1. SE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures 37-39
KFUPM
Read &
CISE301_Topic9
KFUPM

2. Lecture 37 Partial Differential Equations
Partial Differential Equations (PDEs).
What is a PDE?
Examples of Important PDEs.
Classification of PDEs.
CISE301_Topic9
KFUPM

3. Partial Differential Equations
A partial differential equation (PDE) is an equation that involves an unknown function and its partial derivatives.
CISE301_Topic9
KFUPM

4. Notation
CISE301_Topic9
KFUPM

5. Linear PDE Classification
CISE301_Topic9
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
CISE301_Topic9
KFUPM

7. Different curve is used for each value of t
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
CISE301_Topic9
KFUPM

8. Heat Equation Time t ice ice x t1 x1 Temperature T(x,t) Position x
CISE301_Topic9
KFUPM

9. Linear Second Order PDEs Classification
CISE301_Topic9
KFUPM

10. Linear Second Order PDE Examples (Classification)
CISE301_Topic9
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.
CISE301_Topic9
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
CISE301_Topic9
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) .
CISE301_Topic9
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.
CISE301_Topic9
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.
CISE301_Topic9
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.
CISE301_Topic9
KFUPM

17. Biharmonic Equation Used in the study of elastic stress.
CISE301_Topic9
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
CISE301_Topic9
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.
CISE301_Topic9
KFUPM

20. Lecture 38 Parabolic Equations
Heat Conduction Equation
Explicit Method
Implicit Method
Cranks Nicolson Method
CISE301_Topic9
KFUPM

21. Parabolic Equations
CISE301_Topic9
KFUPM

22. Parabolic Problems
ice
ice x
CISE301_Topic9
KFUPM

23. First Order Partial Derivative Finite Difference
Forward
Difference
Method
Backward
Difference
Method
Central
Difference
Method
CISE301_Topic9
KFUPM

24. Finite Difference Methods
CISE301_Topic9
KFUPM

25. Finite Difference Methods New Notation
Superscript for t-axis
and
Subscript for x-axis
Til-1=Ti,j-1=T(x,t-∆t)
CISE301_Topic9
KFUPM

26. Solution of the PDEs t x
CISE301_Topic9
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.
CISE301_Topic9
KFUPM

28. Explicit Method
CISE301_Topic9
KFUPM

29. Explicit Method How Do We Compute?
u(x,t+k)
u(x-h,t) u(x,t) u(x+h,t)
CISE301_Topic9
KFUPM

30. Explicit Method How Do We Compute?
CISE301_Topic9
KFUPM

31. Explicit Method
CISE301_Topic9
KFUPM

32. Crank-Nicolson Method
CISE301_Topic9
KFUPM

33. Explicit Method How Do We Compute?
u(x-h,t) u(x,t) u(x+h,t)
u(x,t - k)
CISE301_Topic9
KFUPM

34. Crank-Nicolson Method
CISE301_Topic9
KFUPM

35. Crank-Nicolson Method
CISE301_Topic9
KFUPM

36. Examples Explicit method to solve Parabolic PDEs.
Cranks-Nicholson Method.
CISE301_Topic9
KFUPM

37. Heat Equation
ice
ice x
CISE301_Topic9
KFUPM

38. Example 1
CISE301_Topic9
KFUPM

39. Example 1 (Cont.)
CISE301_Topic9
KFUPM

40. Example 1 t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π)
t=1.0
t=0.75
t=0.5
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

41. Example 1 t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π)
t=1.0
t=0.75
t=0.5
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

42. Example 1 t=1.0 t=0.75 t=0.5 t=0.25 t=0 Sin(0.25π) Sin(0. 5π)
t=1.0
t=0.75
t=0.5
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

43. Remarks on Example 1
CISE301_Topic9
KFUPM

44. Example 1 t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π)
t=0.10
t=0.075
t=0.05
t=0.025
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

45. Example 1 t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π)
t=0.10
t=0.075
t=0.05
t=0.025
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

46. Example 1 t=0.10 t=0.075 t=0.05 t=0.025 t=0 Sin(0.25π) Sin(0. 5π)
t=0.10
t=0.075
t=0.05
t=0.025
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

47. Example 2
CISE301_Topic9
KFUPM

48. Example 2 Crank-Nicolson Method
CISE301_Topic9
KFUPM

49. Example 2 t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π)
t=1.0
t=0.75
t=0.5
u u u3
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

50. Example 2 t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π)
t=1.0
t=0.75
t=0.5
u u u3
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

51. Example 2 t=1.0 t=0.75 t=0.5 u1 u2 u3 t=0.25 t=0 Sin(0.25π) Sin(0. 5π)
t=1.0
t=0.75
t=0.5
u u u3
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

52. Example 2 Crank-Nicolson Method
CISE301_Topic9
KFUPM

53. Example 2 Second Row t=1.0 t=0.75 u1 u2 u3 t=0.5 t=0.25
t=1.0
t=0.75
u u u3
t=0.5
t=0.25
t=0
Sin(0.25π)
Sin(0. 5π)
Sin(0.75π)
x=0.0
x=0.25
x=0.5
x=0.75
x=1.0
CISE301_Topic9
KFUPM

54. Example 2
The process is continued until the values of u(x,t) on the desired grid are computed.
CISE301_Topic9
KFUPM

55. Remarks The Explicit Method:
One needs to select small k to ensure stability.
Computation per point is very simple but many points are needed.
Cranks Nicolson:
Requires the solution of a Tridiagonal system.
Stable (Larger k can be used).
CISE301_Topic9
KFUPM

56. Lecture 39 Elliptic Equations
Laplace Equation
Solution
CISE301_Topic9
KFUPM

57. Elliptic Equations
CISE301_Topic9
KFUPM

58. Laplace Equation
Laplace equation appears in several engineering problems such as:
Studying the steady state distribution of heat in a body.
Studying the steady state distribution of electrical charge in a body.
CISE301_Topic9
KFUPM

59. Laplace Equation Temperature is a function of the position (x and y)
When no heat source is available f(x,y)=0
CISE301_Topic9
KFUPM

60. Solution Technique A grid is used to divide the region of interest.
Since the PDE is satisfied at each point in the area, it must be satisfied at each point of the grid.
A finite difference approximation is obtained at each grid point.
CISE301_Topic9
KFUPM

61. Solution Technique
CISE301_Topic9
KFUPM

62. Solution Technique
CISE301_Topic9
KFUPM

63. Solution Technique
CISE301_Topic9
KFUPM

64. Example
It is required to determine the steady state temperature at all points of a heated sheet of metal. The edges of the sheet are kept at a constant temperature: 100, 50, 0, and 75 degrees.
100 75 50
The sheet is divided to 5X5 grids.
CISE301_Topic9
KFUPM

65. Example
Known
To be determined
CISE301_Topic9
KFUPM

66. First Equation
Known
To be determined
CISE301_Topic9
KFUPM

67. Another Equation
Known
To be determined
CISE301_Topic9
KFUPM

68. Solution The Rest of the Equations
CISE301_Topic9
KFUPM

69. Convergence and Stability of the Solution
The solutions converge means that the solution obtained using the finite difference method approaches the true solution as the steps approach zero.
Stability:
An algorithm is stable if the errors at each stage of the computation are not magnified as the computation progresses.
CISE301_Topic9
KFUPM