Powers D.L. Boundary Value Problems and Partial Differential Equations
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BOUNDARY
VALUE PROBLEMS
FIFTH EDITION
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BOUNDARY
VALUE PROBLEMS
AND PARTIAL DIFFERENTIAL EQUATIONS
DAVID L. POWERS
Clarkson University
FIFTH EDITION
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Contents
Preface ix
CHAPTER 0 Ordinary Differential Equations 1
0.1 Homogeneous Linear Equations 1
0.2 Nonhomogeneous Linear Equations 14
0.3 Boundary Value Problems 26
0.4 Singular Boundary Value Problems 38
0.5 Green’s Functions 43
Chapter Review 51
Miscellaneous Exercises 51
CHAPTER 1 Fourier Series and Integrals 59
1.1 Periodic Functions and Fourier Series 59
1.2 Arbitrary Period and Half-Range Expansions 64
1.3 Convergence of Fourier Series 73
1.4 Uniform Convergence 79
1.5 Operations on Fourier Series 85
1.6 Mean Error and Convergence in Mean 90
1.7 Proof of Convergence 95
1.8 Numerical Determination of Fourier Coefficients 100
1.9 Fourier Integral 106
1.10 Complex Methods 113
1.11 Applications of Fourier Series and Integrals 117
1.12 Comments and References 124
Chapter Review 125
Miscellaneous Exercises 125
v
vi Contents
CHAPTER 2 The Heat Equation 135
2.1 Derivation and Boundary Conditions 135
2.2 Steady-State Temperatures 143
2.3 Example: Fixed End Temperatures 149
2.4 Example: Insulated Bar 157
2.5 Example: Different Boundary Conditions 163
2.6 Example: Convection 170
2.7 Sturm–Liouville Problems 175
2.8 Expansion in Series of Eigenfunctions 181
2.9 Generalities on the Heat Conduction Problem 184
2.10 Semi-Infinite Rod 188
2.11 Infinite Rod 193
2.12 The Error Function 199
2.13 Comments and References 204
Chapter Review 206
Miscellaneous Exercises 206
CHAPTER 3 The Wave Equation 215
3.1 The Vibrating String 215
3.2 Solution of the Vibrating String Problem 218
3.3 d’Alembert’s Solution 227
3.4 One-Dimensional Wave Equation: Generalities 233
3.5 Estimation of Eigenvalues 236
3.6 Wave Equation in Unbounded Regions 239
3.7 Comments and References 246
Chapter Review 247
Miscellaneous Exercises 247
CHAPTER 4 The Potential Equation 255
4.1 Potential Equation 255
4.2 Potential in a Rectangle 259
4.3 Further Examples for a Rectangle 264
4.4 Potential in Unbounded Regions 270
4.5 Potential in a Disk 275
4.6 Classification and Limitations 280
4.7 Comments and References 283
Chapter Review 285
Miscellaneous Exercises 285
CHAPTER 5 Higher Dimensions and Other Coordinates 295
5.1 Two-Dimensional Wave Equation: Derivation 295
5.2 Three-Dimensional Heat Equation 298
5.3 Two-Dimensional Heat Equation: Solution 303
Contents vii
5.4 Problems in Polar Coordinates 308
5.5 Bessel’s Equation 311
5.6 Temperature in a Cylinder 316
5.7 Vibrations of a Circular Membrane 321
5.8 Some Applications of Bessel Functions 329
5.9 Spherical Coordinates; Legendre Polynomials 335
5.10 Some Applications of Legendre Polynomials 345
5.11 Comments and References 353
Chapter Review 354
Miscellaneous Exercises 354
CHAPTER 6 Laplace Transform 363
6.1 Definition and Elementary Properties 363
6.2 Partial Fractions and Convolutions 369
6.3 Partial Differential Equations 376
6.4 More Difficult Examples 383
6.5 Comments and References 389
Miscellaneous Exercises 389
CHAPTER 7 Numerical Methods 397
7.1 Boundary Value Problems 397
7.2 Heat Problems 403
7.3 Wave Equation 408
7.4 Potential Equation 414
7.5 Two-Dimensional Problems 420
7.6 Comments and References 428
Miscellaneous Exercises 428
Bibliography 433
Appendix: Mathematical References 435
Answers to Odd-Numbered Exercises 441
Index 495
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Preface
This text is designed for a one-semester or two-quarter course in partial dif-
ferential equations given to third- and fourth-year students of engineering and
science. It can also be used as the basis for an introductory course for graduate
students. Mathematical prerequisites have been kept to a minimum — calculus
and differential equations. Vector calculus is used for only one derivation, and
necessary linear algebra is limited to determinants of order two. A reader needs
enough background in physics to follow the derivations of the heat and wave
equations.
The principal objective of the book is solving boundary value problems
involving partial differential equations. Separation of variables receives the
greatest attention because it is widely used in applications and because it pro-
vides a uniform method for solving important cases of the heat, wave, and
potential equations. One technique is not enough, of course. D’Alembert’s so-
lution of the wave equation is developed in parallel with the series solution,
and the distributed-source solution is constructed for the heat equation. In
addition, there are chapters on Laplace transform techniques and on numeri-
cal methods.
The second objective is to tie together the mathematics developed and the
student’s physical intuition. This is accomplished by deriving the mathemati-
cal model in a number of cases, by using physical reasoning in the mathemat-
ical development, by interpreting mathematical results in physical terms, and
by studying the heat, wave, and potential equations separately.
In the service of both objectives, there are many fully worked examples and
now about 900 exercises, including miscellaneous exercises at the end of each
chapter. The level of difficulty ranges from drill and verification of details
to development of new material. Answers to odd-numbered exercises are in
ix