Integral equations and their applications to certain problems in mechanics, mathematical physics, and technology

✍ Scribed by Solomon Grigorʹevich Mikhlin

• Publisher: Pergamon Press
• Year: 1964
• Tongue: English
• Leaves: 344
• Series: International series of monographs in pure and applied mathematics 4.
• Edition: 2nd
• Category: Library

✦ Synopsis

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 Fredholms 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 Ritzs 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

Front Cover
Integral Equations
Copyright Page
Table of Contents
PREFACE TO THE SECOND ENGLISH EDITION
PREFACE TO THE FIRST EDITION
Translator's Note
Part I: METHODS OF SOLUTION OF INTEGRAL EQUATIONS
CHAPTER 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 resolvent10. Equations with a weak singularity
CHAPTER 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. CHAPTER III. SINGULAR INTEGRAL EQUATIONS21. 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 INTEGRALEQUATIONS
CHAPTER IV. DIRICHLET'S PROBLEM AND ITS APPLICATIONS
29. Dirichlet's problem for a simply-connected plane region. 30. Example: conformai transformation of the interior ofan ellipse onto a circle31. Dirichlet's problem for multi-connected regions
32. The modified Dirichlet problem and the Neumannproblem
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
CHAPTER V. THE BIHARMONIC EQUATION: (APPLICATION OF GREEN'S FUNCTION)
39. Problems reducing to the biharmonic equation. 40. Complex representation of a biharmonic function41. 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
CHAPTER 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.

✦ Subjects

Integral equations MATHEMATICS Calculus Mathematical Analysis