| 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 Spaces CHAPTER 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 Principle CHAPTER 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 Problems CHAPTER 5 The Dirichlet Problem APPENDIS A Crash Course in Distribution Theory References Index |
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