Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

T (t) = BeCa2t

t → ∞ u(x, t) C < 0 C = −λ2

T (t) = Be−λ2a2t.

C = −λ2 X (x) + +λ2X(x) = 0

X(x) = C1 cos λx + C2 sin λx,

s2 + λ2 = 0 s1,2 = ±λi

u(t, x) = Be−λ2a2t(C1 cos λx + C2 sin λx).

C1 C2

X(0)T (t) = 0, X(l)T (t) = 0.

T (t)

X(0) = X(l) = 0.

Ck k = 1, 2, 3, . . .

λk = kπ/l sin(kπx/l)

Tk(t)

λk = kπl ; Tk(t) = Bke−λ2k a2t.

Xk(x) Tk(t) uk(x, t)

uk(x, t)

u(x, t) bk(k = 1, 2, . . . )

bk

u|x=0 = h0, u|x=l = h1,

u(x, t) w(x, t) u(x, t) = w(x, t) + γ1x + γ0,

γ0 γ1

w(x, t)

u(x, t)

u(x, t)

w(x, t)

w(x, t)|t=0 = ϕ(x) − h1 − h0 x − h0, l

w|x=0 = 0; w|x=l = 0.

w(x, t) u(x, t)

C

C = −λ2

a0/2 λ0 = 0

ak

ϕ(x) = u0 0 x l u(x, t)

u(x, t) = a0/2 = u0

t > 0

X (0) = 0, X(l) = 0.

(U<0>) m 1

(U<1>) m 0,8

(U<2>) m 0,6

(U<3>) m 0,4

(U<4>) m 0,2

(U<5>) m

u(x) t

t

xm, m = 0, 1, · · ·M x

ak

∂u = a2 ∂2u ∂t ∂x2

u|t=0 = ϕ(x)

u|x=0 = u|x=l = 0. u(x, t) = ∞ bke−(kπ/l)2a2t sin kπx l

k=1

[−1, 1] y = x4