04/12/09
EP 471 – Homework #6: Elliptic PDEs
Each problem is equally weighted:
(1) Consider the following elliptic partial differential equation:
2
2
2
2
2
2
=
∂
∂
−
∂
∂
+
∂
∂
x
u
y
u
x
u
,
Solve this problem over the unit square,
1
0
≤
≤
x
,
1
0
≤
≤
y
, subject to
u
= 0 on all
boundaries.
Approximate all derivatives with centrallydifferenced expressions.
You may use
either a direct solution (simultaneous system of equations) or an iterative solution.
Find the peak
value of
u
within the domain.
(2) In Exercises 16 and 17, we looked at solutions to Laplace’s Equation in a rectangular plate
over the domain
m
5
.
0
0
;
m
1
0
≤
≤
≤
≤
y
x
with the temperature equal to zero
°
C along three
edges and a nonzero temperature of 100
°
C on one edge.
Suppose we modify this problem in
two ways: we’ll incorporate a spatiallydependent heat source to turn Laplace’s Equation into
Poisson’s Equation, and we’ll incorporate convective boundary conditions along the three edges
that were previously specified as zero temperature boundaries.
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 Spring '08
 Witt
 Heat, Heat Transfer, Laplace, Partial differential equation, plate, heat source

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