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D'alembert equation

  1. Mar 29, 2013 #1
    1. The problem statement, all variables and given/known data
    For a wave equation, utt-uxx=0, 0<x< ∞
    u(0,t)=t^2, t>0
    u(x,0)=x^2, 0<x< ∞
    ut(x,0)=6x, 0<x< ∞

    evaulate u(4,1) and u(1,4)

    uxx is taking 2 derivatives in respective of x




    2. Relevant equations

    D'alembert's equation u=(f(x+ct)+f(x-ct))/2 + (1/ct)(∫g(s)ds

    3. The attempt at a solution

    this seems easy, just plug everything into the formula to get u(4,1)=u(1,4)=24, however,
    at u(1,4) when you evaluate f(1-4) that gives you f(-3) which is out of the boundary in the initial condition for x. how do you re-structure another equation so this would work? thanks
     
  2. jcsd
  3. Mar 31, 2013 #2

    rude man

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    Gold Member

    New territory for me, so what follows may be nonsense, but:

    I get u(x,t) = (1/2){(x-t)2 + (x+t)2 + ∫6s ds} with lower limit x-t and upper limit x+t.

    So u(1,4) = (1/2){(1-4)2 + (1+4)2 + 48} = whatever.

    I guess I don't see where x > 0 is violated anywhere. Of course, x - ct can be negative & is in this instance (c = 1).

    Ref: http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node12.html
     
    Last edited: Mar 31, 2013
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