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Having Alot of Trouble

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Let u(x,t) satisfy


    2. Relevant equations


    ([tex]\partial[/tex]u/[tex]\partial[/tex]t) = ([tex]\partial[/tex][tex]^{2}[/tex]u/[tex]\partial[/tex]x[tex]^{2}[/tex])........(0<x<1,t>0)

    u(0,t)=u(1,t)=0........(t[tex]\geq[/tex]0)

    u(x,0)=f(x)........(o[tex]\leq[/tex]x[tex]\leq[/tex]1),

    where f[tex]\in[/tex]C[0.1] show that for any T[tex]\geq[/tex]0

    [tex]\int[/tex] from 0..1 (u(x,T))[tex]^{2}[/tex]dx [tex]\leq[/tex] [tex]\int[/tex] from 0..1 (f(x))[tex]^{2}[/tex]dx


    3. The attempt at a solution

    not sure
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 10, 2009 #2

    berkeman

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    Staff: Mentor

    That's 3 posts and no work shown on any one of them. You *must* show an attempt at a solution, or we cannot be of help to you. Please click on the "Rules" link at the top of the page.
     
  4. Mar 10, 2009 #3
    im sorry, im obviously knew to this forum....

    For this problem, im trying to use the identity as follows

    2u(([tex]\partial[/tex]u/[tex]\partial[/tex]t)-([tex]\partial[/tex][tex]^{2}[/tex]u/[tex]\partial[/tex]x[tex]^{2}[/tex])) = ([tex]\partial[/tex]u[tex]^{2}[/tex]/[tex]\partial[/tex]t)-([tex]\partial[/tex]/[tex]\partial[/tex]x)*(u*([tex]\partial[/tex]u/[tex]\partial[/tex]x))+2*([tex]\partial[/tex]u/[tex]\partial[/tex]x)[tex]^{2}[/tex]
     
  5. Mar 10, 2009 #4

    Dick

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    Science Advisor
    Homework Helper

    It's a diffusion equation, so you might expect this sort of behavior. Your 'identity' is a little messed up. Can you fix it? Once you've done that substitute the PDE in. You should be able to show that ((u^2),t)/2-(u*(u,x)),x=(-(u,x)^2)<=0. I'm using commas for partial derivatives, forgive my laziness. Now integrate dx between 0 and 1. Can you show the (u*(u,x)),x term vanishes? Once you have integal (u^2),t<=0 you are home free.
     
    Last edited: Mar 11, 2009
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