
4.1. Heat Equation
∂w
∂t
= a∆
2
w 421
◮ Domain: 0 ≤ r < ∞, 0 ≤ ϕ ≤ ϕ
0
. Second boundary value problem.
A wedge domain is considered. The following conditions are prescribed:
w = f (r, ϕ) at t = 0 (initial condition),
r
−1
∂
ϕ
w = g
1
(r, t) at ϕ = 0 (boundary condition),
r
−1
∂
ϕ
w = g
2
(r, t) at ϕ = ϕ
0
(boundary condition ).
Solution:
w(r, ϕ, t) =
Z
ϕ
0
0
Z
∞
0
f(ξ, η) G(r, ϕ, ξ, η, t)ξ dξ dη
− a
Z
t
0
Z
∞
0
g
1
(ξ, τ ) G(r, ϕ, ξ, 0, t − τ ) dξ dτ
+ a
Z
t
0
Z
∞
0
g
2
(ξ, τ ) G(r, ϕ, ξ, ϕ
0
, t −τ) dξ dτ.
Here,
G(r, ϕ, ξ, η, t) =
1
aϕ
0
t
exp
−
r
2
+ ξ
2
4at
1
2
I
0
rξ
2at
+
∞
X
n=1
I
nπ/ϕ
0
rξ
2at
cos
nπϕ
ϕ
0
cos
nπη
ϕ
0
,
where I
ν
(r) are modified Bessel functions.
⊙ Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikh ...