Partielle Di erentialgleichungen - Blatt 5 (Prof. Dr. Yu. …kondrat/script1/U4.pdfO.Kutovyi WS...
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O.Kutovyi WS 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 u tt - a 2 u xx =0, u| x=0 =0, u| x=l =0 if a) (1 Point) u| t=0 = A sin πnx l ,n ∈ Z, u t | t=0 =0. b) (1 Point) u| t=0 =0, u t | t=0 = v 0 = const. Exercise 2. Find solution to the equation which describes the motion of bounded string u tt - u xx =0, u| x=0 =0, u| x=l =0 if a) (2 Points) u| t=0 =0, u t | t=0 = A cos π(x-x 0 ) 2α if x ∈ [x 0 - α, x 0 + α] 0 otherwise, where 0 ≤ x 0 - α<x 0 + α ≤ l. b) (2 Points) l = π, u| t=0 = sin 10x, u t | t=0 =0. 1
Transcript of Partielle Di erentialgleichungen - Blatt 5 (Prof. Dr. Yu. …kondrat/script1/U4.pdfO.Kutovyi WS...
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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