Partial Differential Equations: Theory and Completely Solved Problems

693 Pages·2012·21.415 MB·English

by  Thomas Hillen, I. Ed Leonard, Henry van Roessel

PARTIAL DIFFERENTIAL
EQUATIONS

PARTIAL DIFFERENTIAL
EQUATIONS
Theory and Completely
Solved Problems
T. HILLEN
I. E. LEONARD
H. VAN ROESSEL
Department of Mathematical and Statistical Sciences
University of Alberta
~WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
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Library ofC ongress Clltllloging-in-Publiclllion Dlllll:
Hillen, Thomas, 1966--
Partial differential equations : theory and completely solved problems I Thomas Hillen, I. Ed Leonard,
Henry van Roessel.
p.cm.
Includes bibliographical references and index.
ISBN 978-1-118-06330-9 (hardback)
1. Differential equations, Partial. I. Leonard,!. Ed., 1938- II. Van Roessel, Henry, 1956 III. Title.
QA377.H55 2012
515' .353-dc23 2012017382
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 I
CONTENTS
Preface xi
PART I THEORY
1 Introduction 3
1.1 Partial Differential Equations 4
1.2 Classification of Second-Order Linear PDEs 7
1.3 Side Conditions 10
1.3.1 Boundary Conditions on an Interval 12
1.4 LinearPDEs 12
1.4.1 Principle of Superposition 14
1.5 Steady-State and Equilibrium Solutions 16
1.6 First Example for Separation of Variables 19
1.7 Derivation of the Diffusion Equation 24
1.7.1 Boundary Conditions 25
1.8 Derivation of the Heat Equation 26
1.9 Derivation of the Wave Equation 29
1.10 Examples of Laplace's Equation 33
1.11 Summary 37
1.11.1 Problems and Notes 38
v
vi CONTENTS
2 Fourier Series 39
2.1 Piecewise Continuous Functions 39
2.2 Even, Odd, and Periodic Functions 41
2.3 Orthogonal Functions 43
2.4 Fourier Series 48
2.4.1 Fourier Sine and Cosine Series 53
2.5 Convergence of Fourier Series 56
2.5.1 Gibbs' Phenomenon 60
2.6 Operations on Fourier Series 63
2.7 Mean Square Error 74
2.8 Complex Fourier Series 78
2.9 Summary 81
2.9.1 Problems and Notes 82
3 Separation of Variables 83
3.1 Homogeneous Equations 83
3.1.1 General Linear Homogeneous Equations 89
3.1.2 Limitations of the Method of Separation of Variables 93
3.2 Nonhomogeneous Equations 95
3.2.1 Method of Eigenfunction Expansions 100
3.3 Summary 111
3.3.1 Problems and Notes 113
4 Sturm-Liouville Theory 115
4.1 Formulation 115
4.2 Properties of Sturm-Liouville Problems 119
4.3 Eigenfunction Expansions 127
4.4 Rayleigh Quotient 134
4.5 Summary 141
4.5.1 Problems and Notes 143
5 Heat, Wave, and Laplace Equations 145
5.1 One-Dimensional Heat Equation 145
5.2 Two-Dimensional Heat Equation 150
5.3 One-Dimensional Wave Equation 153
5.3.1 d' Alembert's Solution 157
5.4 Laplace's Equation 163
5.4.1 Potential in a Rectangle 163
5.5 Maximum Principle 167
CONTENTS vii
5.6 Two-Dimensional Wave Equation 168
5.7 Eigenfunctions in Two Dimensions 172
5.8 Summary 176
5.8.1 Problems and Notes 177
6 Polar Coordinates 179
6.1 Interior Dirichlet Problem for a Disk 179
6.1.1 Poisson Integral Formula 186
6.2 Vibrating Circular Membrane 188
6.3 Bessel's Equation 191
6.3.1 Series Solutions of ODEs 191
6.4 Bessel Functions 195
6.4.1 Properties of Bessel Functions 201
6.4.2 Integral Representation of Bessel Functions 204
6.5 Fourier-Bessel Series 210
6.6 Solution to the Vibrating Membrane Problem 214
6.7 Summary 218
6.7.1 Problems and Notes 220
7 Spherical Coordinates 221
7.1 Spherical Coordinates 221
7.1.1 Derivation of the Laplacian 222
7.2 Legendre's Equation 224
7.3 Legendre Functions 227
7.3.1 Legendre Polynomials 228
7.3.2 Fourier-Legendre Series 245
7.3.3 Legendre Functions of the Second Kind 248
7.3.4 Associated Legendre Functions 249
7.4 Spherical Bessel Functions 252
7.5 Interior Dirichlet Problem for a Sphere 253
7.6 Summary 257
7.6.1 Problems and Notes 259
8 Fourier Transforms 261
8.1 Fourier Integrals 261
8.1.1 Fourier Integral Representation 261
8.1.2 Examples 264
8.1.3 Fourier Sine and Cosine Integral Representations 268
8.1.4 Proof of Fourier's Theorem 271
viii CONTENTS
8.2 Fourier Transforms 277
8.2.1 Operational Properties of the Fourier Transform 281
8.2.2 Fourier Sine and Cosine Transforms 284
8.2.3 Operational Properties of the Fourier Sine and Cosine
Transforms 288
8.2.4 Fourier Transforms and Convolutions 289
8.2.5 Fourier Transform of a Gaussian Function 294
8.3 Summary 297
8.3.1 Problems and Notes 298
9 Fourier Transform
Methods In PDEs 299
9.1 The Wave Equation 300
9.l.l d' Alembert's Solution to the One-Dimensional Wave
Equation 300
9.2 The Heat Equation 305
9.2.1 Heat Flow in an Infinite Rod 305
9.2.2 Fundamental Solution to the Heat Equation 306
9.2.3 Error Function 308
9.2.4 Heat Flow in a Semi-infinite Rod: Dirichlet Condition 311
9.2.5 Heat Flow in a Semi-infinite Rod: Neumann Condition 317
9.3 Laplace's Equation 319
9.3.1 Laplace's Equation in a Half-Plane 319
9.3.2 Laplace's Equation in a Semi-infinite Strip 324
9.4 Summary 328
9.4.1 Problems and Notes 329
10 Method of Characteristics 331
10.1 Introduction to the Method of Characteristics 331
10.2 Geometric Interpretation 335
10.3 d' Alembert's Solution 344
10.4 Extension to Quasilinear Equations 348
10.5 Summary 350
10.5.1 Problems and Notes 351
CONTENTS iX
PART II EXPLICITLY SOLVED PROBLEMS
11 Fourier Series Problems 355
12 Sturm-Liouville Problems 387
13 Heat Equation Problems 425
14 Wave Equation Problems 481
15 Laplace Equation Problems 533
16 Fourier Transform Problems 567
17 Method of Characteristics Problems 597
18 Four Sample Midterm Examinations 615
18.1 Midterm Exam 1 615
18.2 Midterm Exam 2 619
18.3 Midterm Exam 3 623
18.4 Midtenn Exam 4 626
19 Four Sample Final Examinations 631
19.1 Final Exam 1 631
19.2 Final Exam 2 640
19.3 Final Exam 3 646
19.4 Final Exam 4 653
Appendix A: Gamma Function 661
Bibliography 667
Index 671