Table of contents for Fourier series, transforms, and boundary value problems.


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1. Linear Differential Equations                              1
1.1. Linear Operators, 1
1.2. Ordinary Differential Equations, 2
1.3. Homogeneous Linear ODE with Constant Coefficients, 5
1.4. Euler's ODE, 7
1.5. Series Solutions, 8
1.6. Frobenius Method, 13
1.7. Numerical Solutions, 19
1.8. Linear PDEs, 23
1.9. Classification of a Linear PDE of Second Order, 23
1.10. Boundary Value Problems with PDEs, 24
1.11. Second Order Linear PDEs with Constant Coefficients, 26
1.12. Separation of Variables, 35
2. Orthogonal Sets of Functions                               40
2.1. Orthogonality and Vectors, 40
2.2. Orthogonal Functions, 42
2.3. Complex Functions, 46
2.4. Additional Concepts of Orthogonality, 47
2.5. The Sturm-Liouville Boundary Value Problem, 50
2.6. Uniform Convergence of Series, 58
2.7. Series of Orthogonal Functions, 61
2.8. Approximation by Least Squares, 64
2.9. Completeness of Sets, 65
3. Fourier Series                                             68
3.1. Piecewise Continuous Functions, 68
3.2. A Basic Fourier Series, 72
3.3. Even and Odd Functions, 76
3.4. Fourier Sine and Cosine Series, 77
3.5. Complex Fourier Series, 80
3.6. Harmonic Analysis, 82
3.7. Uniform Convergence of Fourier Series, 89
3.8. Differentiation of Fourier Series, 92
3.9. Integration of Fourier Series, 94
3.10. Double Fourler Series, 97
4. Fourier Integrals                                          102
4.1. Uniform Convergence of Integrals, 102
4.2. A Generalization of the Fourier Series, 107
4.3. Fourier Sine and Cosine Integrals, 109
4.4. The Exponential Fourier Integral, 112
5.. Bessel Functions                                          117
5.1. The Gamma Function and the Bessel Function, 117
5.2. Additional Bessel Functions, 120
5.3. Differential Equations Solvable with Bessel Functions, 122
5.4. Special Bessel Functions and Identities, 124
5.5. An Integral Form for J,(x), 130
5.6. Singular SLPs, 133
5.7. Orthogonality of Bessel Functions, 134
5.8. Orthogonal Series of Bessel Functions, 137
5.9. Bessel Functions and Cylindrical Geometry, 140
6. Legendre Polynomials                                       142
6.1. Solutions to the Legendre Equation, 142
6.2. Rodrigues' Formula for Legendre Polynomials, 146
6.3. A Generating Function for P,(x), 149
6.4. The Legendre Polynomial P,(cos 0), 151
6.5. Orthogonality and Norms of P,(x), 152
6.6. Legendre Series, 154
6.7. Legendre Polynomials and Spherical Geometry, 158
6.8. Spherical Harmonics, 161
6.9. The Generalized Legendre Equation, 162
7. Integral Transforms                                        168
7.1. Laplace Transforms, 168
7.2. Existence of the Transform, 169
7.3. The Gamma Function and Laplace Transforms, 170
7.4. Transforms of Derivatives, 172
7.5. Derivatives of Transforms, 172
7.6. The Inverse Laplace Transform, 173
7.7. Solutions of ODEs and IVPs, 173
7.8. Partial Fractions, 174
7.9. The Unit Step Function, 175
7.10. Shifting Properties, 176
7.11. The Dirac Delta Function, 177
7.12. Convolution, 180
7.13. Laplace Transform Method for PDEs, 186
7.14. Finite Fourier Transforms, 189
7.15. Fourier Transforms, 191
7.16. The Discrete Fourier Transform, 197
7.17. The Fast Fourier Transform, 203
7.18. Fourier Transforms of Functions of Two Variables, 208
7.19. Hankel Transforms, 209
7.20. Legendre Transform, 214
7.21. Mellin Transform, 215
8. Application of BVPs                                       219
8.1. The Vibrating String, 219
8.2. Verification and Uniqueness of the Solution of the
Vibrating String Problem, 225
8.3. The Vibrating String with Two Nonhomogeneous
Conditions, 228
8.4. Longitudinal Vibrations along an Elastic Rod, 230
8.5. Heat Conduction, 236
8.6. Numerical Solution of the Heat Equation, 241
8.7. Verification and Uniqueness of the Solution for the
Heat Problem, 242
8.8. Gravitational Potential, 246
8.9. Laplace's Equation, 247
8.10. Numerical Solution of the Laplace Equation, 251
8.11. Temperature in a Circular Disk with Insulated Faces, 254
8.12. Steady State Temperature in a Right Semicircular Cylinder, 256
8.13. Harmonic Interior of a Right Circular Cylinder, 260
8.14. Steady State Temperature Distribution in a Sphere, 264
8.15. Potential for a Sphere, 267
9. Additional Applications                                   270
9.1. Mechanical and Electrical Oscillations, 270
9.2. The Vibrating Membrane, 273
9.3. Vibrations of a Circular Membrane Dependent on Distance
from Center, 280
9.4. The Vibrating String with an External Force, 283
9.5. Nonhomogeneous End Temperatures in a Rod, 289
9.6. A Rod with Insulated Ends, 291
9.7. A Semi-Infinite Bar, 295
9.8. An Infinite Bar, 297
9.9. Discrete Fourier Transform Solutions, 305
9.10. A Semi-Infinite String, 307
9.11. A Semi-Infinite String with Initial Velocity, 310
References                                                      315
Answers to Exercises                                            317
Appendix 1 Selected Integrals                                   340
Appendix 2 Table of Laplace Transforms                          342
Appendix 3  Tables of Finite Fourier Transforms                 344
Appendix 4  Tables of Fourier Transforms                        346



Library of Congress subject headings for this publication: Boundary value problems, Fourier series