partielle di erentialgleichungen - blatt 5 (prof. dr. yu. …kondrat/script1/u4.pdfo.kutovyi ws...
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O.KutovyiWS 2008/09
Partielle Differentialgleichungen - Blatt 5
(Prof. Dr. Yu. Kondratiev)
(Motion of bounded string without external force)Exercise 1.. Find solution to the equation which describes the motion
of bounded string
utt − a2uxx = 0, u|x=0 = 0, u|x=l = 0
ifa) (1 Point)
u|t=0 = A sinπnx
l, n ∈ Z, ut|t=0 = 0.
b) (1 Point)u|t=0 = 0, ut|t=0 = v0 = const.
Exercise 2. Find solution to the equation which describes the motion ofbounded string
utt − uxx = 0, u|x=0 = 0, u|x=l = 0
ifa) (2 Points)
u|t=0 = 0, ut|t=0 =
{A cos π(x−x0)
2αif x ∈ [x0 − α, x0 + α]
0 otherwise,
where 0 ≤ x0 − α < x0 + α ≤ l.
b) (2 Points)
l = π, u|t=0 = sin 10x, ut|t=0 = 0.
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