2. The DuFort Frankel method for solving the heat equation requires solution of the di±er
ence equation
u
n
+1
j
−
u
n
−
1
j
2∆
t
=
α
(∆
x
)
2
u
n
j
+1
−
u
n
+1
j
−
u
n
−
1
j
+
u
n
j
−
1
Develop the stability requirements necessary for the solution of this equation.
Let
r
=
α
∆
t
(∆
x
)
2
then DuFort Frankel method can be written as
u
n
+1
j
−
u
n
−
1
j
=2
r
u
n
j
+1
−
u
n
+1
j
−
u
n
−
1
j
+
u
n
j
−
1
or
(1 + 2
r
)
u
n
+1
j
r
u
n
j
+1
+
u
n
j
−
1
+(1
−
2
r
)
u
n
−
1
j
Use
λ
n
e
ik
m
j
∆
x
λ
n
+1
(1 + 2
r
)
−
2
r
e
ik
m
∆
x
−
e
−
ik
m
∆
x

{z
}
2cos
β
λ
n
−
(1
−
2
r
)
λ
n
−
1
=0
where
β
=
k
m
∆
x
. This leads to a quadratic equation for
λ
(1 + 2
r
)
λ
2
−
4
r
cos
βλ
−
(1
−
2
r
)=0
and the solutions are
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 Fall '08
 BELL,D
 Quadratic equation, Irrational number, 1 j, DuFort Frankel method

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