Cain G., Meyer G.H. Separation of Variables for Partial Differential Equations: An Eigenfunction Approach
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title: Separation of Variables for Partial Differential Equations :
An Eigenfunction Approach Studies in Advanced
Mathematics
author: Cain, George L.; Meyer, Gunter H.
publisher: CRC Press
isbn10 | asin: 020349878X
print isbn13: 9780203621035
ebook isbn13: 9780203498781
language: English
subject Separation of variables, Eigenfunctions.
publication date: 2006
lcc: QA377.C247 2006eb
ddc: 515/.353
subject: Separation of variables, Eigenfunctions.
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Separation of Variables for Partial Differential Equations
An Eigenfunction Approach
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Cover
title: Separation of Variables for Partial Differential Equations :
An Eigenfunction Approach Studies in Advanced
Mathematics
author: Cain, George L.; Meyer, Gunter H.
publisher: CRC Press
isbn10 | asin: 020349878X
print isbn13: 9780203621035
ebook isbn13: 9780203498781
language: English
subject Separation of variables, Eigenfunctions.
publication date: 2006
lcc: QA377.C247 2006eb
ddc: 515/.353
subject: Separation of variables, Eigenfunctions.
cover next page >
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Studies in Advanced Mathematics
Titles Included in the Series
John P.D’Angelo,
Several Complex Variables and the Geometry of Real Hypersurfaces
Steven R.Bell,
The Cauchy Transform, Potential Theory, and Conformal Mapping
John J.Benedetto,
Harmonic Analysis and Applications
John J.Benedetto and Michael W.Frazier,
Wavelets: Mathematics and Applications
Albert Boggess,
CR Manifolds and the Tangential Cauchy-Riemann Complex
Keith Burns and Marian Gidea,
Differential Geometry and Topology: With a View to Dynamical Systems
George Cain and Gunter H.Meyer,
Separation of Variables for Partial Differential Equations: An
Eigenfunction Approach
Goong Chen and Jianxin Zhou,
Vibration and Damping in Distributed Systems
Vol. 1: Analysis, Estimation, Attenuation, and Design
Vol. 2: WKB and Wave Methods, Visualization, and Experimentation
Carl C.Cowen and Barbara D.MacCluer,
Composition Operators on Spaces of Analytic Functions
Jewgeni H.Dshalalow,
Real Analysis: An Introduction to the Theory of Real Functions and Integration
Dean G.Duffy,
Advanced Engineering Mathematics with MATLAB®, 2nd Edition
Dean G.Duffy,
Green’s Functions with Applications
Lawrence C.Evans and Ronald F.Gariepy,
Measure Theory and Fine Properties of Functions
Gerald B.Folland,
A Course in Abstract Harmonic Analysis
José García-Cuerva, Eugenio Hernández, Fernando Soria, and José-Luis Torrea,
Fourier Analysis and
Partial Differential Equations
Peter B.Gilkey,
Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd Edition
Peter B.Gilkey, John V.Leahy, and Jeonghueong Park,
Spectral Geometry, Riemannian Submersions, and
the Gromov-Lawson Conjecture
Alfred Gray,
Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd Edition
Eugenio Hernández and Guido Weiss,
A First Course on Wavelets
Kenneth B.Howell,
Principles of Fourier Analysis
Steven G.Krantz,
The Elements of Advanced Mathematics, Second Edition
Steven G.Krantz,
Partial Differential Equations and Complex Analysis
Steven G.Krantz,
Real Analysis and Foundations, Second Edition
Kenneth L.Kuttler,
Modern Analysis
Michael Pedersen,
Functional Analysis in Applied Mathematics and Engineering
Clark Robinson,
Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition
John Ryan,
Clifford Algebras in Analysis and Related Topics
John Scherk,
Algebra: A Computational Introduction
Pavel Solín, Karel Segeth, and Ivo Doležel,
High-Order Finite Element Method
André Unterberger and Harold Upmeier,
Pseudodifferential Analysis on Symmetric Cones
James S.Walker,
Fast Fourier Transforms, 2nd Edition
James S.Walker,
A Primer on Wavelets and Their Scientific Applications
Gilbert G.Walter and Xiaoping Shen,
Wavelets and Other Orthogonal Systems, Second Edition
Nik Weaver,
Mathematical Quantization
Kehe Zhu,
An Introduction to Operator Algebras
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Separation of Variables for Partial Differential Equations
An Eigenfunction Approach
George Cain
Georgia Institute of Technology
Atlanta, Georgia, USA
Gunter H.Meyer
Georgia Institute of Technology
Atlanta, Georgia, USA
Chapman & Hall/CRC
Taylor & Francis Group
Boca Raton London New York
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Published in 2006 by
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487–2742
© 2006 by Taylor & Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group
This edition published in the Taylor & Francis e-Library, 2006.
