Integral Equations

Integral Equations

And Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology

1st Edition - January 1, 1957

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  • Author: S. G. Mikhlin
  • eBook ISBN: 9781483226279

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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



Product details

  • No. of pages: 356
  • Language: English
  • Copyright: © Pergamon 1957
  • Published: January 1, 1957
  • Imprint: Pergamon
  • eBook ISBN: 9781483226279

About the Author

S. G. Mikhlin

About the Editors

I. N. Sneddon

M. Stark

S. Ulam

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