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Partial Differential Equations

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Partial Differential Equations

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.

Differential EquationsMathematics
Author
Niels Walet
Publisher
University of Manchester
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 58 sections
  1. 01: Introduction to Partial Differential Equations
  2. 1.01: Ordinary Differential Equations
  3. 1.02: PDE’s
  4. 02: Classication of Partial Dierential Equations
  5. 2.0E: 2.E: Classification of Partial Differential Equations (Exercises)
  6. 2.01: Examples of PDE
  7. 2.02: Second Order PDE
  8. 2.03: More than 2D
  9. 03: Boundary and Initial Conditions
  10. 3.01: Introduction to Boundary and Initial Conditions
  11. 3.02: Explicit Boundary Conditions
  12. 3.03: Implicit Boundary Conditions
  13. 3.04: More Realistic Examples of Boundary and Initial Conditions
  14. 04: Fourier Series
  15. 4.01: Taylor Series
  16. 4.02: Introduction to Fourier Series
  17. 4.03: Periodic Functions
  18. 4.04: Orthogonality and Normalization
  19. 4.05: When is it a Fourier Series
  20. 4.06: Fourier series for even and odd functions
  21. 4.07: Convergence of Fourier series
  22. 05: Separation of Variables on Rectangular Domains
  23. 5.01: Cookbook
  24. 5.02: Parabolic Equation
  25. 5.03: Hyperbolic Equation
  26. 5.04: Laplace’s Equation
  27. Boundary Conditions
  28. 5.06: Inhomogeneous Equations
  29. 06: DAlemberts Solution to the Wave Equation
  30. 6.01: Background to D’Alembert’s Solution
  31. 6.02: New Variables
  32. 6.03: Examples
  33. 07: Polar and Spherical Coordinate Systems
  34. 7.01: Polar Coordinates
  35. 7.02: Spherical Coordinates
  36. 08: Separation of Variables in Polar Coordinates
  37. 8.01: Example
  38. 8.02: Three cases for λ
  39. 8.03: Putting it all together
  40. 09: Series Solutions of ODEs (Frobenius Method)
  41. 9.01: Frobenius’ Method
  42. 9.02: Singular Points
  43. 9.03: Special Cases
  44. 10: Bessel Functions and Two-Dimensional Problems
  45. 10.01: Temperature on a Disk
  46. 10.02: Bessel’s Equation
  47. 10.03: Gamma Function
  48. 10.04: Bessel Functions of General Order
  49. 10.05: Properties of Bessel functions
  50. 10.06: Sturm-Liouville theory
  51. 10.07: Our Initial Problem and Bessel Functions
  52. 10.08: Fourier-Bessel Series
  53. 10.09: Back to our Initial Problem
  54. 11: Separation of Variables in Three Dimensions
  55. 11.01: Modelling the Eye
  56. 11.02: Properties of Legendre Polynomials
  57. 11.03: Fourier-Legendre Series
  58. 11.04: Modelling the eye–revisited