Partial differential equations Function depends on two or more independent variables This is a very...
-
date post
22-Dec-2015 -
Category
Documents
-
view
220 -
download
2
Transcript of Partial differential equations Function depends on two or more independent variables This is a very...
![Page 1: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/1.jpg)
Partial differential equations
Function depends on two or more independent variables
0
y
u
x
u
This is a very simple one - there are many more complicated ones
1252
2
2
3
yu
x
ux
yx
u
![Page 2: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/2.jpg)
Order of the PDE is given by its highest derivative
xyx
u
y
u
yx
u4
2
2
2
2
22
4
is 2nd order
xyx
u
yx
u
y
u42
2
22
2
2
is 4th order
![Page 3: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/3.jpg)
Linear PDE is linear in dependent variable, and all coefficients depend on independent variables only
222
2
3
3
4 yxx
uy
y
ux
Nonlinear PDEs violate these rules
14 222
2
2
2
3
3
uyxx
uy
y
uxu
![Page 4: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/4.jpg)
PDE that often appears in engineering is second order, linear PDE
General form:
02
22
2
2
Dy
uC
yx
uB
x
uA
A, B, C are functions of x and y
D is function of x,y,u and andx
u
y
u
![Page 5: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/5.jpg)
Can use values of coefficients A,B,C to characterize the PDE
B2-4AC Type Example
<0 Elliptic Laplaceequation
=0 Parabolic Heatconduction
>0 Hyperbolic Waveequation
02
2
2
2
y
T
x
T
2
2
x
Tk
t
T
01
2
2
22
2
t
y
cx
y
![Page 6: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/6.jpg)
Why categorize?
Different methods to solve different types
Different types describe different engineering problems
• Elliptic - steady state
• Parabolic - propagation
• Hyperbolic - vibrations
![Page 7: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/7.jpg)
Analytic solutions - there aren’t many
Often can use analytic tools to get idea of behavior of a PDE, especially as parameters are changed
Important for limiting cases
![Page 8: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/8.jpg)
Elliptic PDEs
Steady-state two-dimensional heat conduction equation is prototypical elliptic PDE
02
2
2
2
y
T
x
T
This is the Laplace equation
yxfy
T
x
T,
2
2
2
2
This is the Poisson equation
![Page 9: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/9.jpg)
Think of a small box
qx, in
qy, out
qx, out
qy, in
At steady state, net change in heat is 0, so
0,,,, outyinyoutxinx qqqq
![Page 10: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/10.jpg)
Shrink to differential size
0
y
q
x
q
Fourier’s law of heat conduction
a
TCkq pa
![Page 11: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/11.jpg)
Substitute
02
2
2
2
y
T
x
T
0
y
TCk
yx
TCk
x pp
We will solve with finite differences
![Page 12: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/12.jpg)
Discretize PDE so that we have a mesh of grid points with boundary conditions
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Index for x is i
Index for y is j
![Page 13: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/13.jpg)
Use finite differences for the derivatives
2
,1,,1
2
2 2
x
TTT
x
T jijiji
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Ti-1,j
Ti,j
Ti+1,j
![Page 14: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/14.jpg)
Now the y derivative
2
1,,1,
2
2 2
y
TTT
y
T jijiji
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Ti,j+1
Ti,j
Ti,j-1
![Page 15: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/15.jpg)
Substitute these expressions back into original elliptic PDE
022
2
1,,1,
2
,1,,1
y
TTT
x
TTT jijijijijiji
Assume x=y. Can rearrange to get
04 ,1,1,,1,1 jijijijiji TTTTT
True for all interior points
