
296 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
◮ Domain: 0 ≤ r ≤ R. Third boundary value problem.
The following conditions are prescribed:
w = f (r) at t = 0 (initial condition),
∂
r
w + kw = g(t) at r = R (boundary condition),
|w| 6= ∞ at r = 0 (boundedness condition).
Solution:
w(r, t) =
Z
R
0
f(ξ)G(r, ξ, t) dξ + a
Z
t
0
g(τ)G(r, R, t − τ) dτ
+
Z
t
0
Z
R
0
Φ(ξ, τ )G(r, ξ, t − τ) dξ dτ.
Here,
G(r, ξ, t) =
2
R
2
∞
X
n=1
µ
2
n
ξ
(k
2
R
2
+ µ
2
n
)J
2
0
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
exp
−
aµ
2
n
t
R
2
,
where the µ
n
are positive roots of the transcendental equation
µJ
1
(µ) −kRJ
0
(µ) = 0.
The numerical values of the first six roots µ
n
can be found in Carslaw and Jaeger (1984).
⊙ Literature: A. V. Lyko ...