R × (0, ´), ´ > 0 u
R × {0} Õ
uxx + uyy = 0, u(x, 0) = Õ(x), uy(x, 0) = 0.
Õ " C(R)
Õ
1
Õn(x) = cos nx.
n
1
un(x, y) = cos nx cosh ny,
n
cosh y := (exp y +exp(-y))/2. exp(ny)/(2n2)
y = 0,

Õn 0 =: Õ0 u = 0
Õ0.
"u
Ä…(x) · (x) + ²(x)u(x) = Õ(x), x " "&!,
"½
Ä…, ², Õ &! &!
u
&! = (0, 1) × (0, 1)
uxy = 0.
u|"&! = Åš,
Åš
u(x, y) =
f(x) + g(y), f, g C1 (0, 1), [0, 1]
Åš y = 1
Åš(x, 1) = f(x) + g(1) y = 0 Åš(x, 0) = f(x) + g(0), Åš(x, 1) -
Åš(x, 0) = g(1) - g(0) = const.
Åš