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Partial Differential Equations
Harold Levine, Stanford University, CA
A co-publication of the AMS and International Press.
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AMS/IP Studies in Advanced Mathematics
1997; 706 pp; hardcover
Volume: 6
Reprint/Revision History:
reprinted 1998
ISBN-10: 0-8218-0775-7
ISBN-13: 978-0-8218-0775-0
List Price: US$84
Member Price: US$67.20
Order Code: AMSIP/6
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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, Cambridge, MA.

Readership

Graduate students and research mathematicians interested in partial differential equations.

Reviews

"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

  • Introduction
  • Partial differentiation
  • Solutions of PDE's and their specification
  • PDE's and related arbitrary functions
  • Particular solutions of PDE's
  • Similarity solutions
  • Correctly set problems
  • Some preliminary aspects of linear first order PDE's
  • First order PDE's, linear
  • First order nonlinear PDE's
  • Some technical problems and related PDE's
  • First order PDE's, general theory
  • First order PDE's with multiple independent variables
  • Original detaials of the Fourier approach to boundary value problems
  • Eigenfunctions and eigenvalues
  • Eigenfunctions and eigenvalues, continued
  • Non-orthogonal eigenfunctions
  • Further example of Fourier style analysis
  • Inhomogeneous problems
  • Local heat sources
  • An inhomogeneous configuration
  • Other eigenfunction/eigenvalue problems
  • Uniqueness of solutions
  • Alternative representations of solutions
  • Other differential equations and inferences therefrom
  • Second order ODE's
  • Boundary value problems and Sturm-Liouville theory
  • Green's functions and boundary value problems
  • Green's functions and generalizations
  • PDE's, Green's functions, and integral equations
  • Singular and infinite range problems
  • Orthogonality and its ramifications
  • Fourier expansions: Generalities
  • Fourier expansions: Varied examples
  • Fourier integrals and transforms
  • Applications of Fourier transforms
  • Legendre polynomials and related expansions
  • Bessel functions and related expansions
  • Hyperbolic equations
  • Afterwords
  • Bibliography
  • Index
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