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Heat Equation

  1. Jan 31, 2009 #1
    Problem:IF there is heat radiation within the rod of lenght L , then the 1 dimensional heat equation might take the form
    u_t = ku_xx + F(x,t)

    Find u(x) if F = x , k = 1 , , u(0)=0 , u(L) = 0

    the problem is that i am not sure what this is asking me , how can i find u(x) if i have only u(0)= 0 , k =1 , u(L) = 0 ,and F = x


    this problem becomes just an ordinary differential equation but still i dont fully understand or how to proceed from there

    any hints would be appreciated
     
  2. jcsd
  3. Jan 31, 2009 #2
    have you tried green's function method?
     
  4. Feb 1, 2009 #3
    I assume that a steady-state solution is required, therefore the time derivative vanishes in the pde and you get the following equation to solve:
    [tex]\frac{d^2u}{dx^2}+x=0[/tex]
    which has the solution:
    [tex]u(x)=-\frac{x^3}{6}+Ax+B[/tex]
    Using the boundary conditions, you get:
    [tex]u(x)=\frac{x}{6}\cdot \left[L^2-x^2\right][/tex]
    Hope this is what has been asked for.

    coomast
     
  5. Feb 1, 2009 #4
    THank you so much
     
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