Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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Partial Differential Equations
and Boundary Value Problems
with Maple
Second Edition
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Partial Differential Equations
and Boundary Value Problems
with Maple
Second Edition
George A. Articolo
AMSTERDAM
•
BOSTON
•
HEIDELBERG
•
LONDON
NEW YORK
•
OXFORD
•
PARIS
•
SAN DIEGO
SAN FRANCISCO
•
SINGAPORE
•
SYDNEY
•
TOKYO
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Library of Congress Cataloging-in-Publication Data
Articolo, George A.
Partial differential equations and boundary value problems with Maple/George A. Articolo. – 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-374732-7 (pbk. : alk. paper) 1. Differential equations, Partial—Data processing.
2. Boundary value problems—Data processing. 3. Maple (Computer file) I. Title.
QA377.A82 2009
515’.3530285–dc22
2009010098
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN: 978-0-12-374732-7
For information on all Academic Press publications
visit our Web site at www.elsevierdirect.com
Typeset by: diacriTech, India.
Printed in the United States
091011 987654321
Contents
Preface ............................................................................... ix
Chapter 0: Basic Review ...............................................................1
0.1 Preparation for Maple Worksheets...................................................... 1
0.2 Preparation for Linear Algebra ......................................................... 4
0.3 Preparation for Ordinary Differential Equations ....................................... 8
0.4 Preparation for Partial Differential Equations......................................... 10
Chapter 1: Ordinary Linear Differential Equations .................................. 13
1.1 Introduction .............................................................................13
1.2 First-Order Linear Differential Equations .............................................14
1.3 First-Order Initial-Value Problem .....................................................19
1.4 Second-Order Linear Differential Equations with Constant Coefficients............23
1.5 Second-Order Linear Differential Equations with Variable Coefficients ............28
1.6 Finding a Second Basis Vector by the Method of Reduction of Order............... 32
1.7 The Method of Variation of Parameters—Second-Order Green’s Function .........36
1.8 Initial-Value Problem for Second-Order Differential Equations .....................45
1.9 Frobenius Method of Series Solutions to Ordinary Differential Equations ..........49
1.10 Series Sine and Cosine Solutions to the Euler Differential Equation ................51
1.11 Frobenius Series Solution to the Bessel Differential Equation .......................56
Chapter Summary ...................................................................... 63
Exercises ................................................................................65
Chapter 2: Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series ..... 73
2.1 Introduction .............................................................................73
2.2 The Regular Sturm-Liouville Eigenvalue Problem ...................................73
2.3 Green’s Formula and the Statement of Orthonormality ..............................75
2.4 The Generalized Fourier Series Expansion............................................81
2.5 Examples of Regular Sturm-Liouville Eigenvalue Problems ........................86
v
vi Contents
2.6 Nonregular or Singular Sturm-Liouville Eigenvalue Problems .................... 129
Chapter Summary .................................................................... 146
Exercises .............................................................................. 147
Chapter 3: The Diffusion or Heat Partial Differential Equation.....................161
3.1 Introduction ........................................................................... 161
3.2 One-Dimensional Diffusion Operator in Rectangular Coordinates ................ 161
3.3 Method of Separation of Variables for the Diffusion Equation..................... 163
3.4 Sturm-Liouville Problem for the Diffusion Equation ............................... 165
3.5 Initial Conditions for the Diffusion Equation in Rectangular Coordinates ........ 168
3.6 Example Diffusion Problems in Rectangular Coordinates ......................... 170
3.7 Verification of Solutions—Three-Step Verification Procedure ..................... 186
3.8 Diffusion Equation in the Cylindrical Coordinate System ......................... 190
3.9 Initial Conditions for the Diffusion Equation in Cylindrical Coordinates ......... 194
3.10 Example Diffusion Problems in Cylindrical Coordinates .......................... 196
Chapter Summary .................................................................... 205
Exercises .............................................................................. 206
Chapter 4: The Wave Partial Differential Equation .................................217
