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a0 = 1<br />
π<br />
an = 2<br />
π<br />
π<br />
0<br />
π<br />
<br />
0<br />
x = π<br />
2<br />
x cos(nx) dx = 2<br />
π<br />
x sin(nx)<br />
n<br />
x = π 2<br />
−<br />
2 π<br />
n=1<br />
∂tu(x, 0) = k0 +<br />
k0 = π/2 n √ 3kn = − 2<br />
π<br />
u(x, t) = π<br />
2 t − 2√3 3π<br />
= π<br />
2 t − 4√3 3π<br />
k=0<br />
x=π<br />
x=0<br />
− 2<br />
π<br />
sin(nx) dx = −<br />
nπ 0<br />
2<br />
n2π (1 − (−1)n ),<br />
∞ 1 − (−1) n<br />
n2 cos(nx).<br />
1−(−1) n<br />
n 2<br />
∞<br />
n √ 3kn cos(nx)<br />
n=1<br />
kn = − 2√ 3<br />
3π<br />
1−(−1) n<br />
n3 <br />
∞ 1 − (−1)<br />
n=1<br />
n<br />
n3 sin(n √ 3t) cos(nx)<br />
∞ 1<br />
(2k + 1) 3 sin((2k + 1)√3t) cos((2k + 1)x).<br />
1/n 3 <br />
<br />
<br />
<br />
<br />
⎧<br />
⎪⎨ −∂tu(t, x) + ∂xxu(t, x) + 2∂xu(t, x) − 3u = 0, (t, x) ∈]0, +∞[×[0, π],<br />
u(t, 0) = u(0, π) = 0,<br />
⎪⎩<br />
u(0, x) = xe−x .<br />
u(t, x) = T (t)X(x) <br />
<br />
− ˙<br />
T (t, x) + T (t) ¨ X(x) + 2T (t) ˙ X(x) − 3T (t)X(x) = 0<br />
U(t)X(x) λ ∈ R <br />
−T ˙ (t, x) − 3T (t)<br />
− =<br />
T (t)<br />
¨ X(x) + 2 ˙ X(x)<br />
=: λ<br />
X(x)<br />
λ ∈ R <br />
<br />
¨X(x) + 2X(x) ˙ − λX(x) = 0<br />
− ˙ T (t) + (−3 + λ) T (t) = 0.<br />
<br />
X X(0) = X(π) = 0 <br />
µ 2 + 2µ − λ = 0 ∆ = 4(1 + λ) <br />
λ ∈ R<br />
∆ > 0 µ1 = −2−√∆ 2 µ2 = −2+√∆ 2 <br />
Φ(c1, c2, x) = c1e µ1x +c2e µ2x Φ(c1, c2, 0) =<br />
X(0) = 0 c1 = −c2 Φ(c1, c2, π) = X(π) = 0 <br />
c1(e µ1π − e µ2π ) = 0 ∆ > 0 µ1 = µ2 <br />
c1 = c2 = 0
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