Partial Differential Equations and Boundary Value Problems with Fourier Series, Second Edition

Contents

Preface VII

1 A Preview of Applications and Techniques 1
1.1 What Is a Partial Differential Equation? 2
2 1.2 Solving and Interpreting a Partial Differential Equation 7

2 Fourier Series 17
2.1 Periodic Functions 18
2.2 Fourier Series 26
2.3 Fourier Series of Functions with Arbitrary Periods 38
2.4 Half-Range Expansions: The Cosine and Sine Series 50
2.5 Mean Square Approximation and Parseval\'s Identity 53
2.6 Complex Form of Fourier Series 60
2.7 Forced Oscillations 69
Supplement on Convergence
2.8 Proof of the Fourier Series Representation Theorem 77
2.9 Uniform Convergence and Fourier Series 85
2.10 Dirichlet Test and Convergence of Fourier Series 94

3 Partial Differential Equations in Rectangular Coordinates 103
3.1 Partial Differential Equations in Physics and Engineering 104
3.2 Modeling: Vibrating Strings and the Wave Equation 109
3.3 Solution of the One Dimensional Wave Equation: The Method of Separation of Variables 114
3.4 D\'Alembert\'s Method 126
3.5 The One Dimensional Heat Equation 135
3.6 Heat Conduction in Bars: Varying the Boundary Conditions 146
3.7 The Two Dimensional Wave and Heat Equations 155
3.8 Laplace\'s Equation in Rectangular Coordinates 163
3.9 Poisson\'s Equation: The Method of Eigenfunction Expansions 170
3.10 Neumann and Robin Conditions 180
3.11 The Maximum Principle 187

4 Partial Differential Equations in Polar and Cylindrical Coordinates 193
4.1 The Laplacian in Various Coordinate Systems 194
4.2 Vibrations of a Circular Membrane: Symmetric Case 198
4.3 Vibrations of a Circular Membrane: General Case 207
4.4 Laplace\'s Equation in Circular Regions 216
4.5 Laplace\'s Equation in a Cylinder 228
4.6 The Helmholtz and Poisson Equations 231
Supplement on Bessel Functions
4.7 Bessel\'s Equation and Bessel Functions 237
4.8 Bessel Series Expansions 248
4.9 Integral Formulas and Asymptotics for Bessel Functions 261

5 Partial Differential Equations in Spherical Coordinates 269
5.1 Preview of Problems and Methods 270
5.2 Dirichlet Problems with Symmetry 274
5.3 Spherical Harmonics and the General Dirichlet Problem 281
5.4 The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations 291
Supplement on Legendre Functions
5.5 Legendre\'s Differential Equation 300
5.6 Legendre Polynomials and Legendre Series Expansions 308
5.7 Associated Legendre Functions and Series Expansions 319

6 Sturm-Liouville Theory with Engineering Applications 325
6.1 Orthogonal Functions 326
6.2 Sturm-Liouville Theory 333
6.3 The Hanging Chain 346
6.4 Fourth Order Sturm-Liouville Theory 353
6.5 Elastic Vibrations and Buckling of Beams 360
6.6 The Biharmonic Operator 371
6.7 Vibrations of Circular Plates 377

7 The Fourier Transform and Its Applications 389
7.1 The Fourier Integral Representation 390
7.2 The Fourier Transform 398
7.3 The Fourier Transform Method 411
7.4 The Heat Equation and Gauss\'s Kernel 420
7.5 A Dirichlet Problem and the Poisson Integral Formula 429
7.6 The Fourier Cosine and Sine Transforms 433
7.7 Problems Involving Semi-Infinite Intervals 440
7.8 Generalized Functions 445
7.9 The Nonhomogeneous Heat Equation 461
7.10 Duhamel\'s Principle 471

8 The Laplace and Hankel Transforms with Applications 479
8.1 The Laplace Transform 480
8.2 Further Properties of the Laplace Transform 491
8.3 The Laplace Transform Method 502
8.4 The Hankel Transform with Applications 508

9 Finite Difference Numerical Methods 515
9.1 The Finite Difference Method for the Heat Equation 516
9.2 The Finite Difference Method for the Wave Equation 525
9.3 The Finite Difference Method for Laplace\'s Equation 533
9.4 Iteration Methods for Laplace\'s Equation 541

10 Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations 546
10.1 The Sampling Theorem 547
10.2 Partial Differential Equations and the Sampling Theorem 555
10.3 The Discrete and Fast Fourier Transforms 559
10.4 The Fourier and Discrete Fourier Transforms 567

11 An Introduction to Quantum Mechanics 573
11.1 Schrodinger\'s Equation 574
11.2 The Hydrogen Atom 581
11.3. Heisenberg\'s Uncertainty Principle 590
Supplement on Orthogonal Polynomials
11.4 Hermite and Laguerre Polynomials 597

12 Green\'s Functions and Conformal Mappings 611
12.1 Green\'s Theorem and Identities 612
12.2 Harmonic Functions and Green\'s Identities 622
12.3 Green\'s Functions 629
12.4 Green\'s Functions for the Disk and the Upper Half-Plane 638
12.5 Analytic Functions 645
12.6 Solving Dirichlet Problems with Conformal Mappings 663
12.7 Green\'s Functions and Conformal Mappings 674
12.8 Neumann Functions and the Solution of Neumann Problems 684

APPENDIXES
A Ordinary Differential Equations: Review of Concepts and Methods Al
A.l Linear Ordinary Differential Equations A2
A.2 Linear Ordinary Differential Equations with Constant Coefficients A10
A.3 Linear Ordinary Differential Equations with Nonconstant Coefficients A21
A.4 The Power Series Method, Part I A28
A.5 The Power Series Method, Part II A40
A.6 The Method of Frobenius A51

B Tables of Transforms A65
B.1 Fourier Transforms A66
B.2 Fourier Cosine Transforms A68
B.3 Fourier Sine Transforms A69
B.4 Laplace Transforms A70

References A73

Answers to Selected Exercises A75

Index A99