Inhomogeneous PDE

  1. Hi all,

    1. The problem statement, all variables and given/known data

    Determine the equilibrium temperature distribution (if it exists). For what values of B, are there solutions.

    2. Relevant equations

    a) Ut = Uxx + 1, U(x,0) = f(x), Ux(0,t) = 1, U(L,t) = B

    b) Ut = Uxx + X - B, U(x,0) = f(x), Ux(0,t) = 0, U(L,t) = 0

    3. The attempt at a solution

    a) Assume solution U(x,t) = V(x,t) + K

    => Ut = Vt
    => Uxx = Vxx -- Sub into Ut = Uxx + 1

    Vt = Vxx + 1

    Now to get homogeneous boundary conditions,

    U(0,t) = V(0,t) + K = 1 ?
    but K = 1 => V(0,t) = 1?

    // Trouble at the BC

    b)

    Assume solution U(x,t) = V(x,t) + W(x)

    => Ut = Vt
    => Uxx = Vxx + W'' -- Sub into Ut = Uxx + Q(x)

    Vt = Vxx + W'' + Q

    Now, let W'' + Q = 0 or W'' = -Q
    => Vt = Vxx

    V(0,t) = 0
    V(L,t) = 0

    U(x,0) = V(x,0) + W(x) = f(x)
    => V(x,0) = f(x) - W(x)

    Now Solve transient solution, v(x,t)
    . .
    . .
    . .
    V(x,t) = (2/L)sum(n=1 to inf)(int((f(x)-W(x))sin(npi/L)xdx)... etc

    Now Solve steady state solution, W" = -Q B - X
    W" + X + B = 0
    . .
    . .
    W(x) = Scos(zx) + Tsin(zx) + Yp... etc

    Now U(x,t) = W(x) + V(x,t)

    Am I right at all???
     
  2. jcsd
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