# 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