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And Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology
1st Edition - January 1, 1957
Author: S. G. Mikhlin
Editors: I. N. Sneddon, M. Stark, S. Ulam
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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… Read more
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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.
Preface to Second English editionPreface to First EditionTranslator's NotePart 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 EquationsPart 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 ShapeBibliographyIndex