偏微分方程

內(nèi)容概要

This book is based on a course I have given five times at the University of Michigan, beginning in 1973. The aim is to present an introduction to a sampling of ideas, phenomena, and methods from the subject of partial differential equations that can be presented in one semester and requires no previous knowledge of differential equations. The problems, with hints and discussion, form an important and integral part of the course. In our department, students with a variety of specialties--notably differential geometry, numerical analysis, mathematical physics, complex analysis, physics, and partial differential equations--have a need for such a course.

書籍目錄

Preface CHAPTER l Power Series Methods   1.1 The Simplest Partial Differential Equation   1.2 The lnitial Value Problem for Ordinary Differential Equations   1.3 Power Series and the Initial Value Problem for Partial Differential Equations   1.4 The Fully Nonlinear Cauchy-Kowaleskaya Theorem   1.5 Cauchy-Kowaleskaya with General Initial Surfaces   1.6 The Symbol ora Differential Operator   1.7 Holmgren's Uniqueness Theorem   1.8 Fritz John's Global Holmgren Theorem   1.9 Characteristics and Singular Solutions CHAPTER 2 Some Harmonic Analysis   2.1 The Schwartz Space (Rd)   2.2 The Fourier Transform on (Rd)   2.3 The Fourier Transform on Lp(Rd):1 ≤ p≤ 2   2.4 Tempered Distributions   2.5 Convolution in (Rd) and (Rd)   2.6 L2 Derivatives and Sobolev SpacesCHAPTER 3 Solution of Initial Value Problems by Fourier Synthesis  3.1 Introducion  3.2 Schrodinger's Equation  3.3 Solutions of Schrodinger's Equation with Data in (Rd)  3.4 Generalized Solutions of Schrodinger's Equation  3.5 Alternate Characteriztions of the Generalized Solution  3.6 Fourier Synthesis for the Heat Equation  3.7 Fourier Synthesis for the Wave Equation  3.8 Fourier Synthesis for the Cauchy-Riemann Operator  3.9 The Sidways Heat Equation and Null Solutions  3.10 The Hadamard-Petrowsky Dichotomy  3.11 Inhomogeneous Equation, Duhamels PrincipleCHAPTER 4 Propagators and x-Space Mehods  4.1 Introduction  4.2 Solution Formulas in x Space  4.3 Application of the Heat Propagator  4.4 Application of the Schrodinger Propagator  4.5 The Wave Equation Propagator for d=1  4.6 Rotation-Invariant Smooth Solutions of 1+3u=0  4.7 The Wave Equation Propagator for d=3  4.8 The Method of Descent  4.9 Radiation ProblemsCHAPTER 5 The Dirichlet ProblemAPPENDIS A Crash Course in Distribution TheoryReferencesIndex

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