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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