PROBLEM 4.1
K
NOWN:
Method of separation of variables for twodimensional, steadystate conduction.
FIND:
Show that negative or zero values of
λ
2
, the separation constant, result in solutions which
annot satisfy the boundary conditions.
c
SCHEMATIC:
A
SSUMPTIONS:
(1) Twodimensional, steadystate conduction, (2) Constant properties.
ANALYSIS:
From Section 4.2, identification of the separation constant
λ
2
leads to the two ordinary
differential equations, 4.6 and 4.7, having the forms
2
2
2
2
2
2
d X
d Y
X
0
Y
0
dx
dy
+
=
−
=
λ
λ
(1,2)
and the temperature distribution is
(
)
(
)
(
)
x,y
X x
Y y .
=
⋅
θ
(3)
C
onsider now the situation when
λ
2
= 0.
From Eqs. (1), (2), and (3), find that
(
)
(
)
(
)
1
2
3
4
1
2
3
4
X
C
C x,
Y
C
C y
and
x,y
C
C x C
C y .
=
+
=
+
=
+
+
θ
(4)
Evaluate the constants  C
1
, C
2
, C
3
and C
4
 by substitution of the boundary conditions:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)(
)
(
)
(
)(
)
1
2
3
4
1
2
3
4
3
2
4
2
4
x
0:
0,y
C
C
0
C
C
y
0
C
0
y
0:
x,0
0
C
x
C
C
0
0
C
0
x
L:
L,0
0
C
L
0
C
y
0
C
0
y
W:
x,W
0
0 x
0
C
W
1
=
=
+
⋅
+
⋅
=
=
=
=
+
⋅
+
⋅
=
=
=
+
⋅
+
⋅
=
=
=
+
⋅
+
⋅
=
θ
θ
θ
θ
0
1
=
=
≠
The last boundary condition leads to an impossibility (0
≠
1).
We therefore conclude that a
λ
2
value of
zero will not result in a form of the temperature distribution which will satisfy the boundary
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 Spring '08
 gough
 Boundary value problem, ORDINARY DIFFERENTIAL EQUATIONS, Boundary conditions, C7 cos

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