**Graduate Studies in Mathematics**

Volume: 197;
2018;
205 pp;
Hardcover

MSC: Primary 35; 42;

**Print ISBN: 978-1-4704-4292-7
Product Code: GSM/197**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

**Electronic ISBN: 978-1-4704-5057-1
Product Code: GSM/197.E**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

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#### Supplemental Materials

# A Course on Partial Differential Equations

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*Walter Craig*

Does entropy really increase no matter what we
do? Can light pass through a Big Bang? What is certain about the
Heisenberg uncertainty principle? Many laws of physics are formulated
in terms of differential equations, and the questions above are about
the nature of their solutions. This book puts together the three main
aspects of the topic of partial differential equations, namely theory,
phenomenology, and applications, from a contemporary point of view. In
addition to the three principal examples of the wave equation, the
heat equation, and Laplace's equation, the book has chapters on
dispersion and the Schrödinger equation, nonlinear hyperbolic
conservation laws, and shock waves.

The book covers material for an introductory course that is aimed
at beginning graduate or advanced undergraduate level
students. Readers should be conversant with multivariate calculus and
linear algebra. They are also expected to have taken an introductory
level course in analysis. Each chapter includes a comprehensive set of
exercises, and most chapters have additional projects, which are
intended to give students opportunities for more in-depth and
open-ended study of solutions of partial differential equations and
their properties.

#### Readership

Undergraduate and graduate students and researchers interested in partial differential equations (PDEs).

#### Reviews & Endorsements

The content is developed in a clear and engaging way, the derivations are very well developed, and the author does a nice job connecting the mathematics with the physical motivation. This volume is a very pleasant book written by a renowned expert in this field. The book offers a nice elementary introduction to a fascinating field of mathematics with multiple relevant applications to various fields.

-- Vicenţiu D. Radulescu, Mathematical Reviews

I really enjoyed reading 'A Course on Partial Differential Equations'. The writing is clear and engaging, the derivations are very well-developed, and the author does a nice job connecting the mathematics with the physical motivation that underlies many of the PDEs discussed in the text... I wish that it would have been available when I was a student...'A Course on Partial Differential Equations' is an excellent book that provides a modern introduction to a fascinating field of mathematics and I highly recommend reading it.

-- Jason Graham, MAA Reviews

#### Table of Contents

# Table of Contents

## A Course on Partial Differential Equations

- Cover Cover11
- Title page i2
- Preface vii8
- Chapter 1. Introduction 112
- Chapter 2. Wave equations 920
- 2.1. Transport equations: The Fourier transform 1021
- 2.2. Transport equations: The method of characteristics 1223
- 2.3. Conservation laws 1526
- 2.4. The d’Alembert formula 1627
- 2.5. Duhamel’s principle 1930
- 2.6. The method of images 2132
- 2.7. Separation of variables 2435
- Exercises: Chapter 2 3445
- Projects: Chapter 2 3748

- Chapter 3. The heat equation 4354
- 3.1. The heat kernel 4455
- 3.2. Convolution operators 4657
- 3.3. The maximum principle 4859
- 3.4. Initial and initial-boundary value problems 4960
- 3.5. Conservation laws and the evolution of moments 5667
- 3.6. The heat equation in ℝⁿ 6172
- 3.7. Entropy 6273
- 3.8. Gradient flow 6576
- Exercises: Chapter 3 6677
- Projects: Chapter 3 6980

- Chapter 4. Laplace’s equation 7182
- 4.1. Dirichlet, Poisson, and Neumann boundary value problems 7182
- 4.2. Green’s identities 7283
- 4.3. The fundamental solution 7485
- 4.4. Maximum principle 7788
- 4.5. Green’s functions and Dirichlet–Neumann operators 8091
- 4.6. Poisson kernel on ℝⁿ₊ 8697
- 4.7. Maximum principle again 90101
- 4.8. Oscillation and attenuation estimates 91102
- 4.9. Hadamard variational formula 93104
- Exercises: Chapter 4 95106
- Projects: Chapter 4 99110

- Chapter 5. Properties of the Fourier transform 103114
- Chapter 6. Wave equations on ℝⁿ 121132
- Chapter 7. Dispersion 151162
- Chapter 8. Conservation laws and shocks 177188
- Bibliography 201212
- Index 203214
- Back Cover Back Cover1217