Partial differential equations /
The objective of this book is to present an introduction to the ideas, phenomena, and methods of partial differential equations. This material can be presented in one semester and requires no previous knowledge of differential equations, but assumes the reader to be familiar with advanced calculus,...
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
New York :
Springer-Verlag,
c1991
New York : ©1991 New York : c1991 New York : [1991] |
| Series: | Graduate texts in mathematics ;
128 Graduate texts in mathematics 128 Graduate texts in mathematics ; 128 Graduate texts in mathematics 128 |
| Subjects: |
Table of Contents:
- 1 Power Series Methods
- §1.1. The Simplest Partial Differential Equation
- §1.2. The Initial 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 of a Differential Operator
- §1.7. Holmgren's Uniqueness Theorem
- §1.8. Fritz John's Global Holmgren Theorem
- §1.9. Characteristics and Singular Solutions
- 2 Some Harmonic Analysis
- §2.1. The Schwartz Space $$\mathcal{J}({\mathbb{R}̂d})$$
- §2.2. The Fourier Transform on $$\mathcal{J}({\mathbb{R}̂d})$$
- §2.3. The Fourier Transform onLp$${\mathbb{R}̂d}$$d):1 ?p?2
- §2.4. Tempered Distributions
- §2.5. Convolution in $$\mathcal{J}({\mathbb{R}̂d})$$ and $$\mathcal{J}'({\mathbb{R}̂d})$$
- §2.6. L2Derivatives and Sobolev Spaces
- 3 Solution of Initial Value Problems by Fourier Synthesis
- §3.1. Introduction
- §3.2. Schrödinger's Equation
- §3.3. Solutions of Schrödinger's Equation with Data in $$\mathcal{J}({\mathbb{R}̂d})$$
- §3.4. Generalized Solutions of Schrödinger's Equation
- §3.5. Alternate Characterizations 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 Sideways Heat Equation and Null Solutions
- §3.10. The Hadamard
- Petrowsky Dichotomy
- §3.11. Inhomogeneous Equations, Duhamel's Principle
- 4 Propagators andx-Space Methods
- §4.1. Introduction
- §4.2. Solution Formulas in x Space
- §4.3. Applications of the Heat Propagator
- §4.4. Applications of the Schrödinger Propagator
- §4.5. The Wave Equation Propagator ford = 1
- §4.6. Rotation-Invariant Smooth Solutions of $${\square _{1 + 3}}\mu = 0$$
- §4.7. The Wave Equation Propagator ford =3
- §4.8. The Method of Descent
- §4.9. Radiation Problems
- 5 The Dirichlet Problem
- §5.1. Introduction
- §5.2. Dirichlet's Principle
- §5.3. The Direct Method of the Calculus of Variations
- §5.4. Variations on the Theme
- §5.5.H1 the Dirichlet Boundary Condition
- §5.6. The Fredholm Alternative
- §5.7. Eigenfunctions and the Method of Separation of Variables
- §5.8. Tangential Regularity for the Dirichlet Problem
- §5.9. Standard Elliptic Regularity Theorems
- §5.10. Maximum Principles from Potential Theory
- §5.11. E. Hopf's Strong Maximum Principles
- APPEND
- A Crash Course in Distribution Theory
- References