![Page 16: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/16.jpg)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Need to define values on ALL boundaries - Dirichlet boundary condition
(Neumann BC fix flux at boundary)
![Page 17: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/17.jpg)
Each interior point has an equation - for 9 x 9 interior points - 81 equations
Adds up quickly
Example: 4 x 4 grid - 2 x 2 interior points
0
1
2
3
0 1 2 3
75
20
10
02550
60 45 30
75
75
75
![Page 18: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/18.jpg)
1354
046075
04
1,12,11,2
1,12,11,2
1,10,12,11,01,2
TTT
TTT
TTTTT
0
1
2
3
0 1 2 3
75
20
10
02550
60 45 30
75
75
75
Let i=1, j=1
![Page 19: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/19.jpg)
Fill in the matrix
35
65
125
135
411
141
141
114
2,2
1,2
2,1
1,1
T
T
T
T
Generally, we get a sparse matrix (big, too)
Technique most often used is Gauss-Seidel or some variation of it - matrix is always diagonally dominant - also called Liebmann’s rule
![Page 20: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/20.jpg)
Apply these steps iteratively until T converges
41,1,,1,1
,
jijijijiji
TTTTT
Rewrite equation in Gauss-Seidel form
04 ,1,1,,1,1 jijijijiji TTTTT
Use overrelaxation (if desired)
oldji
newji
newji TTT ,,, 1
![Page 21: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/21.jpg)
Solving our example - the four equations are
0
1
2
3
0 1 2 3
75
20
10
02550
60 45 30
75
75
75
0654
01354
2,12,21,1
1,11,22,1
TTT
TTT
0354
01254
2,21,22,1
1,22,21,1
TTT
TTT
![Page 22: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/22.jpg)
Rewrite them in Gauss Seidel form
4
354
1254
654
135
1,22,12,2
2,21,11,2
2,21,12,1
1,22,11,1
TTT
TTT
TTT
TTT
and assume initial values for T
75
75
75
75
2,2
1,2
2,1
1,1
T
T
T
T
![Page 23: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/23.jpg)
Run without overrelaxation
iterationstart 1 2 3 4 5 6 7 8 9T11 75 71.25 64.38 60.31 58.59 57.58 57.15 56.89 56.79 56.72T12 75 53.75 45.63 42.19 40.16 39.3 38.79 38.57 38.45 38.39T21 75 68.75 60.63 57.19 55.16 54.3 53.79 53.57 53.45 53.39T22 75 46.25 39.38 35.31 33.59 32.58 32.15 31.89 31.79 31.72ea1 0.053 0.107 0.067 0.029 0.018 0.008 0.004 0.002 0.001ea2 0.395 0.178 0.081 0.051 0.022 0.013 0.006 0.003 0.001ea3 0.091 0.134 0.06 0.037 0.016 0.009 0.004 0.002 0.001ea4 0.622 0.175 0.115 0.051 0.031 0.013 0.008 0.003 0.002
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10 12
T11
T12
T21
T22
![Page 24: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/24.jpg)
0
1
2
3
0 1 2 3
75
20
10
02550
60 45 30
75
75
75
End result
56.72 38.39
31.7953.39
![Page 25: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/25.jpg)
What about derivative (flux) boundary conditions
I.E. if we insulate one side of the plate, is 0 there
x
T
Create an imaginary point outside boundary
T0,j+1
T-1,j T1,jT0,j
T0,j-1
![Page 26: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/26.jpg)
Equation becomes
04 ,0,1,11,01,0 jjjjj TTTTT
Now consider finite difference for derivative at 0
0,1,1
,1,1
0
2
2
t
TxTT
x
TT
t
T
jj
jj
![Page 27: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/27.jpg)
Substitute
042 ,00
,1,11,01,0
jjjjj Tt
TxTTTT
Derivative BC now included in equation
![Page 28: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/28.jpg)
Irregular domains (funny shapes)
What do you do with a domain like?
![Page 29: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/29.jpg)
Your book uses , to scale the x, y
Different x, y
0
1
2
3
4
5
0 1 2 3 4 5
1 x
2 x
1 y
2 y
![Page 30: Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d795503460f94a5c8b1/html5/thumbnails/30.jpg)
Can develop equations for edge points
02
2
212
,1,
211
,1,
2
212
,,1
211
,,
2
jijijiji
jijijijii
TTTT
y
TTTT
x
Now use a Gauss-Seidel or other matrix approach