1
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

Partielle Di erentialgleichungen - Blatt 5 (Prof. Dr. Yu. …kondrat/script1/U4.pdfO.Kutovyi WS 2008/09 Partielle Di erentialgleichungen - Blatt 5 (Prof. Dr. Yu. Kondratiev) (Motion

  • Upload
    others

  • View
    4

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Partielle Di erentialgleichungen - Blatt 5 (Prof. Dr. Yu. …kondrat/script1/U4.pdfO.Kutovyi WS 2008/09 Partielle Di erentialgleichungen - Blatt 5 (Prof. Dr. Yu. Kondratiev) (Motion

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.

1

Kutovyi
Text Box
Kutovyi
Typewritten Text
WS 2012/13
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Text Box
Kutovyi
Typewritten Text
4
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text
Kutovyi
Typewritten Text