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Green’s Functions with Applications, Second Edition, 2nd Edition

Book Description

Since publication of the first edition over a decade ago, Green’s Functions with Applications has provided applied scientists and engineers with a systematic approach to the various methods available for deriving a Green’s function. This fully revised Second Edition retains the same purpose, but has been meticulously updated to reflect the current state of the art.

The book opens with necessary background information: a new chapter on the historical development of the Green’s function, coverage of the Fourier and Laplace transforms, a discussion of the classical special functions of Bessel functions and Legendre polynomials, and a review of the Dirac delta function.

The text then presents Green’s functions for each class of differential equation (ordinary differential, wave, heat, and Helmholtz equations) according to the number of spatial dimensions and the geometry of the domain. Detailing step-by-step methods for finding and computing Green’s functions, each chapter contains a special section devoted to topics where Green’s functions particularly are useful. For example, in the case of the wave equation, Green’s functions are beneficial in describing diffraction and waves.

To aid readers in developing practical skills for finding Green’s functions, worked examples, problem sets, and illustrations from acoustics, applied mechanics, antennas, and the stability of fluids and plasmas are featured throughout the text. A new chapter on numerical methods closes the book.

Included solutions and hundreds of references to the literature on the construction and use of Green's functions make Green’s Functions with Applications, Second Edition a valuable sourcebook for practitioners as well as graduate students in the sciences and engineering.