To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of
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No claim to original U.S. Government works
ISBN 0-203-49878-X Master e-book ISBN
ISBN 0-203-62103-4 (OEB Format)
International Standard Book Number-10: 1-58488-420-7 (Print Edition) (Hardcover)
International Standard Book Number-13: 978-1-58488-420-0 (Print Edition) (Hardcover)
Library of Congress Card Number 2005051950
This book contains information obtained from authentic and highly regarded sources. Reprinted material
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Library of Congress Cataloging-in-Publication Data
Cain, George L.
Separation of variables for partial differential equations: an eigenfunction approach/George Cain,
Gunter H.Meyer.
p. cm.—(Studies in advanced mathematics)
Includes bibliographical references and index.
ISBN 1-58488-420-7 (alk. paper)
1. Separation of variables. 2. Eigenfunctions. I. Meyer, Gunter H. II. Title. III. Series.
QA377.C247 2005
515′.353–dc22 2005051950
Taylor & Francis Group is the Academic Division of Informa plc.
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and the CRC Press Web site at http://www.crcpress.com
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Acknowledgments
We would like to thank our editor Sunil Nair for welcoming the project and for his willingness to stay
with it as it changed its scope and missed promised deadlines.
We also wish to express our gratitude to Ms. Annette Rohrs of the School of Mathematics of Georgia
Tech who transformed decidedly low-tech scribbles into a polished manuscript. Without her talents, and
patience, we would not have completed the book.
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Preface
Separation of variables is a solution method for partial differential equations. While its beginnings date
back to work of Daniel Bernoulli (1753), Lagrange (1759), and d’Alembert (1763) on wave motion (see
[2]), it is commonly associated with the name of Fourier (1822), who developed it for his research on
conductive heat transfer. Since Fourier’s time it has been an integral part of engineering mathematics,
and in spite of its limited applicability and heavy competition from numerical methods for partial
differential equations, it remains a well-known and widely used technique in applied mathematics.
Separation of variables is commonly considered an analytic solution method that yields the solution of
certain partial differential equations in terms of an infinite series such as a Fourier series. While it may
be straightforward to write formally the series solution, the question in what sense it solves the problem
is not readily answered without recourse to abstract mathematical analysis. A modern treatment
focusing in part on the theoretical underpinnings of the method and employing the language and
concepts of Hilbert spaces to analyze the infinite series may be found in the text of MacCluer [15]. For
many problems the formal series can be shown to represent an analytic solution of the differential
equation. As a tool of analysis, however, separation of variables with its infinite series solutions is not
needed. Other mathematical methods exist which guarantee the existence and uniqueness of a solution
of the problem under much more general conditions than those required for the applicability of the
method of separation of variables.
In this text we mostly ignore infinite series solutions and their theoretical and practical complexities. We
concentrate instead on the first
N
terms of the series which are all that ever are computed in an
engineering application. Such a partial sum of the infinite series is an approximation to the analytic
solution of the original problem. Alternatively, it can be viewed as the exact analytic solution of a new
problem that approximates the given problem. This is the point of view taken in this book.
Specifically, we view the method of separation of variables in the following context: mathematical
analysis applied to the given problem guarantees the existence and uniqueness of a solution
u
in some
infinite dimensional vector space of functions
X,
but in general provides no means to compute it. By
modifying the problem appropriately, however, an approximating problem results which has a
computable closed form solution
uN
in a subspace
M
of
X
. If
M
is
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