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Homework Help: Heat equation - theta function?

  1. Jul 25, 2010 #1
    du/dt=d2u/dx2

    Show that u(x,t)=(t^a) * theta(xi) where xi=x/sqrt(t) and a is a constant, then theta(xi) satisfys the ODE

    a*theta - 0.5 * xi * dtheta/dxi = d2theta/dxi2

    Not sure how to start this. Any help most appreciated

    (sorry if question isnt easy to ready)
     
  2. jcsd
  3. Jul 25, 2010 #2

    HallsofIvy

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    Have you tried the obvious? Put u(x,t)=(t^a) * theta(xi) into the given equation. Since theta(xi) is the only unknow function the result will be an equation in theta.
     
  4. Jul 25, 2010 #3
    theta(xi) = u(x,t) / t^a

    dtheta/dxi = 1 since the right hand side is just a constant wrt xi so this cant be the approach because it wont satisfy the ODE right?
     
  5. Jul 25, 2010 #4

    vela

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    If the RHS were a constant, the derivative would be 0, not 1. But it's not constant, so it's not relevant.

    The idea is to plug in your expression for u(x,t) into the differential equation:

    [tex]\frac{\partial}{\partial t}[t^a \Theta(\xi(x,t))] = \frac{\partial^2}{\partial x^2}[t^a \Theta(\xi(x,t))][/tex]

    where [itex]\xi(x,t)=x/\sqrt{t}[/itex]. You'll need to use the chain rule to express the derivatives with respect to t and x in terms of the derivative with respect to ΞΎ.
     
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