Equations of mathematical physics.
(Uravneniya matematicheskoj fiziki.)

*(Russian)*Zbl 0954.35001
Moskva: Fiziko-Matematicheskaya Literatura. 398 p. (2000).

In this book the problems for differential equations of classical mathematical physics are studied. The concept of generalized solution is broadly used. This concept is based on the notion of generalized function (distribution) and generalized derivative. The investigation of generalized solutions requires a generalized formulation of boundary value problems.

In Chapter I a classification and the formulation of basic boundary value problems of mathematical physics are given, as well as some required information from mathematical analysis. Chapter II contains elements of generalized functions’ theory, including the Fourier transformation of generalized functions. In Chapter III fundamental solutions for differential operators with constant coefficients are constructed. Generalized and classical Cauchy problems for the wave and heat equations are investigated too. For the generalized Cauchy problem initial conditions are taken into account as instantly acting sources. The Cauchy problem’s solution is written as convolution between source and fundamental solution. Chapter IV contains the theory of integral equations with continuous kernel. Theorems by Fredholm, Hilbert-Schmidt, and Mercer are formulated and some of them are proved. In Chapter V eigenvalue problems for elliptic equations, including the problem of Sturm-Liouville, as well as boundary value problems for Laplace and Poisson equations are considered. Mixed problems for equations of hyperbolic and parabolic types in generalized and classical cases are investigated in Chapter VI. The Fourier method and its motivation are given too. In the Appendix elementary theory of harmonic functions, spherical functions and Bessel functions are given.

A number of book sections are illustrated by examples and exercises are given too.

This book is designed for students of technical institutions with certain mathematical experience since the language of functional analysis is used.

In Chapter I a classification and the formulation of basic boundary value problems of mathematical physics are given, as well as some required information from mathematical analysis. Chapter II contains elements of generalized functions’ theory, including the Fourier transformation of generalized functions. In Chapter III fundamental solutions for differential operators with constant coefficients are constructed. Generalized and classical Cauchy problems for the wave and heat equations are investigated too. For the generalized Cauchy problem initial conditions are taken into account as instantly acting sources. The Cauchy problem’s solution is written as convolution between source and fundamental solution. Chapter IV contains the theory of integral equations with continuous kernel. Theorems by Fredholm, Hilbert-Schmidt, and Mercer are formulated and some of them are proved. In Chapter V eigenvalue problems for elliptic equations, including the problem of Sturm-Liouville, as well as boundary value problems for Laplace and Poisson equations are considered. Mixed problems for equations of hyperbolic and parabolic types in generalized and classical cases are investigated in Chapter VI. The Fourier method and its motivation are given too. In the Appendix elementary theory of harmonic functions, spherical functions and Bessel functions are given.

A number of book sections are illustrated by examples and exercises are given too.

This book is designed for students of technical institutions with certain mathematical experience since the language of functional analysis is used.

Reviewer: V.V.Karachik (Tashkent)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35Kxx | Parabolic equations and parabolic systems |

35Lxx | Hyperbolic equations and hyperbolic systems |