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1.
From the parabolic PDE governing equation:
2
2
x
u
t
u
!
!
=
!
!
"
for
L
x
!
!
0
············ (1)
Approximation
of
the
secondorder
spatial
and
firstorder
time
derivative
using
a
central difference formula is
)
(
)
2
(
)
(
1
2
,
1
,
,
1
2
,
2
2
x
O
u
u
u
x
x
u
n
i
n
i
n
i
n
i
!
+
+
"
!
=
#
#
"
+
············ (2)
)
(
)
(
)
(
2
1
2
1
,
1
,
,
t
O
u
u
t
t
u
n
i
n
i
n
i
!
+
"
!
=
#
#
"
+
············ (3)
Substitute Eqs.(2)&(3) into the governing parabolic PDE, Eq(1),
)
(
)
2
(
)
(
1
)
(
)
(
)
(
2
1
2
,
1
,
,
1
2
2
1
,
1
,
x
O
u
u
u
x
t
O
u
u
t
n
i
n
i
n
i
n
i
n
i
!
+
"
"
#
$
%
%
’
+
(
!
=
!
+
(
!
(
+
(
+
)
Rearranging and solving for the highest point in time and in the form:
)
(
2
2
,
1
,
,
1
1
,
t
x
O
Cu
Bu
Au
u
n
i
n
i
n
i
n
i
!
+
!
+
+
"
=
"
+
+
We get
)
(
)
(
2
)
(
4
)
(
2
2
2
,
1
2
,
2
,
1
2
1
,
1
,
t
x
O
u
x
t
u
x
t
u
x
t
u
u
n
i
n
i
n
i
n
i
n
i
!
+
!
+
!
!
+
!
!
"
!
!
+
=
"
+
"
+
#
Stability requires that all coefficients are greater than zero (A>0, B>0, etc.)
Stability: Unstable, because B will always be less than zero.
2.
The flow and migration of human leukocytes on a substrate in chapter8 of
text is:
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z
C
v
z
C
t
C
eff
D
!
!
"
!
!
=
!
!
2
2
μ
············ (1)
Solve this Eq numerically by expressing the partial derivative into finite difference
approximation
with secondorder
spatial
and
firstorder
time
derivatives
using
a
central difference formula is
)
(
)
(
2
1
2
,
1
,
1
,
z
O
C
C
z
z
C
n
i
n
i
n
i
!
+
"
!
=
#
#
"
+
············ (2)
)
(
)
2
(
1
2
,
1
,
,
1
2
,
2
2
z
O
C
C
C
z
z
C
n
i
n
i
n
i
n
i
!
+
+
"
!
=
#
#
"
+
············ (3)
Solve the time derivative into forward finite difference:
)
(
)
(
1
2
,
1
,
,
t
O
C
C
t
t
C
n
i
n
i
n
i
!
+
"
!
=
#
#
+
Combine the above Eqs(2)&(3) into Eq (1), and rearrange to solve for the highest
point in time and in the form
)
(
2
2
,
1
,
,
1
1
,
t
x
O
Cu
Bu
Au
u
n
i
n
i
n
i
n
i
!
+
!
+
+
"
=
"
+
+
We get:
!
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 Spring '08
 RobertSpilker
 Partial differential equation, Concentration Profiles, Therapeutic Concentration Profiles, central difference formula

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