Fourier Analysis and Boundary Value Problems
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Table of contents
- Table of Contentsv
- Prefaceix
- CHAPTER 1. A HEATED DISCUSSION1
- 1.1 Historical Prologue1
- 1.2 The Heat Equation4
- 1.3 Boundary Value Problems6
- 1.4 The Method of Separation of Variables9
- 1.5 Linearity and Superposition of Solutions11
- 1.6 Historical Epilogue14
- Exercises16
- CHAPTER 2. FOURIER SERIES23
- 2.1 Introduction23
- 2.2 Fourier Series25
- 2.3 The Riemann-Lebesgue Theorem30
- 2.4 The Convergence of Fourier Series36
- 2.5 Fourier Series on Arbitrary Intervals45
- 2.6 The Gibbs Phenomenon49
- 2.7 Fejér Sums53
- 2.8 Integration of Fourier Series58
- 2.9 Historical Epilogue62
- Exercises69
- CHAPTER 3. RETURN TO THE HEATED BAR84
- 3.1 Existence of a Solution84
- 3.2 Uniqueness and Stability of the Solution90
- 3.3 Nonzero Temperature at the Endpoints93
- 3.4 Bar Insulated at the Endpoints95
- 3.5 Mixed Endpoint Conditions97
- 3.6 Heat Convection at One Endpoint99
- 3.7 Time-Independent Problems101
- 3.8 The Steady-State Solution104
- 3.9 The Transient Solution108
- 3.10 The Complete Solution111
- 3.11 Time-Dependent Problems114
- Exercises120
- CHAPTER 4. GENERALIZED FOURIER SERIES133
- 4.1 Sturm-Liouville Problems133
- 4.2 The Eigenvalues and Eigenfunctions139
- 4.3 The Existence of the Eigenvalues141
- 4.4 Generalized Fourier Series151
- 4.5 Approximations154
- 4.6 Historical Epilogue157
- Exercises159
- CHAPTER 5. THE WAVE EQUATION165
- 5.1 Introduction165
- 5.2 The Vibrating String167
- 5.3 D'Alembert's Solution170
- 5.4 A Struck String178
- 5.5 Bernoulli's Solution181
- 5.6 Time-Independent Problems187
- 5.7 Time-Dependent Problems191
- 5.8 Historical Epilogue195
- Exercises198
- CHAPTER 6. ORTHOGONAL SYSTEMS207
- 6.1 Fourier Series and Parseval's Identity207
- 6.2 An Approximation Problem213
- 6.3 The Uniform Convergence of Fourier Series215
- 6.4 Convergence in the Mean217
- 6.5 Applications to the Vibrating String223
- 6.6 The Riesz-Fischer Theorem224
- Exercises230
- CHAPTER 7. FOURIER TRANSFORMS237
- 7.1 The Laplace Equation237
- 7.2 Fourier Transforms242
- 7.3 Properties of the Fourier Transform247
- 7.4 Convolution248
- 7.5 Solution of the Dirichlet Problem for the Half-Plane252
- 7.6 The Fourier Transform Method256
- Exercises259
- CHAPTER 8. LAPLACE TRANSFORMS266
- 8.1 The Laplace Transform and the Inversion Theorem266
- 8.2 Properties of the Laplace Transform271
- 8.3 Convolution278
- 8.4 The Telegraph Equation280
- 8.5 The Method of Residues286
- 8.6 Historical Epilogue293
- Exercises296
- CHAPTER 9. BOUNDARY VALUE PROBLEMS IN HIGHER DIMENSIONS302
- 9.1 Electrostatic Potential in a Charged Box302
- 9.2 Double Fourier Series310
- 9.3 The Dirichlet Problem in a Box319
- 9.4 Return to the Charged Box323
- 9.5 The Multiple Fourier Transform Method324
- 9.6 The Double Laplace Transform Method334
- Exercises340
- CHAPTER 10. BOUNDARY VALUE PROBLEMS WITH CIRCULAR SYMMETRY351
- 10.1 Vibrations of a Circular Membrane351
- 10.2 The Gamma Function355
- 10.3 Bessel Functions of the First Kind357
- 10.4 Recursion Formulas for Bessel Functions361
- 10.5 Bessel Functions of the Second Kind363
- 10.6 The Zeros of Bessel Functions365
- 10.7 Orthogonal Systems of Bessel Functions370
- 10.8 Fourier-Bessel Series and Dini-Bessel Series373
- 10.9 Return to the Vibrating Membrane378
- 10.10 Modified Bessel Functions383
- 10.11 The Skin Effect388
- Exercises394
- CHAPTER 11. BOUNDARY VALUE PROBLEMS WITH SPHERICAL SYMMETRY410
- 11.1 The Potbellied Stove410
- 11.2 Solutions of the Legendre Equation413
- 11.3 The Norms of the Legendre Polynomials417
- 11.4 Fourier-Legendre Series418
- 11.5 Return to the Potbellied Stove425
- 11.6 The Dirichlet Problem for the Sphere427
- 11.7 The Associated Legendre Functions428
- 11.8 Solution of the Dirichlet Problem for the Sphere432
- 11.9 Poisson's Integral Formula for the Sphere434
- 11.10 The Cooling of a Sphere441
- Exercises443
- CHAPTER 12. DISTRIBUTIONS AND GREEN'S FUNCTIONS451
- 12.1 Historical Prologue451
- 12.2 Distributions456
- 12.3 Basic Properties of Distributions460
- 12.4 Differentiation of Distributions463
- 12.5 Sequences and Series of Distributions468
- 12.6 Convolution471
- 12.7 The Poisson Equation on the Sphere480
- 12.8 Distributions Depending on a Parameter485
- 12.9 The Cauchy Problem for Time-Dependent Equations488
- 12.10 Conclusion493
- Exercises495
- APPENDIX A. UNIFORM CONVERGENCE506
- Excercise515
- APPENDIX B. IMPROPER INTEGRALS518
- Exercises533
- APPENDIX C. TABLES OF FOURIER AND LAPLACE TRANSFORMS535
- APPENDIX D. HISTORICAL BIBLIOGRAPHY539
- Index543
Fourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics.
A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.
Key Features
* Topics are covered from a historical perspective with biographical information on key contributors to the field
* The text contains more than 500 exercises
* Includes practical applications of the equations to problems in both engineering and physics
A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.
Key Features
* Topics are covered from a historical perspective with biographical information on key contributors to the field
* The text contains more than 500 exercises
* Includes practical applications of the equations to problems in both engineering and physics