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a = 0 b = 0 c = 0
⎧
⎪⎨ b ∂tu + c ∂xxu + d ∂xu + e u = 0 ]0, π[×]0, +∞[,
u(t, 0) = u(t, π) = 0,
⎪⎩
u(0, x) = f(x).
a = 0 b = 0 λ
λn = − d2 +4c 2 n 2
4c n ∈ N n > 0
X U
d2 + 4c2n2 − 4ce
un(t, x) = bn exp
4bc
d
−
t e 2c x sin nx.
u(0, x) = f(x)
∞
∞
∞
u(0, x) = f(x) = un(0, x) =
bn sin nx
n=1
n=1
d
−
bne 2c x d
−
sin nx = e 2c x
bn f(x)e d
2c x
[−π, π] 2π
bn = 2
π
u(t, x) = 2 d
e− 2c
π x
∞
n=1
π
0
π
0
f(s)e d
2c s sin ns ds
f(x)e d
2c x sin nx dx,
a = 0 b = 0 c = 0 d = 0
⎧⎪⎨
b ∂tu + c ∂xxu + e u = 0 ]0, π[×]0, +∞[,
ux(t, 0) = ux(t, π) = 0,
⎪⎩
u(0, x) = f(x).
n=1
d2 + 4c2n2 − 4ce
exp
t sin nx.
4bc
a = 0 b = 0 d = 0 λ
λn = −cn 2 n ∈ N n > 0
X U
cn2 − e
un(t, x) = an exp
b
t cos nx.
u(0, x) = f(x)
∞
∞
u(0, x) = f(x) = un(0, x) = a0 + an cos nx
n=1
an f(x) [−π, π]
2π
a0 = 1
π
π
f(x) dx,
n=1
0
an = 2
π
f(x) cos nx dx,
π 0