
416 SECOND-ORDER PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES
Solution:
w(x, y, t) =
Z
l
1
0
Z
l
2
0
f(ξ, η) G(x, y, ξ, η, t) dη dξ
+ a
Z
t
0
Z
l
2
0
g
1
(η, τ)
∂
∂ξ
G(x, y, ξ, η, t − τ)
ξ=0
dη dτ
+ a
Z
t
0
Z
l
2
0
g
2
(η, τ) G(x, y, l
1
, η, t − τ ) dη dτ
+ a
Z
t
0
Z
l
1
0
g
3
(ξ, τ )
∂
∂η
G(x, y, ξ, η, t − τ)
η=0
dξ dτ
+ a
Z
t
0
Z
l
1
0
g
4
(ξ, τ ) G(x, y, ξ, l
2
, t −τ) dξ dτ,
where
G(x, y, ξ, η, t)=
4
l
1
l
2
∞
X
n=0
sin
π(2n+1)x
2l
1
sin
π(2n+1)ξ
2l
1
exp
−
aπ
2
(2n+1)
2
t
4l
2
1
×
∞
X
m=0
sin
π(2m+1)y
2l
2
sin
π(2m+1)η
2l
2
exp
−
aπ
2
(2m+1)
2
t
4l
2
2
.
4.1.2 Problems in Polar Coordinates
The sourceless heat equation with two space variables in the polar coordinate system r, ϕ
has the form
∂w
∂t
= a
∂
2
w
∂r
2
+
1
r
∂w
∂r
+
1
r
2
∂
2
w
∂ϕ
2
, r =
p
x
2
+ y
2
.