文档介绍：YuX Egorov . Shubin (Eds.)
Partial Differential
Equations I
Foundations of the Classical Theory
Springer-Verlag
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Encyclopaediaof
Mathematical Sciences
Volume 30
Editor-in-Chief: R. V. Ganikrelidze
Linear Partial DXerential Equations.
Foundations of the Classical Theory
Yu. V. Egorov, M. A. Shubin
Translated from the Russian
by R. Cooke
Contents
Preface ........................... 6
Chapter 1. Basic Concepts ................... 7
31. Basic Definitions and Examples ............... 7
. The Definition of a Linear Partial Differential Equation ... 7
. The Role of Partial Differential Equations in the
Mathematical Modeling of Physical Processes ....... 7
. Derivation of the Equation for the Longitudinal Elastic
Vibrations of a Rod .................. 8
. Derivation of the Equation of Heat Conduction ...... 9
. The Limits of Applicability of Mathematical Models .... 10
. Initial and Boundary Conditions ............. 11
. Examples of Linear Partial Differential Equations ..... 12
. The Concept of Well-Posedness of a Boundary-value
Problem. The Cauchy Problem ............. 21
52. The Cauchy-Kovalevskaya Theorem and Its Generalizations ... 28
. The Cauchy-Kovalevskaya Theorem ........... 28
. An Example of Nonexistence of an Analytic Solution .... 31
. Some Generalizations of the Cauchy-Kovalevskaya Theorem.
Characteristics .................... 31
. Ovsyannikov’s Theorem ................. 33
. Holmgren’s Theorem .................. 35
2 Contents
53. Classification of Linear Differential Equations. Reduction to
Canonical Form and Characteristics . . . . . . . . . . . . . 37
. Classification of Second-Order Equations and Their
Reduction to Canonical Form at a Point . . . . . . . . . 37
. Characteristics of Second-Order Equations and Reduc