# Partial Differential Equations

### Overview

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, real analysis, the rudiments of complex analysis, and thelanguage of functional analysis. Topics discussed in the text include elliptic, hyperbolic, and parabolic equations, the energy method, maximum principle, and ...

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Paperback (Softcover reprint of the original 1st ed. 1991)
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### Overview

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, real analysis, the rudiments of complex analysis, and thelanguage of functional analysis. Topics discussed in the text include elliptic, hyperbolic, and parabolic equations, the energy method, maximum principle, and the Fourier Transform. The text features many historical and scientific motivations and applications. Included throughout are exercises, hints, and discussions which form an important and integral part of the course.

### Editorial Reviews

##### From the Publisher

"...this is an outstanding text presenting a healthy challenge not only to students but also to teachers used to more traditional or more pedestrian developments of the subject.—MATHEMATICAL REVIEWS

### Product Details

• ISBN-13: 9781461269595
• Publisher: Springer New York
• Publication date: 4/30/2013
• Series: Graduate Texts in Mathematics Series , #128
• Edition description: Softcover reprint of the original 1st ed. 1991
• Edition number: 1
• Pages: 266
• Product dimensions: 6.14 (w) x 9.21 (h) x 0.59 (d)

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
$$\mathcal{J}({\mathbb{R}subd})$$.- §2.2. The Fourier Transform on
$$\mathcal{J}({\mathbb{R}subd})$$.- §2.3. The Fourier Transform onLp$${\mathbb{R}subd}$$d):1—p?2.- §2.4. Tempered Distributions.- §2.5. Convolution in
$$\mathcal{J}({\mathbb{R}subd})$$
and
$$\mathcal{J}'({\mathbb{R}subd})$$.- §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
$$\mathcal{J}({\mathbb{R}subd})$$.- §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.

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