
306 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
where
G(r, ξ, t) =
3ξ
2
R
3
+
2ξ
Rr
∞
X
n=1
µ
2
n
+ 1
µ
2
n
sin
µ
n
r
R
sin
µ
n
ξ
R
exp
−
aµ
2
n
t
R
2
.
Here, the µ
n
are positive roots of the transcendental equation tan µ −µ = 0. The values of
the fi rst five roots µ
n
can be found in Section 3.2.3 (see the second boundary value problem
for 0 ≤ r ≤ R).
⊙ Literature: A. V. Lykov (1967), A. V. Bitsadze and D. F. Kalinichenko (1985).
◮ 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