Fourier Series and Boundary Value Problems

Contents

Preface  xv

1 Fourier Series 1
Piecewise Continuous Functions 2
Fourier Cosine Series 4
Examples 6
Fourier Sine Series 8
Examples 9
Fourier Series 13
Examples 15
Adaptations to Other Intervals 18

2 Convergence of Fourier Series 23
One-Sided Derivatives 23
A Property of Fourier Coefficients 26
Two Lemmas 29
A Fourier Theorem 33
Discussion of the Theorem and Its Corollary 36
Convergence on Other Intervals 40
A Lemma 45
Absolute and Uniform Convergence of Fourier Series 47
Differentiation of Fourier Series 50
Integration of Fourier Series 52

3 Partial Differential Equations of Physics 57
Linear Boundary Value Problems 57
One-Dimensional Heat Equation 59
Related Equations 62
Laplacian in Cylindrical and Spherical Coordinates 64
Derivations 66
Boundary Conditions 68
A Vibrating String 73
Vibrations of Bars and Membranes 77
General Solution of the Wave Equation 81
Types of Equations and Boundary Conditions 84

4 The Fourier Method 88
Linear Operators 88
Principle of Superposition 90
A Temperature Problem 94
A Vibrating String Problem 99
Historical Development 102

5 Boundary Value Problems 104
A Slab with Faces at Prescribed Temperatures 105
Related Problems 109
A Slab with Internally Generated Heat 114
Steady Temperatures in a Rectangular Plate 120
Cylindrical Coordinates 124
A String with Prescribed Initial Conditions 129
Resonance 134
An Elastic Bar 137
Double Fourier Series 140
Periodic Boundary Conditions 143

6 Fourier Integrals and Applications 148
The Fourier Integral Formula 148
Dirichlet's Integral 150
Two Lemmas 152
A Fourier Integral Theorem 155
The Cosine and Sine Integrals 159
More on Superposition of Solutions 163
Temperatures in a Semi-Infinite Solid 165
Temperatures in an Unlimited Medium 172

7 Orthonormal Sets 174
Inner Products and Orthonormal Sets 174
Examples 176
Generalized Fourier Series 180
Examples 182
Best Approximation in the Mean 185
Bessel's Inequality and Parseval's Equation 188
Applications to Fourier Series 190

8 Sturm-Liouville Problems and Applications 195
Regular Sturm-Liouville Problems 195
Modifications 197
Orthogonality of Eigenfunctions 198
Real-Valued Eigenfunctions and Nonnegative Eigenvalues 203
Methods of Solution 205
Examples of Eigenfunction Expansions 211
A Temperature Problem in Rectangular Coordinates 217
Another Problem 219
Other Coordinates 224
A Modification of the Method 227
Another Modification 230
A Vertically Hung Elastic Bar 233

9 Bessel Functions and Applications 241
Bessel Functions $J_n(x)$ 242
General Solutions of Bessel's Equation 245
Recurrence Relations 252
Bessel's Integral Form 255
Some Consequences of the Integral Forms 257
The Zeros of $J_n(x)$ 260
Zeros of Related Functions 263
Orthogonal Sets of Bessel Functions 264
Proof of the Theorems 267
The Orthonormal Functions 272
Fourier- Bessel Series 275
Examples 278
Temperatures in a Long Cylinder 283
Internally Generated Heat 288
Vibration of a Circular Membrane 293

10 Legendre Polynomials and Applications 298
Solutions of Legendre's Equation 298
Legendre Polynomials 300
Orthogonality of Legendre Polynomials 305
Rodrigues' Formula and Norms 307
Legendre Series 313
The Eigenfunctions $P_n(\cos\theta)$ 317
Dirichlet Problems in Spherical Regions 319
Steady Temperatures in a Hemisphere 323

11 Verification of Solutions and Uniqueness 328
Abel's Test for Uniform Convergence 328
Verification of Solution of Temperature Problem 331
Uniqueness of Solutions of the Heat Equation 334
Verification of Solution of Vibrating String Problem 338
Uniqueness of Solutions of the Wave Equation 342

Appendixes 345
Bibliography 345
Some Fourier Series Expansions 349
Solutions of Some Regular Sturm-Liouville Problems 351
Index 355