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About the Book

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.

Table of Contents

PART 1

CONTENTS

PROBLEMS, Page 1

DERIVATION OF EQUATIONS AND FORMULATION OF PROBLEMS, Page 3

Mechanics, 3
Heat Conduction, 9
Electricity and Magnetism, 11
SOME SPECIAL METHODS FOR SOLVING HYPERBOLIC AND ELLIPTIC EQUATIONS, Page 19

Hyperbolic Equations, 19
Elliptic Equations: The Green’s Function Method, 27
Elliptic Equations: The Method of Conformal Mapping, 33
STEADY-STATE HARMONIC OSCILLATIONS, Page 42

Elastic Bodies: Free Oscillations, 43
Elastic Bodies: Forced Oscillations, 46
Electromagnetic Oscillations, 49
THE FOURIER METHOD, Page 55

Mechanics: Vibrating Systems, Acoustics, 60
Mechanics: Statics of Deformable Media, Fluid Dynamics, 73
Heat Conduction: Nonstationary Problems, 77
Heat Conduction: Stationary Problems, 83
Electricity and Magnetism, 91
THE EIGENFUNCTION METHOD FOR SOLVING INHOMOGENEOUS PROBLEMS, Page 103

Mechanics: Vibrating Systems, 107
Mechanics: Statics of Deformable Media, 114
Heat Conduction: Nonstationary Problems, 119
Heat Conduction: Stationary Problems, 124
Electricity and Magnetism, 131
INTEGRAL TRANSFORMS, Page 143

The Fourier Transform, 146
The Hankel Transform, 160
The Laplace Transform, 169
The Mellin Transform, 189
Integral Transforms Involving Cylinder Functions of Imaginary Order, 194
CURVILINEAR COORDINATES, Page 203

Elliptic Coordinates, 204
Parabolic Coordinates, 210
Two-Dimensional Bipolar Coordinates, 212
Spheroidal Coordinates, 219
Paraboloidal Coordinates, 231
Toroidal Coordinates, 233
Three-Dimensional Bipolar Coordinates, 242
Some General Problems on Separation of Variables, 247
INTEGRAL EQUATIONS, Page 253

Diffraction Theory, 254
Electrostatics, 259
PART 2 SOLUTIONS, Page 273

MATHEMATICAL APPENDIX, Page 381

Special Functions Appearing in the Text, 381
Expansions in Series of Orthogonal Functions, 384
Some Definite Integrals Frequently Encountered in the Applications, 386
Expansion of Some Differential Operators in Orthogonal Curvilinear Coordinates, 388
Supplement: 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, 401
Related Methods, 404
2.1 Galerkin’s Method, 404
2.2 Collocation, 407
2.3 Least Squares, 411
References, 412

BIBLIOGRAPHY, Page 415

NAME INDEX, Page 423

SUBJECT INDEX, Page 427

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