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Partial Differential Equations

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Partial Differential Equations

Partial differential equations are differential equations that contains unknown multivariable functions and their partial derivatives

Differential EquationsMathematics
Author
Erich Miersemann
Publisher
Universität Leipzig
Source
LibreTexts
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 67 sections
  1. 1: Introduction
  2. 1.0: Introduction
  3. 1.1: Examples
  4. 1.2: Ordinary Differential Equations
  5. 1.3: Partial Differential Equations
  6. 1.E: Introduction (Exercises)
  7. 2: Equations of First Order
  8. 2.0: Prelude to First Order Equations
  9. 2.1: Linear Equations
  10. 2.2.0: Quasilinear Equations
  11. 2.2.1: A Linearization Method
  12. 2.2.2: Initial Value Problem of Cauchy
  13. 2.3.0: Nonlinear Equations in Two Variables
  14. 2.3.1: Initial Value Problem of Cauchy
  15. 2.4: Nonlinear Equations in Rn
  16. 2.5: Hamilton-Jacobi Theory
  17. 2.E: Equations of First Order (Exercises)
  18. 3: Classification
  19. 3.1.0: Linear Equations of Second Order
  20. 3.1.1: Normal Form in Two Variables
  21. 3.2.0: Quasilinear Equations of Second Order
  22. 3.2.1: Quasilinear Elliptic Equations
  23. 3.3.0: Systems of First Order
  24. 3.3.1: Examples
  25. 3.4.0: Systems of Second Order
  26. 3.4.1: Examples
  27. 3.5.0: Theorem of Cauchy-Kovalevskaya
  28. 3.5.1 Appendix: Real Analytic Functions
  29. 3.E: Classification (Exercises)
  30. 4: Hyperbolic Equations
  31. 4.1: One-Dimensional Wave Equation
  32. 4.2.0: Higher Dimensions
  33. 4.2.1: Case n=3
  34. 4.2.2: Case n=2
  35. 4.3: Inhomogeneous Equations
  36. 4.4: A Method of Riemann
  37. 4.5.0: Initial-Boundary Value Problems
  38. 4.5.1: Oscillation of a String
  39. 4.5.2: Oscillation of a Membrane
  40. 4.5.3: Inhomogeneous Wave Equations
  41. 4.E: 4: Hyperbolic Equations (Exercises)
  42. 5: Fourier Transform
  43. 5.1.0: Definition and Properties
  44. 5.1.1: Pseudodifferential Operators
  45. 5.E: Fourier Transform (Exercises)
  46. 6: Parabolic Equations
  47. 6.1: Poisson's Formula
  48. 6.2: Inhomogeneous Heat Equation
  49. 6.3: Maximum Principle
  50. 6.4.0: Initial-Boundary Value Problems
  51. 6.4.1: Fourier's Method
  52. 6.4.2: Uniqueness
  53. 6.5: Black-Scholes Equation
  54. 6.E: Parabolic Equations (Exercises)
  55. 7: Elliptic Equations of Second Order
  56. 7.1: Fundamental Solution
  57. 7.2.0: Representation Formula
  58. 7.2.1: Conclusions from the Representation Formula
  59. 7.3.1: Boundary Value Problems: Dirichlet Problem
  60. 7.3.2: Boundary Value Problems: Neumann Problem
  61. 7.3.3: Boundary Value Problems: Mixed Boundary Value Problem
  62. 7.4.0: Green's Function for Delta
  63. 7.4.1: Green's Function for a Ball
  64. 7.4.2: Green's Function and Conformal Mapping
  65. 7.5: Inhomogeneous Equation
  66. 7.E: Elliptic Equations of Second Order (Exercises)
  67. Bibliography