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ω = n ∈ Z ω λ < 0 λ
−4n 2
λ λ0 = 0 X0(x) = c0 λn = −4n 2
n ∈ N \ {0} Xn(x) = cn cos nx
U λn
Un(t) = dne 3−4n2
6
an = cndn Un(t)Xn(x)
un(t, x) = ane 3−4n2
6
t .
t cos nx.
∞
u(0, x) = x(π − x) = a0 + an cos nx
aj j > 1 x(π − x)
[0, π] [−π, π 2π R a0
0
a0 = 1
2
2 π 0π x(π − x) dx = π2
6 .
an = 2
π
π(x − x
π 0
2 ) cos nx dx = − 2(1 + (−1)n )
n2 .
a2k = −1/k2 k ∈ N \ {0}
u(t, x) = π2
6 et/2 ∞ e
−
3−16k2
t 6
k2 cos(2kx) = e t/2
⎛
⎝ π2
6 −
∞ e −8k2
t 3
k2 ⎞
cos(2kx) ⎠ .
k=1
1/k 2
n=1
k=1