4.1 Introduction ........................................................................... 217
4.2 One-Dimensional Wave Operator in Rectangular Coordinates .................... 217
4.3 Method of Separation of Variables for the Wave Equation ......................... 219
4.4 Sturm-Liouville Problem for the Wave Equation ................................... 221
4.5 Initial Conditions for the Wave Equation in Rectangular Coordinates............. 224
4.6 Example Wave Equation Problems in Rectangular Coordinates ................... 228
4.7 Wave Equation in the Cylindrical Coordinate System .............................. 244
4.8 Initial Conditions for the Wave Equation in Cylindrical Coordinates ............. 249
4.9 Example Wave Equation Problems in Cylindrical Coordinates .................... 251
Chapter Summary .................................................................... 261
Exercises .............................................................................. 262
Chapter 5: The Laplace Partial Differential Equation ...............................275
5.1 Introduction ........................................................................... 275
5.2 Laplace Equation in the Rectangular Coordinate System .......................... 276
5.3 Sturm-Liouville Problem for the Laplace Equation in Rectangular
Coordinates............................................................................ 278
5.4 Example Laplace Problems in the Rectangular Coordinate System ............... 284
5.5 Laplace Equation in Cylindrical Coordinates ....................................... 299
5.6 Sturm-Liouville Problem for the Laplace Equation in Cylindrical
Coordinates............................................................................ 301
Contents vii
5.7 Example Laplace Problems in the Cylindrical Coordinate System ................ 307
Chapter Summary .................................................................... 325
Exercises .............................................................................. 327
Chapter 6: The Diffusion Equation in Two Spatial Dimensions ......................339
6.1 Introduction ........................................................................... 339
6.2 Two-Dimensional Diffusion Operator in Rectangular Coordinates ................ 339
6.3 Method of Separation of Variables for the Diffusion Equation in
Two Dimensions ...................................................................... 341
6.4 Sturm-Liouville Problem for the Diffusion Equation in Two Dimensions ........ 342
6.5 Initial Conditions for the Diffusion Equation in Rectangular Coordinates ........ 347
6.6 Example Diffusion Problems in Rectangular Coordinates ......................... 351
6.7 Diffusion Equation in the Cylindrical Coordinate System ......................... 365
6.8 Initial Conditions for the Diffusion Equation in Cylindrical Coordinates ......... 371
6.9 Example Diffusion Problems in Cylindrical Coordinates .......................... 374
Chapter Summary .................................................................... 394
Exercises .............................................................................. 395
Chapter 7: The Wave Equation in Two Spatial Dimensions..........................409
7.1 Introduction ........................................................................... 409
7.2 Two-Dimensional Wave Operator in Rectangular Coordinates .................... 409
7.3 Method of Separation of Variables for the Wave Equation ..........................411
7.4 Sturm-Liouville Problem for the Wave Equation in Two Dimensions ............. 412
7.5 Initial Conditions for the Wave Equation in Rectangular Coordinates............. 417
7.6 Example Wave Equation Problems in Rectangular Coordinates ................... 420
7.7 Wave Equation in the Cylindrical Coordinate System .............................. 437
7.8 Initial Conditions for the Wave Equation in Cylindrical Coordinates ............. 443
7.9 Example Wave Equation Problems in Cylindrical Coordinates .................... 447
Chapter Summary .................................................................... 466
Exercises .............................................................................. 467
Chapter 8: Nonhomogeneous Partial Differential Equations ........................477
8.1 Introduction ........................................................................... 477
8.2 Nonhomogeneous Diffusion or Heat Equation...................................... 477
8.3 Initial Condition Considerations for the Nonhomogeneous Heat Equation ....... 488
8.4 Example Nonhomogeneous Problems for the Diffusion Equation ................. 490
8.5 Nonhomogeneous Wave Equation ................................................... 510
8.6 Initial Condition Considerations for the Nonhomogeneous Wave Equation ...... 520
8.7 Example Nonhomogeneous Problems for the Wave Equation ..................... 523
Chapter Summary .................................................................... 546
