
6.4. Equations Containing the First Time Derivative 617
Solution:
w(r, t) =
∂
∂t
Z
R
0
f
0
(ξ)G(r, ξ, t) dξ +
Z
R
0
f
1
(ξ) + kf
0
(ξ)
G(r, ξ, t) dξ
− a
2
Z
t
0
g(τ)
∂
∂ξ
G(r, ξ, t − τ)
ξ=R
dτ +
Z
t
0
Z
R
0
Φ(ξ, τ )G(r, ξ, t − τ) dξ dτ.
Here,
G(r, ξ, t) = exp
−
1
2
kt
∞
X
n=1
2ξ
R
2
J
2
1
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
sin
t
√
λ
n
√
λ
n
,
λ
n
=
a
2
µ
2
n
R
2
+ b −
k
2
4
,
where the µ
n
are positive zeros of the Bessel function, J
0
(µ) = 0. The numerical values
of the first ten µ
n
are specified in Section 3.2.1 (see the first boundary value problem for
0 ≤ r ≤ R).
3
◦
. Domain: 0 ≤ r ≤ R. Second boundary value problem.
The following conditions are prescribed:
w = f
0
(r) at t = 0 (initial condition),
∂
t
w = f
1
(r) at t = 0 (initial conditi ...