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Partial differential equation help

  1. Apr 8, 2008 #1

    I'm new here.

    given the pde:

    u(t) = Uxx - U

    u(0,t) = 0
    u(1,t) = 0

    u(x,0) = sin(pi*x) + 0.5(sin(3*pi*x))
    when 0<x<1

    can anyone help me with the solution?
  2. jcsd
  3. Apr 8, 2008 #2


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    Staff Emeritus
    Science Advisor

    When you say "u(t)" do you mean Ut?

    If so, you should be able to use standard "separation of variables". Assume we can write U(x,t)= X(x)T(t). Then the equation is XT'= TX"+ XT. Dividng through by XT,
    T'/T= (X"+ X)/X. Since the left side is a function of t only and the right side a function of x only, they must each be equal to the same constant:

    T'/T= [itex]\alpha[/itex] and (X"+ X)/X= [itex]alpha[/itex].

    (X"+ X)/X= [itex]\alpha[/itex] gives X"+ X= [itex]\alpha[/itex]X or X"+ (1-[itex]\alpha[/itex])X= 0. If 1-[itex]\alpha[/itex]> 0, that gives exponential solutions which cannot satisfy the boundary conditions. If 1- [itex]\alpha[/itex] = 0, that gives a linear functions which cannot satisfy the boundary conditions. In order to satisfy the boundary conditions, 1- [itex]\alpha[/itex] must equal a negative multiple of [itex]\pi[/itex]: [itex]1- \alpha= -n\pi[/itex] so [itex]\alpha= 1+ n\pi[/itex]. Put that into the equation for T and solve. The solution to the original problem is the sum, over n, of those solutions.
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