**AMS/IP Studies in Advanced Mathematics**

Volume: 6;
1997;
706 pp;
Hardcover

MSC: Primary 35;

**Print ISBN: 978-0-8218-0775-0
Product Code: AMSIP/6**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-3797-8
Product Code: AMSIP/6.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

# Partial Differential Equations

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*Harold Levine*

A co-publication of the AMS and International Press of Boston

The subject matter, partial differential equations (PDEs), has a long history (dating from the 18th century) and an active contemporary phase. An early phase (with a separate focus on taut string vibrations and heat flow through solid bodies) stimulated developments of great importance for mathematical analysis, such as a wider concept of functions and integration and the existence of trigonometric or Fourier series representations. The direct relevance of PDEs to all manner of mathematical, physical and technical problems continues. This book presents a reasonably broad introductory account of the subject, with due regard for analytical detail, applications and historical matters.

Titles in this series are co-published with International Press of Boston, Inc., Cambridge, MA.

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### Reviews & Endorsements

A large variety of examples and problems for solutions is given … The book will be certainly of great value with respect to applications.

-- Monatshefte für Mathematik

Although the scope is large—there are 706 pages—the chapters tend to be short and to the point, with the detailed work developed in the problems set at the end of each chapter. These problem sets should ideally provide instructors delivering an advanced mathematical methods course with plenty of ideas for tutorial material for their students. The book is comprehensive in its background coverage, including, for example, an introductory chapter on partial differentiation, which at the same time brings in and manipulates a couple of well-known canonical forms, by way of illustration. In all, this text is a useful addition to the extensive literature on PDEs.

-- Mathematical Reviews

Naturally the book will be helpful for a very wide audience which would benefit from reading it–from students and Ph.D. candidates (not only of mathematical direction) to specialists. This book can be recommended as a well written handbook containing an original approach to the description of basic and advanced methods of the theory of PDE.

-- Zentralblatt MATH

# Table of Contents

## Partial Differential Equations

- Cover Cover11
- Title page v6
- Contents vii8
- Preface xvii18
- Introduction 120
- Partial differentiation 322
- Solutions of PDE’s and their specification 1736
- PDE’s and related arbitrary functions 3150
- Particular solutions of PDE’s 3958
- Similarity solutions 5170
- Correctly set problems 6786
- Some preliminary aspects of linear first order PDE’s 7190
- Linear first order PDE’s with two independent variables 7796
- First order nonlinear PDE’s 91110
- Some technical problems and related PDE’s 109128
- Linear first order PDE’s with two independent variables, general theory 119138
- First order PDE’s with multiple independent variables 129148
- Original details of the Fourier approach to boundary value problems 135154
- Eigenfunctions and eigenvalues 143162
- Eigenfunctions and eigenvalues, continued 161180
- Non-orthogonal eigenfunctions 169188
- Further example of Fourier style analysis 175194
- Inhomogeneous problems 183202
- Local heat sources 203222
- An inhomogeneous configuration 213232
- Other eigenfunction/eigenvalue problems 221240
- Uniqueness of solutions 241260
- Alternative representations of solutions 253272
- Other differential equations and inferences therefrom 269288
- Second order ODE’s 285304
- Boundary value problems and Sturm-Liouville theory 307326
- Green’s functions and boundary value problems 331350
- Green’s functions and generalizations 361380
- PDE’s, Green’s functions, and integral equations 401420
- Singular and infinite range problems 425444
- Orthogonality and its ramifications 457476
- Fourier expansions: Generalities 483502
- Fourier expansions: Varied examples 515534
- Fourier integrals and transforms 555574
- Applications of Fourier transforms 577596
- Legendre polynomials and related expansions 603622
- Bessel functions and related expansions 623642
- Hyperbolic equations 665684
- Afterwords 699718
- Bibliography 701720
- Index 703722
- Back Cover Back Cover1726