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by K. D. Cole and D. H. Y. Yen
The GF for the 2D strip satisfies:
+
=  (x  x^{})(y  y^{}) 
(18) 
Figure:
Geometry of (a) infinite and (b) semiinfinite strip.

For the infinite strip, shown in Fig. 1a, the domain is
(
< x < , 0 < y < W) and the boundary conditions are
k_{i}
+ h_{i}G = 0 

at y = 0 and y = W 

G,
are bounded 

as x

(19) 
For the semiinfinite strip, shown in Fig. 1b, the domain is
(0 < x < , 0 < y < W)and the homogeneous boundary conditions are
k_{i}
+ h_{i}G = 0 

at x = 0, y = 0, and y = W 

G,
are bounded 

as x
+ 
(20) 
The GF of infinite and semiinfinite strip are described by number XI0YKL
which represents 36 different GF for I = 0, 1, 2, or 3 and K, L =
1, 2, or 3. Note that the GF for I = 0 are for the infinite strip
geometry only.
The GF for the strip has a singlesummation form with
eigenfunction Y_{n}, eigenvalue
, norm
N_{y}, and kernel function P_{n}, as follows:
G(x, y x^{}, y^{}) = P_{0}(x, x^{}) + P_{n}(x, x^{}) 
(21) 
The first term with kernel function P_{0} is needed only when
Y22 is part of the GF number (i.e., when zero is an eigenvalue).
There are nine different eigenfunctions associated with the nine
possible boundary condition combinations YKL (K, L = 1, 2, or 3).
Table 1 contains the eigenfunctions and norms, and Table 2
contains the associated eigenconditions (and eigenvalues for
simple cases).
Kernel functions P_{n} for n
0 are given by
P_{n}(x, x^{}) = S^{+}exp( x  x^{}) + S^{}exp( x + x^{}) 
(22) 
where the values for S^{+} and S^{} are given in Table 3.
Kernel functions P_{n} for n
0 have four different forms,
one for each of the boundary condition combinations XI0 for I = 0,
1, 2, and 3.
Kernel functions P_{0} satisfy
=  (x  x^{}) 
(23) 
as well as appropriate homogeneous boundary conditions.
Functions P_{0}, listed in Table 4, must be included in the GF
whenever zero is an eigenvalue (XI0Y22 for I = 0, 1, 2, and 3).
Table 1. Eigenfunctions and inverse norm ^{a, b}
Geometry 
Y_{n}(y) 
N_{y}^{1} 
Y11 
sin(y) 
2/W 
Y12 
sin(y) 
2/W 
Y13 
sin(y) 
2/W 
Y21 
cos(y) 
2/W 
Y22 
cos(y);
0 
2/W for
0 

1;
= 0 
1/W for
= 0 
Y23 
cos(y) 
2/W 
Y31 
Wcos(y) + (h_{1}W/k)sin(y) 
2/W 
Y32 
Wcos(y) + (h_{1}W/k)sin(y) 
2/W 
Y33 
Wcos(y) + (h_{1}W/k)sin(y) 
2/W 
^{a} Index
n = 1, 2,... for all cases except
Y22 with
n = 0, 1, 2,...
^{b}
= [(W)^{2} + (h_{i}W/k)^{2}] ÷ [(W)^{2} + (h_{i}W/k)^{2} + h_{i}W/k]
=
÷ [(W)^{2} + (h_{i}W/k)^{2} + (h_{1}W/k)]
Table 2. Eigencondition and eigenvalues for
Y_{n}(y)^{a}
Geometry 
Eigencondition 
Eigenvalues 
Y11 
sin(W) = 0 
n/W 
Y12 
cos(W) = 0 
(2n  1)/2W 
Y13 
Wcot(W) =  h_{2}W/k 

Y21 
cos(W) = 0 
(2n  1)/2W 
Y22 
sin(W) = 0 
n/W, n = 0, 1, 2,... 
Y23 
Wtan(W) = h_{2}W/k 

Y31 
Wcot(W) =  h_{1}W/k 

Y32 
Wtan(W) = hW/k 

Y33 
tan(W) = [(h_{1} + h_{2})/k]/[
 h_{1}h_{2}k^{2}] 

^{a} Index
n = 1, 2,... for all cases except
Y22 with
n = 0, 1, 2,...
Table 3. Coefficients for P_{n} for n
0, strip and semistrip.
Geometry 
S^{+} 
S^{} 
X00 
1 
0 
X10 
1 
1 
X20 
1 
1 
X30 
W + hW/k 
W  hW/k 
Table 4. Kernel function P_{0} for strip and semistrip.
Geometry 
P_{0}(x, x^{}) 
X00 
 x  x^{} 
X10 
 x  x^{}
+ x + x^{} 
X20 
 x  x^{}
 x + x^{} 
X30 
 x  x^{}
+ x + x^{}
+ k/h 
Next: RadialCylindrical Coordinates. Steady 1D.
Up: Laplace Equation. Steady Heat
Previous: Pseudo GF. Rectangular coordinate,
Kevin D. Cole
20021231