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T ˙
Tn(t) = −5n 2 Tn(t) Tn(t) = Tn(0)e −5n2 t
n ∈ N
un(t, x) = Tn(t)Xn(x) = ane −5n2 t cos (nx) ,
an = T (0)c1 an ∈ R \ {0}
u(0, x) = 2x =
∞
∞
un(0, x) = a0 + cos (nx) ,
j=0
an
f(x) = 2x a0 0 f
f [−π, π] 2π R
a0 = 1
π
an = 2
π
π
0
π
0
2x dx = π
2x cos nx dx = 4
π
x
= 4
n2π [cos(nx)]π0 = 4 (−1)n − 1
n2π n=1
π sin nx
−
n 0
4
π
sin(nx) dx
nπ 0
a0 = π a2k = 0 a2k−1 = −8/(π(2k − 1) 2 ) k ∈ N k ≥ 1
u(t, x) = π − 8
∞ e
π
−5(2k−1)2
t cos (2k − 1)x
(2k − 1) 2 .
∞
e
sup
−5(2k−1)2
t cos (2k − 1)x
(2k − 1) 2
≤
∞ 1
< +∞,
(2k − 1) 2
k=1
k=1
1/k 2
⎧
−ut + 2uxx + 3ux + u = 0, t > 0, x ∈]0, π[,
⎪⎨
u(0, t) = u(π, t) = 0,
⎪⎩
3
−
u(x, 0) = e 4 x π
2 −
x − π
,
2
u(x, t) = T (t)X(x)
− ˙ T (t)X(x) + 2T (t) ¨ X(x) + 3T (t) ˙ X(x) + T (t)X(x) = 0
T (t)X(x)
k=1
− ˙ T (t) + T (t)
= −
T (t)
2 ¨ X(x) + 3 ˙ X(x)
X(x)
λ ∈ R
− ˙ T (t) + (1 − λ)T (t) = 0,
2 ¨ X(x) + 3 ˙ X(x) + λX(x) = 0.