
11.3. Two-Dimensional Nonstationary Fourth-Order Equations 991
3
◦
. Domain: 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
. Mixed boundary conditions are set on the sides:
w = ∂
xx
w = 0 at x = 0, w = ∂
xx
w = 0 at x = l
1
,
∂
y
w = ∂
y yy
w = 0 at y = 0, ∂
y
w = ∂
y yy
w = 0 at y = l
2
.
Green’s function:
G(x, y, ξ, η, t) =
2
al
1
l
2
∞
X
n=1
∞
X
m=0
ε
m
sin(p
n
x) cos(q
m
y) sin(p
n
ξ) cos(q
m
η)
sin(λ
nm
at)
λ
nm
,
p
n
=
πn
l
1
, q
m
=
πm
l
2
, λ
nm
= p
2
n
+ q
2
m
, ε
m
=
(
1 for m = 0,
2 for m 6= 0.
◮ Solution of the boundary value problem on transverse vibrations of a circular
plate.
Domain: 0 ≤r ≤R, 0 ≤ ϕ ≤2π. Initial and boundary conditions for symmetric transverse
vibrations of a circular plate of radius R with clamped contour in the ...