Biomedical Engineering Reference
In-Depth Information
Using the method of variable separation, the function
h
2
can be written as
h
2
(
r
,
t
) =
R
(
r
)
T
(
t
)
(2.96)
Then, Equation (2.93)
2
becomes
2
1
()
dRr
dr
()
1
()
dR r
dr
()
1
k
Tt
dT t
dt
()
2
+
− =−
=
m
(2.97)
2
2
Rr
rR r
r
()
or, equivalently,
k
dT t
dt
()
2
+
mTt
()
=
0
(2.98)
with the solution
2
−
(
mt k
/)
Tt
()
=
Be
(2.99)
1
and
2
dRr
dr
()
r
dR r
dr
()
2
22
r
+
−
(
rm
+
1)()0
Rr
=
(2.100)
2
with the solution
R
(
r
) =
B
2
I
1
(
mr
) +
B
3
K
1
(
mr
)
(2.101)
where
I
1
(
mr
) and
K
1
(
mr
) are the modified Bessel functions of the first and
second kind, respectively, of order one.
Using the solutions (2.99) and (2.101), Equation (2.96) can be written as
2
−
(
mt k
/)
hrt
(,)(
=
BImrBKmre
()
+
( )
(2.102)
2
6
1
7
1
Noting that the functions
u
and
U
r
can be expressed as
r
uQRhh
=+ℵ+ =− ++ℵ+
(
) (
),
U
(2
NAQhh
) (
2
)
(2.103)
r
1
2
r
1
and
∂
∂
h
t
2
2
2(
−
mt k
/)
ℵ=−
h
k
=
(
BI mr
(
)
+
BK mr
(
))
me
(2.104)
2
61
71
