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## SE301:Numerical Methods Topic 9 Partial Differential Equations

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### 2. Lect 27: Partial Differential Equations. Partial Differential ... 2. Crank-Nicolson Method: involves solution of Tridiagonal system of equations, stable. ... – PowerPoint PPT presentation

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Title: SE301:Numerical Methods Topic 9 Partial Differential Equations

1
SE301Numerical MethodsTopic 9Partial
Differential Equations
• Dr. Samir Al-Amer
• Term 071

2
Lect 27 Partial Differential Equations
• Partial Differential Equations (PDE)
• What is a PDE
• Examples of Important PDE
• Classification of PDE

3
Partial Differential Equations
813
A partial differential equation (PDE) is an
equation that involves an unknown function and
its partial derivatives.
4
Notation
5
Linear PDEClassification
813
6
Representing the solution of PDE(two independent
variables)
• Three main ways to represent the solution

T5.2
t1
T3.5
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
7
Heat Equation
Different curve is used for each value of t
ice
ice
Temperature at different x at t0
Temperature
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 th
8
Heat Equation
Temperature T(x,t)
Time t
ice
ice
x
t1
Position x
x1
9
Linear Second Order PDEClassification
814
10
Linear Second Order PDEExamples ( Classification)
11
Classification of PDE
• Linear Second order PDE are important set 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

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

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) .
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.
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
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.
17
Biharmonic Equation
Used in the study of elastic stress.
18
Boundary conditions for PDE
• 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
19
The solution Methods for PDE
• 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 PDE.
• The methods discussed here are based on finite
difference technique

20
Elliptic Equations
• Elliptic Equations
• Laplace Equation
• Solution

21
Elliptic Equations
22
Laplace Equation
• Laplace equation appears is several
engineering problems such as
• Studying the steady state distribution of heat in
a body
• Studying the steady state distribution of
electrical in a body

23
Laplace Equation
• Temperature is function of the position (x and y)
• When no heat source is available ?f(x,y)0

24
Solution Technique
• A grid is used to divide 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.

25
Solution Technique
26
Solution Technique
27
Solution Technique
28
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
constant temperature 100,50, 0 and 75 degrees.

100
50
75
The sheet is divided by 5X5 grids
0
29
Example
Known To be determined
30
First equation
Known To be determined
31
Example
32
Another Equation
Known To be determined
33
Solution The rest of the equations
34
Convergence and stability of solution
• Convergence
• The solutions converge means that the solution
obtained using finite difference method
approaches the true solution as the steps
approaches zero.
• Stability
• An algorithm is stable if the errors at each
stage of the computation are not magnified as the
computation progresses.

35
Parabolic Equations
• Parabolic Equations
• Heat Conduction Equation
• Explicit Method
• Implicit Method
• Cranks Nicolson Method

36
Parabolic Equations
37
Parabolic Problems
ice
ice
x
38
First order Partial derivative Finite Difference
Forward difference Method
Backward difference Method
Central difference Method
39
Finite Difference Methods
40
Finite Difference MethodsNew Notation
Superscript for t-axis And Subscript for
x-axis Til-1T(x,t-?t)
41
Solution of the PDE
t
x
42
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 solution of
Tridiagonal system of equations, stable.
43
Explicit Method
44
Explicit MethodHow do we compute
u(x,tk)
u(x-h,t) u(x,t)
u(xh,t)
45
Explicit MethodHow do we compute
46
Explicit Method
47
Crank-Nicolson Method
48
Explicit MethodHow do we compute
u(x-h,t) u(x,t)
u(xh,t)
u(x,t - k)
49
Crank-Nicolson Method
50
Crank-Nicolson Method
51
Outlines
• Examples
• Explicit method to solve Parabolic PDE
• Cranks-Nicholson Method

52
Heat Equation
ice
ice
x
53
Example 1
54
Example 1 (cont.)
55
Example 1
0
0
t1.0
0
0
t0.75
t0.5
0
0
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
56
Example 1
0
0
t1.0
0
0
t0.75
t0.5
0
0
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
57
Example 1
0
0
t1.0
0
0
t0.75
t0.5
0
0
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
58
Remarks on Example 1
59
Example 1
0
0
t0.10
0
0
t0.075
t0.05
0
0
t0.025
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
60
Example 1
0
0
t0.10
0
0
t0.075
t0.05
0
0
t0.025
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
61
Example 1
0
0
t0.10
0
0
t0.075
t0.05
0
0
t0.025
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
62
Example 2
63
Example 2 Crank-Nicolson Method
64
Example 2Crank-Nicolson Method
65
Example 2
0
0
t1.0
0
0
t0.75
t0.5
0
0
u1 u2 u3
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
66
Example 2
0
0
t1.0
0
0
t0.75
t0.5
0
0
u1 u2 u3
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
67
Example 2
0
0
t1.0
0
0
t0.75
t0.5
0
0
u1 u2 u3
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
68
Example 2Crank-Nicolson Method
69
Example 2Second Row
0
0
t1.0
0
0
t0.75
u1 u2 u3
t0.5
0
0
t0.25
0
0
0.2115 0.2991 0.2115
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
70
Example 2
The process is continued until the values of
u(x,t) on the desired grid are computed.
71
Remarks
• The explicit metod
• one need to select small k to ensure stability
• Computation per point is very simple but many
points are needed.
• Cranks Nicolson
• Requires solution of Tridiagonal system
• Stable (larger k can be used).