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Having Alot of Trouble

  • Thread starter modeiry88
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  • #1
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Homework Statement



Let u(x,t) satisfy


Homework 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


The Attempt at a Solution



not sure

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
berkeman
Mentor
57,313
7,306
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.
 
  • #3
4
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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]
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
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:

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