Fourier analysis and boundary value problems [[electronic resource] /] / by Enrique A. González-Velasco

• Record Nr.: UNINA9910784644503321
• Autore: González-Velasco Enrique A
• Pubbl/distr/stampa: San Diego, : Academic Pres, c1995
• ISBN: 1-281-03822-9; 9786611038229; 0-08-053193-8
• Descrizione fisica: 1 online resource (565 p.)
• Disciplina: 515/.353
• Soggetti: Fourier analysis; Boundary value problems - Numerical solutions
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references (p. 539-542) and index.
• Nota di contenuto: Front Cover; Fourier Analysis and Boundary Value Problems; Copyright Page; Table of Contents; Preface; CHAPTER 1. A HEATED DISCUSSION; 1.1 Historical Prologue; 1.2 The Heat Equation; 1.3 Boundary Value Problems; 1.4 The Method of Separation of Variables; 1.5 Linearity and Superposition of Solutions; 1.6 Historical Epilogue; Exercises; CHAPTER 2. FOURIER SERIES; 2.1 Introduction; 2.2 Fourier Series; 2.3 The Riemann-Lebesgue Theorem; 2.4 The Convergence of Fourier Series; 2.5 Fourier Series on Arbitrary Intervals; 2.6 The Gibbs Phenomenon; 2.7 Fejér Sums; 2.8 Integration of Fourier Series; 2.9 Historical EpilogueExercises; CHAPTER 3. RETURN TO THE HEATED BAR; 3.1 Existence of a Solution; 3.2 Uniqueness and Stability of the Solution; 3.3 Nonzero Temperature at the Endpoints; 3.4 Bar Insulated at the Endpoints; 3.5 Mixed Endpoint Conditions; 3.6 Heat Convection at One Endpoint; 3.7 Time-Independent Problems; 3.8 The Steady-State Solution; 3.9 The Transient Solution; 3.10 The Complete Solution; 3.11 Time-Dependent Problems; Exercises; CHAPTER 4. GENERALIZED FOURIER SERIES; 4.1 Sturm-Liouville Problems; 4.2 The Eigenvalues and Eigenfunctions; 4.3 The Existence of the Eigenvalues; 4.4 Generalized Fourier Series4.5 Approximations; 4.6 Historical Epilogue; Exercises; CHAPTER 5. THE WAVE EQUATION; 5.1 Introduction; 5.2 The Vibrating String; 5.3 D'Alembert's Solution; 5.4 A ; Struck String; 5.5 Bernoulli's Solution; 5.6 Time-Independent Problems; 5.7 Time-Dependent Problems; 5.8 Historical Epilogue; Exercises; CHAPTER 6. ORTHOGONAL SYSTEMS; 6.1 Fourier Series and Parseval's Identity; 6.2 An Approximation Problem; 6.3 The Uniform Convergence of Fourier Series; 6.4 Convergence in the Mean; 6.5 Applications to the Vibrating String; 6.6 The Riesz-Fischer Theorem; Exercises; CHAPTER 7. FOURIER TRANSFORMS7.1 The Laplace Equation; 7.2 Fourier Transforms; 7.3 Properties of the Fourier Transform; 7.4 Convolution; 7.5 Solution of the Dirichlet Problem for the Half-Plane; 7.6 The Fourier Transform Method; Exercises; CHAPTER 8. LAPLACE TRANSFORMS; 8.1 The Laplace Transform and the Inversion Theorem; 8.2 Properties of the Laplace Transform; 8.3 Convolution; 8.4 The Telegraph Equation; 8.5 The Method of Residues; 8.6 Historical Epilogue; Exercises; CHAPTER 9. BOUNDARY VALUE PROBLEMS IN HIGHER DIMENSIONS; 9.1 Electrostatic Potential in a Charged Box; 9.2 Double Fourier Series9.3 The Dirichlet Problem in a Box; 9.4 Return to the Charged Box; 9.5 The Multiple Fourier Transform Method; 9.6 The Double Laplace Transform Method; Exercises; CHAPTER 10. BOUNDARY VALUE PROBLEMS WITH CIRCULAR SYMMETRY; 10.1 Vibrations of a Circular Membrane; 10.2 The Gamma Function; 10.3 Bessel Functions of the First Kind; 10.4 Recursion Formulas for Bessel Functions; 10.5 Bessel Functions of the Second Kind; 10.6 The Zeros of Bessel Functions; 10.7 Orthogonal Systems of Bessel Functions; 10.8 Fourier-Bessel Series and Dini-Bessel Series; 10.9 Return to the Vibrating Membrane
• Sommario/riassunto: 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 hav