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PDE — lost on this separation of variables problem

  1. Aug 13, 2016 #1
    << Mentor Note -- thread moved from the technical math forums >>

    I am getting stuck on this partial differential equation.
    Ut = Uxx - U + x ; 0<x<1
    U(0,t) = 0
    U(1,t) = 1
    U(x,0) = 0

    Here is my work so far :
    U = e-tw + x
    gives the new eq wt=wxx
    to get rid of boundary conditions :
    w=x+W
    Wt=Wxx
    W(0,t) = 0
    W(1,t)=0
    W(x,0)=-x

    W=X(x)T(t)
    Plug that in, and I get these :
    T'=μT
    X''=μx

    w = e-(nπ)2t[ansin(nπx)]
    an = -2∫xsin(nπx) = 2cos(nπ)/nπ
    w = x + W
    w = x +(2/π)Σ(1/n)cos(nπ)sin(nπx)e-(nπ)2t
    u = e-tw + x
    u = x + e-t(x +(2/π)Σ(1/n)cos(nπ)sin(nπx)e-(nπ)2t)

    But the books answer is :
    u(x,t) = x - (2/π)e-t* [ e2tsin(πx) - (1/2) e-2π2tsin(2πx)+...]

    What am I doing wrong?
     
    Last edited by a moderator: Aug 13, 2016
  2. jcsd
  3. Aug 13, 2016 #2
    I figured out the answer. I was not plugging U = we-t+x into the boundary conditions properly. I did not need the extra substitution. Cant find a delete button for the thread though !
     
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