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SE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures 37-39

SE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures 37-39. KFUPM Read 29.1-29.2 & 30.1-30.4. L ecture 37 Partial Differential Equations. Partial Differential Equations (PDEs). What is a PDE? Examples of Important PDEs. Classification of PDEs.

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SE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures 37-39

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  1. SE301: Numerical MethodsTopic 9Partial Differential Equations (PDEs)Lectures 37-39 KFUPM Read 29.1-29.2 & 30.1-30.4 KFUPM

  2. Lecture 37Partial Differential Equations Partial Differential Equations (PDEs). What is a PDE? Examples of Important PDEs. Classification of PDEs. KFUPM

  3. Partial Differential Equations A partial differential equation (PDE) is an equation that involves an unknown function and its partial derivatives. KFUPM

  4. Notation KFUPM

  5. Linear PDEClassification KFUPM

  6. Representing the Solution of a PDE(Two Independent Variables) • Three main ways to represent the solution T=5.2 t1 T=3.5 x1 Different curves are used for different values of one of the independent variable Three dimensional plot of the function T(x,t) The axis represent the independent variables. The value of the function is displayed at grid points KFUPM

  7. Heat Equation Different curve is used for each value of t ice ice Temperature Temperature at different x at t=0 x Thin metal rod insulated everywhere except at the edges. At t =0 the rod is placed in ice Position x Temperature at different x at t=h KFUPM

  8. Heat Equation Temperature T(x,t) Time t ice ice x t1 x1 Position x KFUPM

  9. Linear Second Order PDEsClassification KFUPM

  10. Linear Second Order PDEExamples (Classification) KFUPM

  11. Classification of PDEs Linear Second order PDEs are important sets of equations that are used to model many systems in many different fields of science and engineering. Classification is important because: • Each category relates to specific engineering problems. • Different approaches are used to solve these categories. KFUPM

  12. Examples of PDEs PDEs are used to model many systems in many different fields of science and engineering. Important Examples: • Wave Equation • Heat Equation • Laplace Equation • Biharmonic Equation KFUPM

  13. Heat Equation The function u(x,y,z,t)is used to represent the temperature at time t in a physical body at a point with coordinates (x,y,z) . KFUPM

  14. Simpler Heat Equation x u(x,t)is used to represent the temperature at time t at the point x of the thin rod. KFUPM

  15. Wave Equation The function u(x,y,z,t) is used to represent the displacement at time t of a particle whose position at rest is (x,y,z) . Used to model movement of 3D elastic body. KFUPM

  16. Laplace Equation Used to describe the steady state distribution of heat in a body. Also used to describe the steady state distribution of electrical charge in a body. KFUPM

  17. Biharmonic Equation Used in the study of elastic stress. KFUPM

  18. Boundary Conditions for PDEs • To uniquely specify a solution to the PDE, a set of boundary conditions are needed. • Both regular and irregular boundaries are possible. t region of interest x 1 KFUPM

  19. The Solution Methods for PDEs • Analytic solutions are possible for simple and special (idealized) cases only. • To make use of the nature of the equations, different methods are used to solve different classes of PDEs. • The methods discussed here are based on the finite difference technique. KFUPM

  20. Lecture 38Parabolic Equations Parabolic Equations Heat Conduction Equation Explicit Method Implicit Method Cranks Nicolson Method KFUPM

  21. Parabolic Equations KFUPM

  22. Parabolic Problems ice ice x KFUPM

  23. First Order Partial Derivative Finite Difference Forward Difference Method Backward Difference Method Central Difference Method KFUPM

  24. Finite Difference Methods KFUPM

  25. Finite Difference MethodsNew Notation Superscript for t-axis and Subscript for x-axis Til-1=Ti,j-1=T(x,t-∆t) KFUPM

  26. Solution of the PDEs t x KFUPM

  27. Solution of the Heat Equation • Two solutions to the Parabolic Equation (Heat Equation) will be presented: • 1. Explicit Method: • Simple, Stability Problems. • 2. Crank-Nicolson Method: • Involves the solution of a Tridiagonal system of equations, Stable. KFUPM

  28. Explicit Method KFUPM

  29. Explicit MethodHow Do We Compute? u(x,t+k) u(x-h,t) u(x,t) u(x+h,t) KFUPM

  30. Explicit MethodHow Do We Compute? KFUPM

  31. Explicit Method KFUPM

  32. Crank-Nicolson Method KFUPM

  33. Explicit MethodHow Do We Compute? u(x-h,t) u(x,t) u(x+h,t) u(x,t - k) KFUPM

  34. Crank-Nicolson Method KFUPM

  35. Crank-Nicolson Method KFUPM

  36. Examples • Explicit method to solve Parabolic PDEs. • Cranks-Nicholson Method. KFUPM

  37. Heat Equation ice ice x KFUPM

  38. Example 1 KFUPM

  39. Example 1 (Cont.) KFUPM

  40. Example 1 0 0 t=1.0 0 0 t=0.75 t=0.5 0 0 t=0.25 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  41. Example 1 0 0 t=1.0 0 0 t=0.75 t=0.5 0 0 t=0.25 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  42. Example 1 0 0 t=1.0 0 0 t=0.75 t=0.5 0 0 t=0.25 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  43. Remarks on Example 1 KFUPM

  44. Example 1 0 0 t=0.10 0 0 t=0.075 t=0.05 0 0 t=0.025 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  45. Example 1 0 0 t=0.10 0 0 t=0.075 t=0.05 0 0 t=0.025 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  46. Example 1 0 0 t=0.10 0 0 t=0.075 t=0.05 0 0 t=0.025 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  47. Example 2 KFUPM

  48. Example 2Crank-Nicolson Method KFUPM

  49. Example 2 0 0 t=1.0 0 0 t=0.75 t=0.5 0 0 u1 u2 u3 t=0.25 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

  50. Example 2 0 0 t=1.0 0 0 t=0.75 t=0.5 0 0 u1 u2 u3 t=0.25 0 0 0 0 t=0 Sin(0.25π) Sin(0. 5π) Sin(0.75π) x=0.0 x=1.0 x=0.25 x=0.5 x=0.75 KFUPM

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