Vladimirov V.S. (ed.) A Collection of Problems on the Equations of Mathematical Physics

Moscow: Mir Publishers; Berlin etc.: Springer, 1988. — 285 p.The extensive application of modern mathematical techniques to theoretical and mathematical physics requires a fresh approach to the course of equations of mathematical physics. This is especially true with regards to such a fundamental concept as the solution of a boundary value problem. The concept of a generalized solution considerably broadens the field of problems and enables solving from a unified position the most interesting problems that cannot bo solved by applying classical methods. To this end two new courses have been written at the Department of High« Mathematics at the Moscow Physics and Technology Institute, namely, "Equations of Mathematical Physics" by V.S. Vladimirov and “Partial Differential Equations” by V.P. Mikhailov (both books have been translated Id to English by Mir Publishers, the first in 1984 and the second in 1978). The present collection of problems is based on these courses and amplifies Diem considerably. Besides the classical boundary value problems, we have included a large number of boundary value problems that have only generalized solutions. Solution of these requires using the methods and results of various branches of modern analysis. For this reason we have included problems in Lebesgue integration, problems involving function spaces (especially spaces of generalized differentiable functions) and generalized functions (with Fourier and Laplace transforms), and integral equations. The book Is aimed at undergraduate and graduate students in the physical sciences, engineering, and applied mathematics who have taken the typical ‘‘methods" course that includes vector analysis, elementary complex variables, and an introduction to Fourier series and boundary value problems. Asterisks denote the more difficult problems.We would like to express our gratitude to all who helped with constructive comment to improve this book, our colleagues at the Department of Higher Mathematics at the Moscow Physics and Technology Institute, and especially T.F. Volkov, Yu.N. Drozhzhi- nov, A.F. Nikiforov, and V.I. Chekhiov.
V.S. Vladimirov V.P. Mikhailov A.A. Vasharin Kh.Kh. Karimova Yu.V. Sidorov M.I. ShabPreface S Symbols and Definitions 0 Chapter I Statement of Boundary Value Problems to Mathematical Physics 1 Deriving Equations of Mathematical Physics 2 Classification of Second-order Equations Chapter II Function Spaces and Integral Equations 3 Measurable Functions. The Lebesgne Integral 4 Function Spaces 5 Integral Equations Chapter III Generalized Functions 6 Test and Generalized Functions 7 Differentiation of Generalized Functions 8 The Direct Product aod Convolution of Generalized Functions 9 The Fourier Transform of Generalized Functions of Slow Growth 10 The Laplace Transform of Generalized Functions 11 Fundamental Solutions of Linear Differential Operators Chapter IV The Cauchy Problem 12 The Cauchy Problem for Second-order Equations of Hyperbolic Type 13 The Cauchy Problem for the Heat Conduction Equation 14 The Cauchy Problem for Other Equations sod Goursat’s Problem Chapter V Boundary Value Problems for Equations of Elliptic Type 15 The Sturm-Liouville Problem 16 Fourier’s Method for Laplace’s and Poisson’s Equations 17 Green's Functions of the Dirichlet Problem 18 The Method of Potentials 19 Variational Methods Chapter VI M ind Problems 20 Fourier's Method 21 Other Methods a 32 2B5S 2 asssssiassAppendix Examples oi Solution Techniques for Some Typical Problems A1 Method of Characteristics A2 Fourier’s Method A3 integral Equations with a Degenerate Kernel A4 Variational Problems References Subject Index