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Homework Help: Pde problem: inspiration needed

  1. Apr 29, 2006 #1
    [tex]u_{a}+u_{t}=-\mu t_{u}[/tex]
    [tex]u(a,0)=u_{0}(a)[/tex]
    [tex]u(0,t)=b\int_0^\infty \left u(a,t)da[/tex]

    Solve [tex]u(a,t)[/tex] for the region [tex]a<t[/tex]

    Got this question from assignment. My solution is incomplete though, need some inspirations! I have shown that the general solution is

    [tex]u(a,t)=F(a-t)e^{-1/2{\mu}t^{2}}[/tex]

    So for u(0, t) we have

    [tex]u(0,t)=F(-t)^{-1/2{\mu}t^{2}}=b\int_0^\infty \left u(a,t)da[/tex]

    Substitute the general solution u(a,t) we get

    [tex]F(-t)=b\int_0^\infty \left F(a-t)da[/tex]

    Now -t -> a-t, and we obtain F(a-t) to be

    [tex]F(a-t)=b \int_0^\infty \left F(2a-t)da[/tex]

    Hence our solution u(a,t) for a<t is

    [tex]u(a,t)=be^{-1/2{\mu}t^{2}}\int_0^\infty\left F(2a-t)da[/tex]

    However, when I talked to my lecturer, he told me I can simplify this further. I need some inspirations:redface:
     
    Last edited: Apr 29, 2006
  2. jcsd
  3. Apr 30, 2006 #2
    Your general solution has an arbitrary function F.
    But as you can see in your steps there, the boundary conditions on u have given you quite a bit of information on F. So you can restrict it further.

    So here is some inspiration:
    -What is F(0) equal to?
    - Note you obtained one equation with F on both sides. Can you solve for F given what F(0) is?

    Good luck with your homework.
     
    Last edited: Apr 30, 2006
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