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1. Use the simple explicit method to solve the 1D heat equation on the computational grid
(fgure 59) with the boundary conditions
u
n
1
=2=
u
n
3
and initial conditions
u
1
1
u
1
3
,u
1
2
=1
.
u
n
+1
j
=
u
n
j
+
r
(
u
n
j
+1
−
2
u
n
j
+
u
n
j
−
1
)=(1
−
2
r
)
u
n
j
+
r
(
u
n
j
+1
+
u
n
j
−
1
)
u
1
1
1
2
1
3
are known  initial condition
u
2
1
2
3
are known  boundary conditions
ThereFore
u
2
2
=(1
−
2
r
)
u
1
2
+
r
(
u
1
3
+
u
1
1
)
j
=2
,n
u
2
2
−
2
r
+
r
(2 + 2) = 1 + 2
r
So
u
2
2
=1+2
r
u
3
2
−
2
r
)
u
2
2
{z}
1+2
r
+
r
(
u
2
3
+
u
2
1
)
j
u
3
2
−
2
r
)(1 + 2
r
)+
r
(2 + 2)
So
u
3
2
=1+4
r
−
4
r
2
u
4
2
−
2
r
)
u
3
2
{z}
1+4
r
−
4
r
2
+
r
(
u
3
3
+
u
3
1
)
j
=3
u
4
2
−
2
r
)(1 + 4
r
−
4
r
2
r
(2 + 2)
So
u
4
2
=1+6
r
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.
 Fall '08
 BELL,D

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