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
22
dX
dY
X
0
Y
0
dx
dy
+=
−=
λλ
(1,2)
and the temperature distribution is
( ) ( ) ( )
x,y
X x
Y y .
=⋅
θ
(3)
Consider now the situation when
λ
2
= 0.
From Eqs. (1), (2), and (3), find that
( ) ( ) ( )
12
34
X
C
C x,
Y
C
C y
and
x,y
C
C x
C
C y .
=+
=
+
+
(4)
Evaluate the constants  C
1
, C
2
, C
3
and C
4
 by substitution of the boundary conditions:
( ) ( )( )
()
(
)
(
)
(
)
(
)
(
)
1
23
4
3
24
2
4
x
0:
0,y
C
C
0 C
C
y
0
C
0
y
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
=
=
+⋅
=
=
==
+
⋅
+
⋅
=
+
⋅
+
⋅
=
+
⋅
+
⋅
=
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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