Separation of Variables for Partial Differential Equations: An Eigenfunction Approach

300 Pages·2005·9.719 MB·English

by  George Cain, Gunter H. Meyer

Separation
of Variables
for Partial
Differential
Equations
An Eigenfunction
Approach
STUDIES IN ADVANCED MATHEMATICS
Separation
of Variables
for Partial
Differential
Equations
An Eigenfunction
Approach
Studies in Advanced Mathematics
Titles Included in the Series
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Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping
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Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex
Keith Bums 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
Goo11g Chen and Jianxi11 Zhou, Vibration and Damping in Distributed Systems
Vol. I: 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
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Fourier Analysis and Partial Differential Equations
Peter 8. Gilkey. Invariance Theory, the Heat Equation,_and the Atiyah·Singer Index Theorem,
2nd Edition
Peter B. Gilke.v, John V. Leahy, and Jeonghueong Park, Spectral Geometry, Riemannian Submersions,
and the Gromov-Lawson Conjecture
Alfred Gray, Modem Differential Geometry of Curves and Surfaces with Mathematica, 2nd Edition
Eugenio Hemd.ndez 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. Kurt/er, Modern Analysis
Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering
Ciark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition
Jolm Rya11. Clifford Algebras in Analysis and Related Topics
Joh11 Sclierk. Algebra: A Computational Introduction
Pai·ei Soffn. Karel Segeth, and lvo Doletel, High-Order Finite Element Method
Andr<i Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones
James S. lfolker, Fast Fourier Transforms, 2nd Edition
James S. H'Cilker, A Primer on Wavelets and Their Scientific Applications
Gilbert G. U'i?lter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition
Nik Weaver. Mathematical Quantization
Kehe Zhu. An Introduction to Operator Algebras
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
A.r!anta, Georgia, USA
Boc<1 Raton London New York
Published in 2006 by
Chapman & HaIVCRC
Taylor & Francis Group
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© 2006 by Taylor & Francis Group, LLC
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International Standard Book Number-IO: 1-58488-420-7 (Hardcover)
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Library of Congress Cataloging-in-Publication Data
Cain, George L.
Separation of variables for partial differential equations : an eigenfunction approach I George Cain,
Gunter H. Meyer.
p. cm. --(Studies in advanced mathematics)
Includes bibliographical references and index.
ISBN 1-58488-420-7 (alk. paper)
l. Separation of variables. 2. Eigenfunctions. L Meyer, Gunter H. IL Title. III. Series.
QA377.C247 2005
5 l 5'.353--dc22 2005051950
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Acknowledgments
\Ve 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.

