1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inhomogeneous PDE

  1. Apr 7, 2008 #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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted