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T
1
2
()
(. /) (
=
1153 233115
0
+
0
/
)
=
0 0
.
T
2
2
()
=
(
112115 2321275
0
/ )( ./)
+ +=
0
/
00
.
(2)
T
= + +=
(101.5 / 2)
(301 / 2) 3 / 2
202.750
3
T
4
2
()
(. /) (
=
1153 833315
0
+
0
/
)
=
0 0
.
After repeated applications of the iterative process up to 10 and 20 iterations, the
nodal temperatures have advanced to
(
)
(
)
(
)
10
20
40
T
T
T
137.247
140.3632
140.500
1
1
1
(
)
(
)
(
)
10
20
40
T
208.837
T
218.092
T
218.500
2
2
2
=
and
=
and
=
(
)
286.388
(
)
293.200
(
)
293.500
365.500
10
20
40
T
T
T
3
3
3
361.080
365.313
(
)
(
)
(
)
10
20
40
T
T
T
4
4
4
From the previous Example 4.3, we obtained the exact direct solution by the TDMA
algorithm. The nodal temperatures after 20 iterations converge towards the exact
nodal temperature values.
The Gauss-Siedel Method:
We begin as in the Jacobi method with the set of
equations
TT
1
=
(/ )( /)
2
3233
+
0
TT
=
(
/2) (
+
T
/2)
+
3/2
2
1
3
TT
=
(
/2)
+
(
T
/2)
+
3/2
3
2
4
TT
4
=
(/)( / )
3
3833
+
00
Employing the same initial guesses, the first iteration yields
T
1
1
13233 11
()
=
(
00
/ )( /) .
+
0
=
0 000
(1)
T
= + +=
(101 / 2) (100 / 2)
3 / 2
102.000
2
T
3
1
()
=
(
1221232125
0
/ )( /) /
+ +=
00
0
.
00
T
4
1
()
(. /) (
=
1253 83331833
0
+
0
/
)
=
0
.
After performing 10 iterations, the nodal temperatures have advanced to
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