Green's Functions with Applications, 2nd Edition

Book description

This book provides a systematic approach to the various methods available for deriving a Green’s function. It begins by reviewing the historical development of the Green’s function, the Fourier and Laplace transforms, the classical special functions of Bessel functions and Legendre polynomials, and the Dirac delta function. It 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, including worked examples, problem sets, and illustrations.

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 < x &lt; < L
    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

Product information

  • Title: Green's Functions with Applications, 2nd Edition
  • Author(s): Dean G. Duffy
  • Release date: March 2015
  • Publisher(s): Chapman and Hall/CRC
  • ISBN: 9781498798549