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PDE: Initial Conditions Contradicting Boundary Conditions

  1. Feb 3, 2013 #1
    Suppose we have the following IBVP:

    PDE: [itex]u_{t}=α^{2}u_{xx}[/itex] [itex]0<x<1[/itex] [itex]0<t<∞[/itex]
    BCs: [itex]u(0,t)=0, u_{x}(1,t)=1[/itex] [itex]0<t<∞[/itex]
    IC: [itex]u(x,0)=sin(πx)[/itex] [itex]0≤x≤1[/itex]

    It appears as though the BCs and the IC do not match. The derivative of temperature with respect to x at position x=1 is a constant 1 while with the initial condition, the derivative is equal to -π. Do I conclude that the problem is incorrect or is there another way to reconcile this error?
  2. jcsd
  3. Feb 3, 2013 #2


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    I don't think there is an error. You can think of a bar with the given temperature distribution inserted into a situation with those bc's. I would begin by using a substitution like ##u(x,t) = v(x,t)+\psi(x)## to make a homogeneous system in ##v(x,t)## and let the ##\psi(x)## take care of the ##u_x(1,t)=1## nonhomogeneous term.
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