Partial Differential Equations
-
Upload
ragasree02 -
Category
Documents
-
view
154 -
download
4
Transcript of Partial Differential Equations
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Topic 9: Partial Differential Equations
•An equation involving partial derivatives of an unknown function of two or more independent variables is called a partial differential equation – PDE.
•An example of a PDE is
•The general form of a linear second order PDE is written as:
0
12
2
22
2
2
2
2
2
2
Dy
uC
yx
uB
x
uA
uy
uxy
x
u
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•Here, A, B, and C are functions of the independent variables x and y only.
•The part D is a function of x, y, u, u/x, and u/y.
•Depending on the values of A, B, and C, the 2–PDE is classified into one of three categories.
B2 – 4AC Category
< 0 Elliptic
= 0 Parabolic
> 0 Hyperbolic
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Finite Difference: Elliptic Equations
•Elliptic equations in engineering are typically used to characterize steady-state, boundary value problems.
THE LAPLACE EQUATION
•Elliptic equations are typically used to characterize steady–state systems.
•We will illustrate the solution of elliptic PDEs in the context of an example.
•This example is called the headed plate.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•The plate has a thickness of z.
•The plate is insulated everywhere except its edges.
•The edges are fixed at prescribed temperatures.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•At the steady state, over time period t
heat flow–in = heat flow–out
q(x) y z t + q(y) x z t = q(x + x) y z t
+ q(y + y) x z t
where q is the heat flux.
q(y)
q(x) q(x + x)
q(y + y)
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•After simplification, we arrive to the PDE
•We need a PDE in terms of temperature.
•The relation between heat flux q and temperature T is given by
•This equation is called Fourier’s law of heat conduction.
i
TCq
y
q
x
q
i
0
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•Substituting the value of q in the PDE, we get
•It is called the Laplace equation.
•The Laplace equation is an elliptic equation because
B2 – 4 A C = –4 < 0
02
2
2
2
y
T
x
T
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
SOLUTION TECHNIQUES
•The solution method works by substituting the partial derivatives by the finite difference formulas.
The Laplace Difference Equation
•The centered difference formulas for the second derivatives are
2
1,,1,
2
2
2
,1,,1
2
2
2
and
2
y
TTT
y
T
x
TTT
x
T
jijiji
jijiji
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•After the substitution, the differential equation becomes
•Assume x = y and simplify:
•This relationship is called the Laplace difference equation.
04
022
,1,1,,1,1
2
1,,1,
2
,1,,1
jijijijiji
jijijijijiji
TTTTT
y
TTT
x
TTT
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
41,1,,1,1
,
jijijijiji
TTTTT
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•Consider the simplest case where the boundary temperatures along the edges are set to fixed values:
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•At note (1, 1), the equation is written as
•Note T0,1 = 75 and T1,0 = 0. The equation becomes
•Similarly, the relationship can be written for all interior points and the result is a system of linear equations.
754
04
1,22,11,1
1,10,12,11,01,2
TTT
TTTTT
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
1504
1004
1754
504
04
754
504
04
754
332332
33231322
231312
33322231
2332221221
13221211
323121
22132111
122111
TTT
TTTT
TTT
TTTT
TTTTT
TTTT
TTT
TTTT
TTT
•The system has 9 equations and 9 unknowns:
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
The Liebmann Method
•Most numerical solutions of Laplace equation involve systems that are very large.
•For larger size grids, a significant number of terms will be zero.
•For this reason, approximation methods provide a viable approach for obtaining solutions.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•Steps of the Liebmann method:
1. Set Toldi,j = 0 for the interior points.
2. Compute Tnewi,j:
3. Apply the over-relaxation formula
for 1 ≤ ≤ 2.
4. Go to (2) if .ε%100
)λ1(λ
4
new,
old,
new,
old,
new,
new,
old1,
old1,
old,1
old,1new
,
sji
jiji
jijiji
jijijijiji
T
TT
TTT
TTTTT
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Example
See the example and the solution in the book.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Finite Difference: Parabolic Equations
• Parabolic equations in engineering are typically used to characterize time variable problems.
THE HEAT CONDUCTION EQUATION
• Conservation of energy can be used to develop an transit-state energy balance for the differential element in a long, thin insulated rod.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•The heat conduction equation for the insulated rod is
•The temperature depends on location and time.
t
T
x
Tk
2
2
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
EXPLICIT METHODS
•We can substitute the partial derivatives by their finite difference approximations:
•Substituting in the heat conduction equation:
t
TT
x
TTTk
t
TT
t
T
x
TTT
x
T
li
li
li
li
li
li
li
li
li
li
1
211
1
211
2
2
2
and
2
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
•Simplifying:
2
111
where
2
x
tk
TTTTT li
li
li
li
li
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Example
See the example and the solution in the book.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Convergence and Stability
•Convergence means that as x and t approach zero, the results of the finite difference method approach the true solution.•Stability means that errors at any stage of the computation are not amplified but are attenuated as the computation progresses.•The explicit method is both convergent and stable if ≤ ½, or
k
xt
2
2
1
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Solution to the previous example for = 0.735.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
A SIMPLE IMPLICIT METHOD
• The difference between the explicit and implicit approximations is illustrated in this figure.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
li
li
li
li
li
li
li
li
li
li
li
li
TTTT
t
TT
x
TTTk
x
TTT
x
T
11
111
1
2
11
111
2
11
111
2
2
λλ21λ
2
2
• In the implicit method, the derivative is approximated by:
• The heat conduction equation becomes
• Simplifying:
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
• The solution will involve the solution of a system of linear equations.
Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM
Example
See the example and the solution in the book.