This book aims to equip readers with the mathematical physics skills necessary to solve problems in mechanics, heat conduction, and electromagnetism. It covers a wide range of topics, from basic to advanced, and is intended for both students and researchers. The book provides hints for solving problems and includes detailed solutions to selected ones. Readers should have a solid background in applied mathematics to fully benefit from the book, but most problems in the earlier chapters are accessible to those with a basic understanding of mathematical physics methods.
Translated from the Russian by Richard A. Silverman.
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Contents
PART 1
CONTENTS
PROBLEMS, Page 1
DERIVATION OF EQUATIONS AND FORMULATION OF PROBLEMS, Page 3
Mechanics, 3Heat Conduction, 9Electricity and Magnetism, 11SOME SPECIAL METHODS FOR SOLVING HYPERBOLIC AND ELLIPTIC EQUATIONS, Page 19
Hyperbolic Equations, 19Elliptic Equations: The Green’s Function Method, 27Elliptic Equations: The Method of Conformal Mapping, 33STEADY-STATE HARMONIC OSCILLATIONS, Page 42
Elastic Bodies: Free Oscillations, 43Elastic Bodies: Forced Oscillations, 46Electromagnetic Oscillations, 49THE FOURIER METHOD, Page 55
Mechanics: Vibrating Systems, Acoustics, 60Mechanics: Statics of Deformable Media, Fluid Dynamics, 73Heat Conduction: Nonstationary Problems, 77Heat Conduction: Stationary Problems, 83Electricity and Magnetism, 91THE EIGENFUNCTION METHOD FOR SOLVING INHOMOGENEOUS PROBLEMS, Page 103
Mechanics: Vibrating Systems, 107Mechanics: Statics of Deformable Media, 114Heat Conduction: Nonstationary Problems, 119Heat Conduction: Stationary Problems, 124Electricity and Magnetism, 131INTEGRAL TRANSFORMS, Page 143
The Fourier Transform, 146The Hankel Transform, 160The Laplace Transform, 169The Mellin Transform, 189Integral Transforms Involving Cylinder Functions of Imaginary Order, 194CURVILINEAR COORDINATES, Page 203
Elliptic Coordinates, 204Parabolic Coordinates, 210Two-Dimensional Bipolar Coordinates, 212Spheroidal Coordinates, 219Paraboloidal Coordinates, 231Toroidal Coordinates, 233Three-Dimensional Bipolar Coordinates, 242Some General Problems on Separation of Variables, 247INTEGRAL EQUATIONS, Page 253
Diffraction Theory, 254Electrostatics, 259PART 2 SOLUTIONS, Page 273
MATHEMATICAL APPENDIX, Page 381
Special Functions Appearing in the Text, 381Expansions in Series of Orthogonal Functions, 384Some Definite Integrals Frequently Encountered in the Applications, 386Expansion of Some Differential Operators in Orthogonal Curvilinear Coordinates, 388Supplement: VARIATIONAL AND RELATED METHODS, Page 391
Variational Methods, 392
1.1 Formulation of Variational Problems, 392
1.2 The Ritz Method, 396
1.3 Kantorovich’s Method, 401Related Methods, 404
2.1 Galerkin’s Method, 404
2.2 Collocation, 407
2.3 Least Squares, 411References, 412
BIBLIOGRAPHY, Page 415
NAME INDEX, Page 423
SUBJECT INDEX, Page 427
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