Partial Differential Equations

by Niels Walet

Read here Read at LibreTextsPDF

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics.

Description from LibreTexts.

Publisher
University of Manchester
Subjects
Mathematics, Differential Equations
Licence
CC BY-NC-SA 2.0, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 1: Introduction to Partial Differential Equations
  7. 1.1: Ordinary Differential Equations
  8. 1.2: PDE’s
  9. 2: Classification of Partial Differential Equations
  10. 2.1: Examples of PDE
  11. 2.2: Second Order PDE
  12. 2.3: More than 2D
  13. 2.E: Classification of Partial Differential Equations (Exercises)
  14. 3: Boundary and Initial Conditions
  15. 3.1: Introduction to Boundary and Initial Conditions
  16. 3.2: Explicit Boundary Conditions
  17. 3.3: Implicit Boundary Conditions
  18. 3.4: More Realistic Examples of Boundary and Initial Conditions
  19. 4: Fourier Series
  20. 4.1: Taylor Series
  21. 4.2: Introduction to Fourier Series
  22. 4.3: Periodic Functions
  23. 4.4: Orthogonality and Normalization
  24. 4.5: When is it a Fourier Series?
  25. 4.6: Fourier series for even and odd functions
  26. 4.7: Convergence of Fourier series
  27. 5: Separation of Variables on Rectangular Domains
  28. 5.1: Cookbook
  29. 5.2: Parabolic Equation
  30. 5.3: Hyperbolic Equation
  31. 5.4: Laplace’s Equation
  32. 5.5: More Complex Initial/Boundary Conditions
  33. 5.6: Inhomogeneous Equations
  34. 6: D’Alembert’s Solution to the Wave Equation
  35. 6.1: Background to D’Alembert’s Solution
  36. 6.2: New Variables
  37. 6.3: Examples
  38. 7: Polar and Spherical Coordinate Systems
  39. 7.1: Polar Coordinates
  40. 7.2: Spherical Coordinates
  41. 8: Separation of Variables in Polar Coordinates
  42. 8.1: Example
  43. 8.2: Three cases for λ
  44. 8.3: Putting it all together
  45. 9: Series Solutions of ODEs (Frobenius’ Method)
  46. 9.1: Frobenius’ Method
  47. 9.2: Singular Points
  48. 9.3: Special Cases
  49. 10: Bessel Functions and Two-Dimensional Problems
  50. 10.1: Temperature on a Disk
  51. 10.2: Bessel’s Equation
  52. 10.3: Gamma Function
  53. 10.4: Bessel Functions of General Order
  54. 10.5: Properties of Bessel functions
  55. 10.6: Sturm-Liouville Theory
  56. 10.7: Our Initial Problem and Bessel Functions
  57. 10.8: Fourier-Bessel Series
  58. 10.9: Back to our Initial Problem
  59. 11: Separation of Variables in Three Dimensions
  60. 11.1: Modelling the Eye
  61. 11.2: Properties of Legendre Polynomials
  62. 11.3: Fourier-Legendre Series
  63. 11.4: Modelling the eye–revisited
  64. Back Matter
  65. Index
  66. Detailed Licensing