholm's equation and, in particular, their application to the solution of basic problems in mathematical physics, including certain problems in hydrodynamics and the theory of elasticity. Other topics include the equations of Volterra type, determination of the first eigenvalue by Ritz's method, and systems of singular integral equations. The generalized method of Schwarz, convergence of successive approximations, stability of a rod in compression, and mixed problem of the theory of elasticity are also elaborated. This publication is recommended for mathematicians, students, and researchers concerned with singular integral equations.
Table of Contents
Preface to Second English edition
Preface to First Edition
Translator's Note
Part I Methods of Solution of Integral Equations
I. Equations of Fredholm Type
1. Classification of Integral Equations
2. Method of Successive Approximations: Notion of the Resolvent
3. Equations of Volterra Type
4. Integral Equations with Degenerate Kernels
5. General Case of Fredholm's Equation
6. Systems of Integral Equations
7. Application of Approximate Formulae of Integration
8. Fredholm's Theorems
9. Fredholm's Resolvent
10. Equations with a Weak Singularity
II. Symmetric Equations (Theory of Hilbert-Schmidt)
11. Symmetric Kernels
12. Fundamental Theorems for Symmetric Equations
13. Hilbert-Schmidt Theorem
14. Determination of the First Eigenvalue by Ritz's Method
15. Determination of the First Eigenvalue Using the Trace of the Kernel
16. Kellogg's Method
17. Determination of Subsequent Eigenvalues
18. Kernels Reducible to Symmetric Kernels
19. Solution of Symmetric Integral Equations
20. Theorem of the Existence of an Eigenvalue
III. Singular Integral Equations
21. Principal Value of an Integral
22. The Kernels of Cauchy and Hilbert
23. Formulae for the Compounding of Singular Integrals
24. Singular Integral Equations with Hubert's Kernel
25. Singular Integral Equations with Cauchy's Kernel
26. The Case of the Unclosed Continuous Contour
27. The Case of the Unclosed Discontinuous Contour
28. Systems of Singular Integral Equations
Part II Applications of Integral Equations
IV. Dirichlet's Problem and its Application
29. Dirichlet's Problem for a Simply-Connected Plane Region
30. Example: Conformal Transformation of the Interior of an Ellipse Onto a Circle
31. Dirichlet's Problem for Multi-Connected Regions
32. The Modified Dirichlet Problem and the Neumann Problem
33. Torsion of Solid and Hollow Cylinders
34. Torsion of a Cylinder with Square Section
35. The Problem of Flow
36. Flow Past Two Elliptic Cylinders
37. Conformal Transformation of Multi-Connected Regions
38. Dirichlet's and Neumann's Problems in Three Dimensions
V. The Biharmonic Equation (Application of Green's Function)
39. Problems Reducing to the Biharmonic Equation
40. Complex Representation of a Biharmonic Function
41. Green's Function and Schwarz's Kernel
42. Reduction of the First and Third Problems to an Integral Equation
43. Analysis of the Integral Equation
44. The Case of a Simply-Connected Region
45. Confocal Elliptical Ring
46. Exterior of Two Ovals
47. On the Convergence of the Series of Successive Approximations
VI. The Generalized Method of Schwarz
48. Dirichlet's Problem for a Multi-Connected Plane Region
49. The Case of a Three-Dimensional Region
50. Generalized Method of Schwarz
51. Air Flow Past an Aeroplane Wing Close to the Ground
52. Application to the Problem of the Theory of Elasticity
54. Application of Cauchy Integrals to the Plane Theory of Elasticity (N. I. Muskhelishvili's Equation)
VII. Certain Applications of Integrals Analogous to Potentials
53. Eccentric Circular Ring, Uniformly Compressed at the Outer Circumference
55. Elastic Plane with an Infinite Series of Holes
56. Lauricella's Equation
57. Dirichlet's Problem for the Helmholtz Equation
58. Heat Potentials and their Applications
59. Convergence of Successive Approximations
VIII. Application of the Theory of Symmetric Integral Equations
60. The Problem of the Fundamental Vibrations of a String
61. Vibrations of a String Whose Density Varies According to a Linear Law
62. The Influence Function (Green's Function)
63. Torsional Vibrations of a Rod. Allowance for Concentrated Masses
64. The Stability of a Rod in Compression (Longitudinal Bending of a Rod)
65. The Pressure of a Rigid Stamp on an Elastic Half-Space
IX. Certain Applications of the Theory of Singular Integral Equations
66. Hubert's Problem
67. Hubert's Problem for a Half-Plane
68. The Problem of Two Elastic Half-Planes in Contact
69. The Problem of Two Elastic Half-Planes in Contact (General Case)
70. The Pressure of a Rigid Stamp on an Elastic Half-Plane
71. The Case of Several Stamps
72. The Mixed Problem of the Theory of Elasticity
73. The Case of a Region Which Can be Mapped by a Rational Transformation Onto a Circle
74. The Problem of Flow Past an Arc of Given Shape
Bibliography
Index
Product details
- No. of pages: 356
- Language: English
- Copyright: © Pergamon 1957
- Published: January 1, 1957
- Imprint: Pergamon
- eBook ISBN: 9781483226279