Solution Techniques for Elementary Partial Differential Equations

441 Pages·2022·11.65 MB·English

by  Christian Constanda

Solution Techniques
for Elementary Partial
Differential Equations
“In my opinion, this is quite simply the best book of its kind that I have seen thus far.”
—Professor Peter Schiavone, University of Alberta, from the Foreword to the Fourth
Edition
Praise for the previous editions
“An ideal tool for students taking a first course in PDEs, as well as for the lecturers who
teach such courses.”
—Marian Aron, Plymouth University, UK
“This is one of the best books on elementary PDEs this reviewer has read so far. Highly
recommended.”
—CHOICE
Solution Techniques for Elementary Partial Differential Equations, Fourth Edition
remains a top choice for a standard, undergraduate-level course on partial differential
equations (PDEs). It provides a streamlined, direct approach to developing students’ com-
petence in solving PDEs, and offers concise, easily understood explanations and worked
examples that enable students to see the techniques in action.
New to the Fourth Edition
• Two additional sections
• A larger number and variety of worked examples and exercises
• A companion pdf file containing more detailed worked examples to supplement
those in the book, which can be used in the classroom and as an aid to online
teaching
Christian Constanda, MS, PhD, DSc, is the Charles W. Oliphant Endowed Chair in
Mathematical Sciences and director of the Center for Boundary Integral Methods at the
University of Tulsa. He is also an emeritus professor at the University of Strathclyde and
chairman of the International Consortium on Integral Methods in Science and Engineer-
ing. He is the author/editor of more than 30 books and more than 150 journal papers. His
research interests include boundary value problems for elastic plates with transverse shear
deformation, direct and indirect integral equation methods for elliptic problems and time-
dependent problems, and variational methods in elasticity.

Solution Techniques
for Elementary Partial
Differential Equations
Fourth Edition
Christian Constanda
The University of Tulsa, USA
Fourth edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2023 Christian Constanda
First edition published by CRC Press 2002
Second edition published by CRC Press 2010
Third edition published by CRC Press 2016
CRC Press is an imprint of Taylor & Francis Group, LLC
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LCCN no. 2022007080
ISBN: 978-1-032-00166-1 (hbk)
ISBN: 978-1-032-00031-2 (pbk)
ISBN: 978-1-003-17304-5 (ebk)
DOI: 10.1201/9781003173045
Typeset in CMR10
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Access the Support Material: www.Routledge.com/9781032001661
For Lia.
Also for Jack, Emily, and Archie.

Contents
Foreword xiii
Preface to the Fourth Edition xv
Preface to the Third Edition xvii
Preface to the Second Edition xix
Preface to the First Edition xxi
Chapter 1(cid:4) ORDINARY DIFFERENTIAL EQUATIONS:
BRIEF REVIEW 1
1.1 First-Order Equations 1
1.2 Homogeneous Linear Equations with Constant
Coefficients 3
1.3 Nonhomogeneous Linear Equations with Constant
Coefficients 6
1.4 Cauchy–Euler Equations 9
1.5 Functions and Operators 10
Chapter 2(cid:4) FOURIER SERIES 13
2.1 The Full Fourier Series 13
2.2 Fourier Sine and Cosine Series 20
2.2.1 Fourier Sine Series 21
2.2.2 Fourier Cosine Series 26
2.3 Convergence and Differentiation 32
2.4 Series Expansion of More General Functions 33
vii
viii (cid:4) Contents
Chapter 3(cid:4) STURM–LIOUVILLE PROBLEMS 37
3.1 Regular Sturm–Liouville Problems 37
3.1.1 Eigenvalues and Eigenfunctions of Some
Basic Problems 41
3.1.2 Generalized Fourier Series 51
3.2 Other Problems 58
3.3 Bessel Functions 60
3.4 Legendre Polynomials 66
3.5 Spherical Harmonics 70
Chapter 4(cid:4) SOME FUNDAMENTAL EQUATIONS OF
MATHEMATICAL PHYSICS 77
4.1 The Heat Equation 78
4.2 The Laplace Equation 84
4.3 The Wave Equation 90
4.4 Other Equations 94
Chapter 5(cid:4) THE METHOD OF SEPARATION OF
VARIABLES 99
5.1 The Heat Equation 99
5.1.1 Rod with Zero Temperature at the Endpoints 99
5.1.2 Rod with Insulated Endpoints 103
5.1.3 Rod with Mixed Boundary Conditions 107
5.1.4 Rod with an Endpoint in a
Zero-Temperature Medium 111
5.1.5 Thin Uniform Circular Ring 115
5.2 The Wave Equation 118
5.2.1 Vibrating String with Fixed Endpoints 118
5.2.2 Vibrating String with Free Endpoints 122
5.2.3 Vibrating String with Other Types of
Boundary Conditions 126
5.3 The Laplace Equation 132
5.3.1 Laplace Equation in a Rectangle 132
5.3.2 Laplace Equation in a Circular Disc 137
5.4 Other Equations 142
5.5 Equations with More than Two Variables 149
Contents (cid:4) ix
5.5.1 Vibrating Rectangular Membrane 149
5.5.2 Vibrating Circular Membrane 154
5.5.3 Equilibrium Temperature in a Solid Sphere 158
Chapter 6(cid:4) LINEAR NONHOMOGENEOUS PROBLEMS 163
6.1 Equilibrium Solutions 163
6.2 Nonhomogeneous Problems 168
6.2.1 Time-Independent Sources and Boundary
Conditions 168
6.2.2 The General Case 169
Chapter 7(cid:4) THE METHOD OF EIGENFUNCTION
EXPANSION 175
7.1 The Nonhomogeneous Heat Equation 175
7.1.1 Rod with Zero Temperature at the Endpoints 175
7.1.2 Rod with Insulated Endpoints 181
7.1.3 Rod with Mixed Boundary Conditions 184
7.2 The Nonhomogeneous Wave Equation 186
7.2.1 Vibrating String with Fixed Endpoints 186
7.2.2 Vibrating String with Free Endpoints 190
7.2.3 Vibrating String with Mixed Boundary
Conditions 192
7.3 The Nonhomogeneous Laplace Equation 194
7.3.1 Equilibrium Temperature in a Rectangle 194
7.3.2 Equilibrium Temperature in a Circular Disc 203
7.4 Other Nonhomogeneous Equations 207
Chapter 8(cid:4) THE FOURIER TRANSFORMATIONS 215
8.1 The Full Fourier Transformation 215
8.1.1 Cauchy Problem for an Infinite Rod 219
8.1.2 Vibrations of an Infinite String 223
8.1.3 Equilibrium Temperature in an Infinite Strip 226
8.2 The Fourier Sine and Cosine Transformations 230
8.2.1 Heat Conduction in a Semi-Infinite Rod 231
8.2.2 Vibrations of a Semi-Infinite String 234