The aim of the present book is to ‘help the reader acquirethe proficiency needed to successfully apply themethods of mathematical physics to a variety of problemsdrawn from mechanics, the theory of heat conduction,and the theory of electric and magnetic phenomena.A wide range of topics is covered, including not onlyproblems of the simpler sort, but also problems of amore complicated nature involving such things ascurvilinear coordinates, integral transforms, certainkinds of integral equations, etc. The book is intendedboth for students concomitantly studying the correspondingtopics in courses of mathematical physics,and for research scientists who in their work find itnecessary to carry out calculations using the methodsdescribed here. We also think that quite apart from itsvalue as a tool for acquiring technique, the book canalso serve as a handbook, especially in view of the factthat answers to the problems are included.A rather solid background in applied mathematics isneeded to profit from the book in its entirety. However,most of the problems appearing in Chapters 2 to 5 willbe accessible to those who have taken only the usualfirst course in methods of mathematical physics. Chapters6 to 8 are more specialized, and presuppose somefamiliarity with special functions, integral transforms,integral equations, and so on.To make the book easier to use, each section beginswith a brief introduction describing its contents andpresenting a certain amount of relevant backgroundinformation. However, it is not claimed that this informationis complete in any sense, and the reader desiring further details must consult the literature, e.g., the books and monographs cited at the end of each chapter.The majority of problems in this collection are accompaniedby hints, facilitating the choice of meaningfulmethods of solution. In addition, certain problems,whose numbers are equipped with asterisks (e.g., *52,*148, etc.), are solved in detail in a special section atthe end of the book. The problems singled out in thisway have been selected either because they illustrate theapplication of certain specific methods, or because oftheir special difficulty or particular importance in theapplications. Because of the applied character of thebook, we restrict ourselves to formal solutions, whoserigorous justification can be supplied by the interestedreader.In compiling the collection, we have consulted notonly the classic works on mathematical physics, butalso a number of journal articles. Material accumulatedduring years of teaching and research in the Departmentof Mathematical Physics at the Leningrad PolytechnicInstitute, as well as work done in connection with industrialprojects, plays a role in the material presentedhere.Translated and Edited by Richard A. SilvermanPART 1CONTENTSPROBLEMS, Page 1DERIVATION OF EQUATIONS AND FORMULATION OF PROBLEMS, Page 3Mechanics, 3Heat Conduction, 9Electricity and Magnetism, 11SOME SPECIAL METHODS FOR SOLVING HYPERBOLIC AND ELLIPTIC EQUATIONS, Page 19Hyperbolic Equations, 19Elliptic Equations: The Green’s Function Method, 27Elliptic Equations: The Method of Conformal Mapping, 33STEADY-STATE HARMONIC OSCILLATIONS, Page 42Elastic Bodies: Free Oscillations, 43Elastic Bodies: Forced Oscillations, 46Electromagnetic Oscillations, 49THE FOURIER METHOD, Page 55Mechanics: 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 103Mechanics: Vibrating Systems, 107Mechanics: Statics of Deformable Media, 114Heat Conduction: Nonstationary Problems, 119Heat Conduction: Stationary Problems, 124Electricity and Magnetism, 131INTEGRAL TRANSFORMS, Page 143The Fourier Transform, 146The Hankel Transform, 160The Laplace Transform, 169The Mellin Transform, 189Integral Transforms Involving Cylinder Functions of Imaginary Order, 194CURVILINEAR COORDINATES, Page 203Elliptic 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 253Diffraction Theory, 254Electrostatics, 259PART 2 SOLUTIONS, Page 273MATHEMATICAL APPENDIX, Page 381Special 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 391Variational Methods, 3921.1 Formulation of Variational Problems, 3921.2 The Ritz Method, 3961.3 Kantorovich’s Method, 401Related Methods, 4042.1 Galerkin’s Method, 4042.2 Collocation, 4072.3 Least Squares, 411References, 412BIBLIOGRAPHY, Page 415NAME INDEX, Page 423SUBJECT INDEX, Page 427