Exercises .............................................................................. 547
viii Contents
Chapter 9: Infinite and Semi-infinite Spatial Domains...............................557
9.1 Introduction ........................................................................... 557
9.2 Fourier Integral ....................................................................... 557
9.3 Fourier Sine and Cosine Integrals ................................................... 561
9.4 Nonhomogeneous Diffusion Equation over Infinite Domains ..................... 564
9.5 Convolution Integral Solution for the Diffusion Equation ......................... 568
9.6 Nonhomogeneous Diffusion Equation over Semi-infinite Domains ............... 570
9.7 Example Diffusion Problems over Infinite and Semi-infinite Domains ........... 573
9.8 Nonhomogeneous Wave Equation over Infinite Domains .......................... 586
9.9 Wave Equation over Semi-infinite Domains ........................................ 588
9.10 Example Wave Equation Problems over Infinite and Semi-infinite Domains ..... 594
9.11 Laplace Equation over Infinite and Semi-infinite Domains ........................ 606
9.12 Example Laplace Equation over Infinite and Semi-infinite Domains .............. 612
Chapter Summary .................................................................... 619
Exercises .............................................................................. 621
Chapter 10: Laplace Transform Methods for Partial Differential Equations.........639
10.1 Introduction ........................................................................... 639
10.2 Laplace Transform Operator ......................................................... 639
10.3 Inverse Transform and Convolution Integral ........................................ 641
10.4 Laplace Transform Procedures on the Diffusion Equation ......................... 642
10.5 Example Laplace Transform Problems for the Diffusion Equation ................ 646
10.6 Laplace Transform Procedures on the Wave Equation ............................. 666
10.7 Example Laplace Transform Problems for the Wave Equation .................... 671
Chapter Summary .................................................................... 693
Exercises .............................................................................. 694
References..........................................................................709
Index ...............................................................................711
Preface
This is the second edition of the text Partial Differential Equations and Boundary Value
Problems with Maple, Academic Press, 1998. The text has been updated from Maple release 4
to release 12. In addition, based on recommendations and suggestions of the many helpful
reviewers of the first edition, the text incorporates more of the macro commands in Maple to be
used as a means of checking solutions. Similar to what was done in the first edition, I continued
the presentation of the solutions to problems using the traditional, fundamental, mathematical
approach so that the student gets a firm understanding of the mathematical basis of the
development of the solutions. The macro commands are not intended to be used as a means of
teaching the mathematics—they are used only as a quick means of checking.
If there ever were to be a perfect union in computational mathematics, one between partial
differential equations and powerful software, Maple would be close to it. This text is an
attempt to join the two together.
Many years ago, I recall sitting in a partial differential equations class when the professor was
discussing a heat-flow boundary value problem. Using a piece of chalk at the blackboard, he
was making a seemingly desperate attempt to get his students to visualize the spatial-time
development of the three-dimensional surface temperature of a plate that was allowed to cool
down to a surrounding equilibrium temperature. You can imagine the frustration that he, and
many professors before him, experienced at doing this task. Now, with the powerful
computational tools and graphics capabilities at hand, this era of difficulty is over.
This text presents the formal mathematical concepts needed to develop solutions to boundary
value problems, and it demonstrates the capabilities of Maple software as being a powerful
computational tool. The graphics and animation commands allow for accurate visualization of
the spatial-time development of the solutions on the computer screen—what students could
only imagine many years ago can now be viewed in real time.
The text is targeted for use by senior-graduate level students and practitioners in the disciplines
of physics, mathematics, and engineering. Typically, these people have already had some
exposure to courses in basic physics, calculus, linear algebra, and ordinary differential
equations. The need for previous exposure to the Maple software is not necessary. In Chapter 0,
we provide an introduction to some simple Maple commands, which is all that is necessary for
ix