Table of Contents

  1. Preliminaries
  2. Acknowledgments
  3. Author
  4. Preface
  5. Definitions of the Most Commonly Used Functions
  6. Chapter 1 Historical Development
    1. 1.1 Mr. Green’s Essay
    2. 1.2 Potential Equation
    3. 1.3 Heat Equation
    4. 1.4 Helmholtz’s Equation
    5. 1.5 Wave Equation
    6. 1.6 Ordinary Differential Equations
      1. Figure 1.2.1
      2. Figure 1.3.1
      3. Figure 1.3.2
      4. Figure 1.3.3
      5. Figure 1.5.1
      6. Figure 1.5.2
      7. Figure 1.6.1
  7. Chapter 2 Background Material
    1. 2.1 Fourier Transform
    2. 2.2 Laplace Transform
    3. 2.3 Bessel Functions
    4. 2.4 Legendre Polynomials
    5. 2.5 The Dirac Delta Function
    6. 2.6 Green’s Formulas
    7. 2.7 What is a Green’s Function?
    8. Problems
      1. Figure 2.1.1
      2. Figure 2.2.1
      3. Figure 2.2.2
      4. Figure 2.3.1
      5. Figure 2.3.2
      6. Figure 2.3.3
      7. Figure 2.3.4
      8. Figure 2.3.5
      9. Figure 2.3.6
      10. Figure 2.4.1
      11. Figure 2.4.2
      12. Figure 2.5.1
      13. Figure 2.5.2
      14. Figure 2.5.3
      15. Figure 2.7.1
      16. Figure 2.7.2
      17. Figure 2.7.3
      18. Figure 2.7.4
      1. Table 2.1.1
      2. Table 2.2.1
      3. Table 2.3.1
      4. Table 2.3.2
      5. Table 2.4.1
      6. Table 2.4.2
      7. Table 2.5.1
      8. Table 2.5.2
      1. Example 2.1.1
      2. Example 2.1.2
      3. Example 2.1.3
      4. Example 2.2.1
      5. Example 2.2.2
      6. Example 2.2.3
      7. Example 2.2.4
      8. Example 2.2.5
      9. Example 2.2.6
      10. Example 2.2.7
      11. Example 2.2.8
      12. Example 2.2.9
      13. Example 2.2.10
      14. Example 2.2.11
      15. Example 2.3.1
      16. Example 2.3.2
      17. Example 2.4.1
      18. Example 2.4.2
      19. Example 2.4.3
      20. Example 2.4.4
      21. Example 2.5.1
      22. Example 2.5.2
      23. Example 2.5.3
      24. Example 2.5.4
      25. Example 2.5.5
      26. Example 2.5.6
      27. Example 2.6.1
      28. Example 2.6.2
  8. Chapter 3 Green’s Functions for Ordinary Differential Equations
    1. 3.1 Initial-Value Problems
    2. 3.2 The Superposition Integral
    3. 3.3 Regular Boundary-Value Problems
    4. 3.4 Eigenfunction Expansion for Regular Boundary-Value Problems
    5. 3.5 Singular Boundary-Value Problems
    6. 3.6 Maxwell’s Reciprocity
    7. 3.7 Generalized Green’s Function
    8. 3.8 Integro-Differential Equations
    9. Problems
      1. Figure 3.1.1
      2. Figure 3.1.2
      3. Figure 3.2.1
      4. Figure 3.3.1
      5. Figure 3.3.2
      6. Figure 3.4.1
      7. Figure 3.4.2
      8. Figure 3.5.1
      9. Figure 3.5.2
      10. Figure 3.5.3
      11. Figure 3.6.1
      12. Figure 3.6.2
      13. Figure 3.8.1
      14. Figure 3.8.2
      1. Table 3.3.1
      2. Table 3.3.2
      1. Example 3.1.1
      2. Example 3.1.2
      3. Example 3.3.1
      4. Example 3.3.2
      5. Example 3.3.3
      6. Example 3.3.4
      7. Example 3.3.5
      8. Example 3.3.6
      9. Example 3.4.1
      10. Example 3.4.2
      11. Example 3.4.3
      12. Example 3.5.1
      13. Example 3.5.2
      14. Example 3.5.3
      15. Example 3.5.4
      16. Example 3.5.5
      17. Example 3.6.1
      18. Example 3.7.1
      19. Example 3.7.2
      20. Example 3.7.3
      21. Example 3.7.4
      22. Example 3.8.1
      23. Example 3.8.2
  9. Chapter 4 Green’s Functions for the Wave Equation
    1. 4.1 One-Dimensional Wave Equation in an Unlimited Domain
    2. 4.2 One-Dimensional Wave Equation on the Interval 0 <<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic"> x</span> &lt; < <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">L</span>
    3. 4.3 Axisymmetric Vibrations of a Circular Membrane
    4. 4.4 Two-Dimensional Wave Equation in an Unlimited Domain
    5. 4.5 Three-Dimensional Wave Equation in an Unlimited Domain
    6. 4.6 Asymmetric Vibrations of a Circular Membrane
    7. 4.7 Thermal Waves
    8. 4.8 Diffraction of a Cylindrical Pulse by a Half-Plane
    9. 4.9 Leaky Modes
    10. 4.10 Water Waves
    11. Problems
      1. Figure 4.1.1
      2. Figure 4.1.2
      3. Figure 4.1.3
      4. Figure 4.1.4
      5. Figure 4.1.5
      6. Figure 4.1.6
      7. Figure 4.2.1
      8. Figure 4.2.2
      9. Figure 4.3.1
      10. Figure 4.6.1
      11. Figure 4.8.1
      12. Figure 4.9.1
      13. Figure 4.9.2
      14. Figure 4.9.3
      15. Figure 4.9.4
      16. Figure 4.9.5
      17. Figure 4.9.6
      18. Figure 4.9.7
      19. Figure 4.9.8
      20. Figure 4.9.9
      21. Figure 4.9.10
      22. Figure 4.9.11
      23. Figure 4.9.12
      24. Figure 4.9.13
      25. Figure 4.10.1
      1. Example 4.1.1
      2. Example 4.1.2
      3. Example 4.1.3
      4. Example 4.1.4
      5. Example 4.1.5
      6. Example 4.1.6
      7. Example 4.1.7
      8. Example 4.1.8
      9. Example 4.2.1