Preface
3-=paration of variables is a solution method for partial differential equations.
\\liile its beginnings date back to work of Daniel Bernoulli (1753), Lagrange
, 1759), and d'Alembert (1763) on wave motion (see [2]), it is commonly asso
,ciated with the name of Fourier (1822), who developed it for his research on
·:-Jnductive heat transfer. Since Fourier's time it has been an integral part of
e!:gineering mathematics, and in spite of its limited applicability and heavy com
petition from numerical methods for partial differential equations, it remains a
-;;.·ell-known and widely used technique in applied mathematics.
Separation of variables is commonly considered an analytic solution
::nethod that yields the solution of certain partial differential equations in terms
cf an infinite series such as a Fourier series. While it may be straightforward to
·:uite formally the series solution, the question in what sense it solves the prob
lem 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
che infinite series may be fonnd in the text of MacCluer [15]. For many problems
•he formal series can be shown to represent an analytic solution of the differ
ential equation. As a tool of analysis, however, separation of variables with its
:nfirtite series solutions is not needed. Other mathematical methods exist which
guarantee the existence and uniqueness of a solution of the problem under much
::nore general conditions than those required for the applicability of the method
;:,f 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
c,f 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
o:,f 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 fl!! is
vii
viii PREFACE
suitably chosen, then UN is a good approximation to the unknown solution u.
As we shall see, M will be defined such that UN is just the partial sum of the
first N terms of the infinite series traditionally associated with the method of
separation of variables.
The reader may recognize this view as identical to the setting of the finite
element, collocation, and spectral methods that have been developed for the
numerical solution of differential equations. All these methods differ in how the
subspace Mis chosen and in what sense the original problem is approximated.
These choices dictate how hard it is to compute the approximate solution UN
and how well it approximates the ai!alytic solution u.
Given the almost universal applicability of numerical methods for the solu
tion of partial differential equations, the question arises whether separation of
variables with its severe restrictions on the type of equation and the geometry of
the problem is still a viable tool and deserves further exposition. The existence
of this text ·reflects our view that the method of separation of variables still
belongs to the core of applied mathematics. There are a number of reasons.
Closed form (approximate) solutions show structure and exhibit explicitly
the influence of the problem parameters on the solution. We think, for example,
of the decomposition of wave motion into standing waves, of the relationship
between driving frequency and resonance in sound waves, of the influence of
diffusivity on the rate of decay of temperature in a heated bar, or of the gen
eration of equipotential and stream lines for potential flow. Such structure and
insight are not readily obtained frnm purely numerical solutions of the underly
ing differential equation. Moreover, optimization, control, and inverse problems
tend to be easier to solve when an analytic representation of the (approximate)
solution is available. In addition, the method is not as limited in its applicability
as one might infer from more elementary texts on separation of variables. Ap
proximate solutions are readily computable for problems with time-dependent
data, for diffusion with convection and wave motion with dissipation, problems
seldom seen in introductory textbooks. Even domain restrictions can sometimes
be overcome with embedding and domain decomposition techniques. Finally,
there is the class of singularly perturbed and of higher dimensional problems
where numerical methods are not easily applied while separation of variables
still yieltb an analytic approximate solution.
Our rationale for offering a new exposition of separation of variables is then
twofold. First, although quite common in more advanced treatments (such as
[15]), interpreting the separation of variables solution as an eigenfunction expan
sion is a point of view r.arely taken when introducing the method to students.
Usually the formalism is based on a product solution for the partial differential
equation, and this limits the applicability of the method to homogeneous partial
differential equations. When source terms do appear, then a reformulation of
problems for the heat and wave equation with the help of Duhamel's superposi
tion principle and an approximation of the source term in the potential equation
with the help of an eigenfunction approximation become necessary. In an expo
sition based from the beginning on an eigenfunction expansion, the presence of
source terms in the differential equation is only a technical, but not a conceptual
PREFACE ix
complication, regardless of the type of equation under consideration. A concise
algorithmic approach results.
Equally important to us is the second reason for a new exposition of the
method of separation of variables. We wish to emphasize the power of the
method by solving a great variety of problems which often go well beyond the
usual textbook examples. Many of the applications ask questions which are
not as easily resolved with numerical methods as with analytic approximate
solutions. Of course, evaluation of these approximate solutions usually relies on
numerical methods to integrate, solve linear systems or nonlinear equations, and
to find values of special functions, but these methods by now may be considered
universally available "black boxes." We are, however, mindful of the gap between
the concept of a solution in principle and a demonstrably computable solution
and try .to convey our experience with how well the eigenfunction approach
actually solves the sample problems.
The method of separation of variables from a spectral expansion view is
presented in nine chapters.
Chapter 1 collects some background information on the three dominant equa
tions of this text, the potential equation, the heat equation, and the wave equa
tion. We refer to these results when applying and analyzing the method of
separation of variables.
Chapter 2 contains a discussion of orthogonal projections which are used time
and again to approximate given data functions in a specified finite-dimensional
but parameter-dependent subspace.
Chapter 3 introduces the subspace whose basis consists of the eigenfunctions
of a so-called Sturm-Liouville problem associated with the application under
consideration. These are the eigenfunctions of the title of this text. We cite
results from the Sturm-Liouville theory and provide a table of eigenvalues and
eigenfunctions that arise in the method of separation of variables.
Chapter 4 treats the case in which the eigenfunctions are sine and cosine
functions with a common period. In this case the projection into the subspace is
closely related to the Fourier series representation of the data functions. Precise
information about the convergence of the Fourier series is known. We cite those
results which are helpful later on for the application of separation of variables.
Chapter 5 constitutes the heart of the text. We consider a partial differen
tial equation in two independent variables with a source term and subject to
boundary and initial conditions. We give the algorithm for approximating such
a problem and for solving it in a finite-dimensional space spanned by eigen
functions determined by the "spacial part" of the equation and its boundary
conditions. We illustrate in broad outline the application of this approach to
the heat, wave, and potential equations.
Chapter 6 gives an expansive exposition of the algorithm for the one-dimen
sional heat equation. It contains many worked examples with comments on
the numerical performance of the method, and concludes with a rudimentary
analysis of the error in the approximate solution.
Chapter 7 parallels the previous chapter but treats the wave equation.