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Numerical Solution
Sudarshan Balakrishnan
August 24, 2011
Logistic Equation
Given the logistic equation
u
0
=
u
(
u

1) we notice that the associated operators were
A
(
u
) =

u
and
B
(
u
) =
u
2
[1].The exact solutions associated with
A
,
B
and
C
=
A
+
B
are given by:
Φ
c
(
t
)
u
0
=
u
0
u
0
+
e
t
(1

u
0
)
,
Φ
A
(
t
)
u
0
=
u
0
e

t
,
Φ
B
(
t
)
u
0
=
u
0
1

u
0
t
.
(1)
Splitting Algorithm for Linear Ordinary Diﬀerential Equations
In order to understand the splitting associated with the logistic equation, let us ﬁrst
consider the linear ODE given by:
du
dt
=
Au,
du
dt
=
Bu,
du
dt
=
Cu.
(2)
In (2), we have that
C
=
A
+
B
. Solving the each of them we notice that the solution
for a given time
l
, using the separation of variables is:
ϕ
l
(
u
) =
e
Al
e
Bl
.
(3)
Using the solution, we can estimate the local error
ρ
(
u
;
l
) by:
ρ
(
u
;
l
) =
ϕ
l
(
u
)

e
Cl
.
(4)
In order to estimate (4), we have that:
ρ
(
u
;
l
) = (
I
+
Cl
+
1
2
C
2
l
2
+
...
)

(
I
+
Al
+
1
2
A
2
l
2
+
...
)(
I
+
Bl
+
1
2
B
2
l
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