      10. Example 4.2.2
      11. Example 4.4.1
      12. Example 4.4.2
      13. Example 4.4.3
      14. Example 4.4.4
      15. Example 4.5.1
      16. Example 4.5.2
      17. Example 4.5.3
      18. Example 4.5.4
      19. Example 4.5.5
      20. Example 4.5.6
      21. Example 4.5.7
      22. Example 4.6.1
      23. Example 4.10.1
      24. Example 4.10.2
      25. Example 4.10.3
  10. Chapter 5 Green’s Functions for the Heat Equation
    1. 5.1 Heat Equation Over Infinite or Semi-Infinite Domains
    2. 5.2 Heat Equation Within a Finite Cartesian Domain
    3. 5.3 Heat Equation Within a Cylinder
    4. 5.4 Heat Equation Within a Sphere
    5. 5.5 Product Solution
    6. 5.6 Absolute and Convective Instability
    7. Problems
      1. Figure 5.1.1
      2. Figure 5.1.2
      3. Figure 5.1.3
      4. Figure 5.1.4
      5. Figure 5.1.5
      6. Figure 5.2.1
      7. Figure 5.2.2
      8. Figure 5.3.1
      9. Figure 5.3.2
      10. Figure 5.3.3
      11. Figure 5.6.1
      12. Figure 5.6.2
      13. Figure 5.6.3
      1. Table 5.1.1
      2. Table 5.3.2
      3. Table 5.3.3
      4. Table 5.3.4
      5. Table 5.3.5
      6. Table 5.3.6
      7. Table 5.3.7
      1. Example 5.0.1
      2. Example 5.1.1
      3. Example 5.1.2
      4. Example 5.1.3
      5. Example 5.1.4
      6. Example 5.1.5
      7. Example 5.1.6
      8. Example 5.1.7
      9. Example 5.1.8
      10. Example 5.1.9
      11. Example 5.2.1
      12. Example 5.2.2
      13. Example 5.2.3
      14. Example 5.2.4
      15. Example 5.2.5
      16. Example 5.2.6
      17. Example 5.2.7
      18. Example 5.3.1
      19. Example 5.3.2
      20. Example 5.3.3
      21. Example 5.3.4
      22. Example 5.3.5
      23. Example 5.5.1
      24. Example 5.5.2
      25. Example 5.5.3
      26. Example 5.6.1
      27. Example 5.6.2
  11. Chapter 6 Green’s Functions for the Helmholtz Equation
    1. 6.1 Free-Space Green’s Functions for Helmholtz’s and Poisson’s Equations
    2. 6.2 Method of Images
    3. 6.3 Two-Dimensional Poisson’s Equation Over Rectangular and Circular Domains
    4. 6.4 Two-Dimensional Helmholtz Equation Over Rectangular and Circular Domains
    5. 6.5 Poisson’s and Helmholtz’s Equations on a Rectangular Strip
    6. 6.6 Three-Dimensional Problems in a Half-Space
    7. 6.7 Three-Dimensional Poisson’s Equation in a Cylindrical Domain
    8. 6.8 Poisson’s Equation for a Spherical Domain
    9. 6.9 Improving the Convergence Rate of Green’s Functions
    10. 6.10 Mixed Boundary Value Problems
    11. Problems
      1. Figure 6.1.1
      2. Figure 6.2.1
      3. Figure 6.2.2
      4. Figure 6.2.3
      5. Figure 6.3.1
      6. Figure 6.3.2
      7. Figure 6.3.3
      8. Figure 6.4.1
      9. Figure 6.4.2
      10. Figure 6.4.3
      11. Figure 6.4.4
      12. Figure 6.4.5
      13. Figure 6.4.6
      14. Figure 6.4.7
      15. Figure 6.5.1
      16. Figure 6.5.2
      17. Figure 6.6.1
      18. Figure 6.6.2
      19. Figure 6.8.1
      20. Figure 6.9.1
      21. Figure 6.9.2
      22. Figure 6.10.1
      23. Figure 6.10.2
      24. Figure 6.10.3
      25. Figure 6.10.4
      26. Figure 6.10.5
      27. Figure 6.10.6
      1. Table 6.1.1
      2. Table 6.1.2
      3. Table 6.1.3
      1. Example 6.1.1
      2. Example 6.1.2
      3. Example 6.1.3
      4. Example 6.1.4
      5. Example 6.1.5
      6. Example 6.1.6
      7. Example 6.1.7
      8. Example 6.2.1
      9. Example 6.2.2
      10. Example 6.2.3
      11. Example 6.2.4
      12. Example 6.2.5
      13. Example 6.3.1
      14. Example 6.5.1
      15. Example 6.6.1
      16. Example 6.7.1
      17. Example 6.8.1
      18. Example 6.8.2
      19. Example 6.8.3
      20. Example 6.9.1
      21. Example 6.10.1
      22. Example 6.10.2
      23. Example 6.10.3
      24. Example 6.10.4
      25. Example 6.10.5
      26. Example 6.10.6
  12. Chapter 7 Numerical Methods
    1. 7.1 Discrete Wavenumber Representation
    2. 7.2 Laplace Transform Method
    3. 7.3 Finite Difference Method
    4. 7.4 Hybrid Method
    5. 7.5 Galerkin Method
    6. 7.6 Evaluation of the Superposition Integral
    7. 7.7 Mixed Boundary Value Problems
      1. Figure 7.1.1
      2. Figure 7.2.1
      3. Figure 7.2.2
      4. Figure 7.2.3
      5. Figure 7.2.4
      6. Figure 7.3.1
      7. Figure 7.3.2
      8. Figure 7.4.1
      9. Figure 7.5.1
      10. Figure 7.6.1
      11. Figure 7.6.2
      12. Figure 7.7.1
      13. Figure 7.7.2
      1. Table 7.5.1
      2. Table 7.6.1
      1. Example 7.2.1
      2. Example 7.2.2
      3. Example 7.3.1
      4. Example 7.3.2
      5. Example 7.5.1
      6. Example 7.5.2
      7. Example 7.5.3
      8. Example 3.6.1
      9. Example 7.6.2
      10. Example 7.6.3
  13. Appendix: Relationship between Solutions of Helmholtz’s and Laplace’s Equations in Cylindrical and Spherical Coordinates
    1. Problems
  14. Answers to Some of the Problems