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Nonhomogenous PDE

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data
    du/dt=(d^2 u)/dx^2+1
    u(x,0)=f(x)
    du/dx (0,t)=1
    du/dx (L,t)=B
    du/dt=0
    Determine an equilibrium temperature distribution. For what value of B is there a solution?
    2. Relevant equations

    Not really sure what to put here.

    3. The attempt at a solution
    I started by trying to separate variables, with u(x,t)=phi(x)*g(t), and got to
    g'(t)/g(t)=phi''(x)/phi(x)+1/(phi(x)g(t))=0.
    So g(t) is constant based on the above, but then I get a little lost while trying to solve for phi. I tried letting g(t)=lamda (abbreviated lm from now on), and got

    phi''(t)+1/lm=0, which yields a quadratic solution of [(-lm*x^2)/2+x/lm+C/lm] after using the condition du/dx (0,t)=1. Then, since this is phi(x),
    u(x,t)= -lm^2*x^2/2+ x + lm*C.
    Using the other condition du/dx (L,t)=B, and assuming some quick mental algebra was correct,
    B=-lm^2*L+1.
    First off, is all of the above a correct approach as far as you can tell? And secondly, do I need to find u(x,t) or a specific value of lm?
     
    Last edited: Jan 29, 2013
  2. jcsd
  3. Jan 30, 2013 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    I am confused by the question. You say "Determine an equilibrium temperature distribution. For what value of B is there a solution?"

    It is very easy to determine the general equilibrium solution. Is the second independent of that or is it asking for a value of B for which there is an equilibrium solution?
     
  4. Jan 30, 2013 #3
    I found the equilibrium solution easily enough after posting that, I realized I could solve for lm. Another student and I clarified with our professor today, he wants both, which I now have. Thanks though!
     
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