Gakhov / Sneddon / Stark Boundary Value Problems

1. Auflage 2014
ISBN: 978-1-4831-6498-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 584 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Boundary Value Problems is a translation from the Russian of lectures given at Kazan and Rostov Universities, dealing with the theory of boundary value problems for analytic functions. The emphasis of the book is on the solution of singular integral equations with Cauchy and Hilbert kernels. Although the book treats the theory of boundary value problems, emphasis is on linear problems with one unknown function. The definition of the Cauchy type integral, examples, limiting values, behavior, and its principal value are explained. The Riemann boundary value problem is emphasized in considering the theory of boundary value problems of analytic functions. The book then analyzes the application of the Riemann boundary value problem as applied to singular integral equations with Cauchy kernel. A second fundamental boundary value problem of analytic functions is the Hilbert problem with a Hilbert kernel; the application of the Hilbert problem is also evaluated. The use of Sokhotski's formulas for certain integral analysis is explained and equations with logarithmic kernels and kernels with a weak power singularity are solved. The chapters in the book all end with some historical briefs, to give a background of the problem(s) discussed. The book will be very valuable to mathematicians, students, and professors in advanced mathematics and geometrical functions.

Autoren/Hrsg.

Gakhov, F. D.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Boundary Value Problems;4
3;Copyright Page
;5
4;Table of Contents;6
5;FOREWORD TO THE FIRST EDITION;14
6;FOREWORD TO THE SECOND EDITION;18
7;INTRODUCTION;20
8;CHAPTER I. INTEGRALS OF THE CAUCHY TYPE;22
8.1;§ 1. Definition of the Cauchy type integral and examples;22
8.2;§ 2. Functions satisfying the Hölder condition;26
8.3;§ 3. Principal value of the Cauchy type integral;28
8.4;§ 4. Limiting values of the Cauchy type integral. Integrals over the real axis;41
8.5;§ 5. Properties of the limiting values of the Cauchy type integral;59
8.6;§ 6. The Hilbert formulae for the limiting values of the real and imaginary parts of an analytic function;65
8.7;§ 7. The change of the order of integration in a repeated singular integral;67
8.8;§ 8. Behaviour of the Cauchy type integral at the ends of the contour of integration and at the points of density discontinuities;74
8.9;§ 9· Limiting values of generalized integrals and double Cauchy integrals;87
8.10;§ 10. Integral of the Cauchy type and potentials;94
8.11;§ 11. Historical notes;96
8.12;Problems on Chapter I;98
9;CHAPTER II. RIEMANN BOUNDARY VALUE PROBLEM;106
9.1;§ 12. The index;106
9.2;§ 13. Some auxiliary theorems;111
9.3;§ 14. The Riemann problem for a simply-connected domain;111
9.4;§ 15. Exceptional cases of the Riemann problem;128
9.5;§16. Riemann problem for multiply-connected domain. Some new results;134
9.6;§ 17. Riemann boundary value problem with shift;142
9.7;§ 18. Other generalized problems;154
9.8;§ 19. Historical notes;158
9.9;Problems on Chapter II;159
10;CHAPTER III. SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL;164
10.1;§ 20. Basic concepts and notation;164
10.2;§ 21. The dominant equation;169
10.3;§ 22. Regularization of the complete equation;182
10.4;§ 23. Fundamental properties of singular equations;192
10.5;§ 24. Equivalent regularization. The third method of regularization;199
10.6;§ 25. Exceptional cases of singular integral equations;215
10.7;§ 26. Historical notes;220
10.8;Problems on Chapter III;222
11;CHAPTER IV. HILBERT BOUNDARY VALUE PROBLEM AND SINGULAR INTEGRAL EQUATIONS WITH HILBERT KERNEL;228
11.1;§ 27. Formulation of the Hilbert problem and some auxiliary formulae;228
11.2;§ 28. Regularizing factor;234
11.3;§ 29. The Hilbert boundary value problem for simply-connected domains;241
11.4;§ 30. Relation between the Hilbert and Riemann problems;249
11.5;§ 31. Singular integral equation with Hilbert kernel;257
11.6;§ 32. Boundary value problems for polyharmonic and polyanalytic functions, reducible to the Hilbert boundary value problem;270
11.7;§ 33. The inverse boundary value problem for analytic functions;286
11.8;§ 33*. Historical notes;305
11.9;Problems on Chapter IV;307
12;CHAPTER V. VARIOUS GENERALIZED BOUNDARY VALUE PROBLEMS;311
12.1;§ 34. Boundary value problem of Hilbert type, with the boundary condition containing derivatives;312
12.2;§ 35. Boundary value problem of Riemann type with the boundary condition containing derivatives;337
12.3;§ 36. The Hilbert boundary value problem for multiply-connected domains;347
12.4;§ 36*. Inverse boundary value problem for a multiply-connected domain;379
12.5;§ 37. General boundary value problem of Riemann type for multiply-connected domains;383
12.6;§ 38. Boundary value problems for equations of elliptic type;387
12.7;§ 39. Boundary value problems for systems of elliptic equations;396
12.8;§ 40. Historical notes;419
12.9;Problems on Chapter V;420
13;CHAPTER VI. BOUNDARY VALUE PROBLEMS AND SINGULAR INTEGRAL EQUATIONS WITH DISCONTINUOUS COEFFICIENTS AND OPEN CONTOURS;427
13.1;§ 41. Solution of the Riemann problem with discontinuous coefficients by reduction to a problem with continuous coefficients;428
13.2;§ 42. Riemann boundary value problem for open contours;441
13.3;§ 43. Direct solution of the Riemann problem;449
13.4;§ 44. Riemann problem for a complicated contour;457
13.5;§ 45. Exceptional cases and the general concept of index;463
13.6;§ 46. Hilbert boundary value problem with discontinuous coefficients;470
13.7;§ 47. The dominant equation for open contours;483
13.8;§ 48. Complete equation for open contours;493
13.9;§ 49. The general case;501
13.10;§ 50. Historical notes;506
13.11;Problems on Chapter VI;508
14;CHAPTER VII. INTEGRAL EQUATIONS SOLUBLE IN CLOSED FORM;515
14.1;§ 51. Equations with automorphic kernels and a finite group;516
14.2;§ 52. Continuation. The case of an infinite group;533
14.3;§ 53. Some types of integral equations with power and logarithmic kernels;547
14.4;§ 54. Historical notes;565
14.5;Problems on Chapter VII;567
15;REFERENCES;572
16;INDEX;580
17;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;583