Mikhlin / Sneddon / Stark Integral Equations

And Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology
1. Auflage 2014
ISBN: 978-1-4832-2627-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 4, 356 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

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.

Autoren/Hrsg.

Mikhlin, S. G.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Integral Equations;6
3;Copyright Page;7
4;Table of Contents;8
5;PREFACE TO THE SECOND ENGLISH EDITION;12
6;PREFACE TO THE FIRST EDITION;13
7;Translator's Note;13
8;Part I: METHODS OF SOLUTION OF INTEGRAL EQUATIONS;14
8.1;CHAPTER I. EQUATIONS OF FREDHOLM TYPE;16
8.1.1;§1. Classification of integral equations;16
8.1.2;§2. Method of successive approximations: Notion of the resolvent;20
8.1.3;§3. Equations of Volterra type;28
8.1.4;§4. Integral equations with degenerate kernels;32
8.1.5;§5. General case of Fredholm's equation;35
8.1.6;§6. Systems of integral equations;43
8.1.7;§7. Application of approximate formulae of integration;44
8.1.8;§8. Fredholm's theorems;47
8.1.9;§9. Fredholm's resolvent;59
8.1.10;§10. Equations with a weak singularity;72
8.2;CHAPTER II. SYMMETRIC EQUATIONS: (THEORY OF HILBERT-SCHMIDT);80
8.2.1;§11. Symmetric kernels;80
8.2.2;§12. Fundamental theorems for symmetric equations;86
8.2.3;§13. Hilbert-Schmidt Theorem;89
8.2.4;§14. Determination of the first eigenvalue by Ritz's method;94
8.2.5;§15. Determination of the first eigenvalue using the trace of the kernel;101
8.2.6;§16. Kellogg's method;107
8.2.7;§17. Determination of subsequent eigenvalues;112
8.2.8;§18. Kernels reducible to symmetric kernels;116
8.2.9;§19. Solution of symmetric integral equations;116
8.2.10;§20. Theorem of the existence of an eigenvalue;118
8.3;CHAPTER III. SINGULAR INTEGRAL EQUATIONS;126
8.3.1;§21. Principal value of an integral;126
8.3.2;§22. The kernels of Cauchy and Hilbert;130
8.3.3;§23. Formulae for the compounding of singular integrals;132
8.3.4;§24. Singular integral equations with Hubert's kernel;135
8.3.5;§25. Singular integral equations with Cauchy's kernel;139
8.3.6;§26. The case of the unclosed continuous contour;139
8.3.7;§27. The case of the unclosed discontinuous contour;144
8.3.8;§28. Systems of singular integral equations;146
9;Part II: APPLICATIONS OF INTEGRALEQUATIONS;148
9.1;CHAPTER IV. DIRICHLET'S PROBLEM AND ITS APPLICATIONS;150
9.1.1;§29. Dirichlet's problem for a simply-connected plane region;150
9.1.2;§30. Example: conformai transformation of the interior ofan ellipse onto a circle;154
9.1.3;§31. Dirichlet's problem for multi-connected regions;158
9.1.4;§32. The modified Dirichlet problem and the Neumannproblem;163
9.1.5;§33. Torsion of solid and hollow cylinders;166
9.1.6;§34. Torsion of a cylinder with square section;168
9.1.7;§35. The problem of flow;170
9.1.8;§36. Flow past two elliptic cylinders;172
9.1.9;§37. Conformal transformation of multi-connected regions;178
9.1.10;§38. Dirichlet's and Neumann's problems in three dimensions;182
9.2;CHAPTER V. THE BIHARMONIC EQUATION: (APPLICATION OF GREEN'S FUNCTION);189
9.2.1;§39. Problems reducing to the biharmonic equation;189
9.2.2;§40. Complex representation of a biharmonic function;192
9.2.3;§41. Green's function and Schwarz's kernel;197
9.2.4;§42. Reduction of the first and third problems to an integral equation;204
9.2.5;§43. Analysis of the integral equation;208
9.2.6;§44. The case of a simply-connected region;211
9.2.7;§45. Confocal elliptical ring;213
9.2.8;§46. Exterior of two ovals;218
9.2.9;§47. On the convergence of the series of successive approximations;225
9.3;CHAPTER VI. THE GENERALIZED METHOD OF SCHWARZ;233
9.3.1;§48. Dirichlet's problem for a multi-connected plane region;233
9.3.2;§49. The case of a three-dimensional region;238
9.3.3;§50. Generalized method of Schwarz;239
9.3.4;§51. Air flow past an aeroplane wing close to the ground;244
9.3.5;§52. Application to the problem of the theory of elasticity;246
9.3.6;§53. Eccentric circular ring, uniformly compressed at the outer circumference;252
9.4;CHAPTER VII. CERTAIN APPLICATIONS OF INTEGRALS ANALOGOUS TO POTENTIALS;256
9.4.1;§54. Application of Cauchy integrals to the plane theory of elasticity (N. I. Muskhelishvili's equation);256
9.4.2;§55. Elastic plane with an infinite series of holes;263
9.4.3;§56. Lauricella's equation;268
9.4.4;§57. Dirichlet's problem for the Helmholtz equation;274
9.4.5;§58. Heat potentials and their applications;280
9.4.6;§59. Convergence of successive approximations;285
9.5;CHAPTER VIII. APPLICATION OF THE THEORY OF SYMMETRIC INTEGRAL EQUATIONS;288
9.5.1;§60. The problem of the fundamental vibrations of a string;288
9.5.2;§61. Vibrations of a string, whose density varies according to a linear law;292
9.5.3;§62. The influence function (Green's function);295
9.5.4;§63. Torsional vibrations of a rod. Allowance for concentrated masses;300
9.5.5;§64 The stability of a rod in compression (longitudinal bending of a rod);302
9.5.6;§65. The pressure of a rigid stamp on an elastic half-space;305
9.6;CHAPTER IX. CERTAIN APPLICATIONS OF THE THEORY OF SINGULAR INTEGRAL EQUATIONS;311
9.6.1;§66. Hubert's problem;311
9.6.2;§67. Hubert's problem for a half-plane;313
9.6.3;§68. The problem of two elastic half-planes in contact;317
9.6.4;§69. The problem of two elastic half-planes in contact (general case);323
9.6.5;§70. The pressure of a rigid stamp on an elastic half-plane;325
9.6.6;§71. The case of several stamps;328
9.6.7;§72. The mixed problem of the theory of elasticity;329
9.6.8;§73. The case of a region which can be mapped by a rational transformation onto a circle;334
9.6.9;§74. The problem of flow past an arc of given shape;338
10;BIBLIOGRAPHY;348
11;INDEX;352