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Partial Differential Equations

Partial Differential Equations. Chapter 12 Burden and Faires. General 2 nd Order Form. Discriminant. Classification. Conic Sections. Classification. Elliptical PDE. Elliptical PDE. FDM Molecule. ODE Second Order Formula. x i+1. x i. x i-1.

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Partial Differential Equations

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  1. Partial Differential Equations Chapter 12 Burden and Faires PDE's

  2. General 2nd Order Form PDE's

  3. Discriminant PDE's

  4. Classification PDE's

  5. Conic Sections PDE's

  6. Classification PDE's

  7. Elliptical PDE PDE's

  8. Elliptical PDE PDE's

  9. FDM Molecule PDE's

  10. ODE Second Order Formula xi+1 xi xi-1 PDE's

  11. ODE Second Order Central Difference Formulas PDE's

  12. ODE Second Order Central Difference Formulas PDE's

  13. PDE Second Order Central Difference Formulas PDE's

  14. PDE Second Order Central Difference Formulas PDE's

  15. Example Problem Consider a square plate 0.5 m on a side. Two sides are held at 0o and two sides vary from 0 to 100o Find the steady state temperature. PDE's

  16. Boundary Conditions PDE's

  17. Mesh 4 3 2 1 j = 0 i = 0 1 2 3 4 PDE's

  18. Mesh y4 y3 y2 y1 y0 x0 x1 x2 x3 x4 PDE's

  19. Mesh y4 y3 y2 y1 y0 x0 x1 x2 x3 x4 PDE's

  20. Notation PDE's

  21. PDE Second Order Central Difference Formulas PDE's

  22. PDE Second Order Central Difference Formulas PDE's

  23. Equations: PDE's

  24. PDE's

  25. Solution of Equations • For smaller systems use a direct method. • Gaussian Elimination • Matrix Factorization • For larger systems use an iterative method. • Gauss-Seidel • Successive Over Relaxation PDE's

  26. Symmetric Block Tridiagonal Matrix Form PDE's

  27. Symmetric Block Tridiagonal Matrix Form PDE's

  28. Theorem PDE's

  29. Matrix Form PDE's

  30. Matrix Form PDE's

  31. Poisson’s Equation Problem PDE's

  32. PDE Second Order Central Difference Formulas PDE's

  33. PDE Second Order Central Difference Formulas PDE's

  34. Alternative Notation N W E S PDE's

  35. Other Topics • Alternative Solution Techniques. • Derivative B.C. • Nonuniform Boundaries. • Alternative Coordinate systems. PDE's

  36. Alternative Solution Procedure PDE's

  37. Alternating Direct Implicit Method PDE's

  38. Alternating Direct Implicit Method PDE's

  39. Example 12 0 12 PDE's

  40. Example PDE's

  41. Equations PDE's

  42. Gauss Elimination PDE's

  43. Gauss-Seidel PDE's

  44. PDE's

  45. Alternating Direct Implicit Method- Rows PDE's

  46. Alternating Direct Implicit Method- Columns PDE's

  47. Derivative B.C. • Function value specified on boundary are called Dirichlet B.C. • Derivative values specified on boundary are called Neumann Boundary Conditions PDE's

  48. Example: PDE's

  49. Example PDE's

  50. Example PDE's

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