Results 1 to 4 of 4

Math Help - PDE: Finding the steady-state temp

  1. #1
    Junior Member
    Joined
    Apr 2009
    Posts
    70

    PDE: Finding the steady-state temp

    Hey guys,

    I am trying to find the steady state temp for the following Partial Diff. Eqn:

    Ut = (alpha)^2 * Uxx - (beta)U

    IC: 0<x<1

    BCs: U(0,t) = 1 and U(1,t) = 1.

    I know how to approach the problem, but I am getting stuck on the Integrating U. Here is what I have:

    1.) SETUP OF PROBLEM
    steady-state => Ut = 0, so
    (alpha)^2 * Uxx - (beta)U = 0
    (alpha)^2 * Uxx = (beta)U

    2.) INTEGRATION/SOLVE
    Integral((alpha)^2 * Uxx)dx = Integral ((beta)U)dx
    ((alpha)^2 * Ux = (beta) * How do I do [Integral(U) dx] ????

    This is where I get stuck, so if you could please show me how to Integrate(U) dx or what I need to do to fix this? Thanks for any and all help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by spearfish View Post
    Hey guys,

    I am trying to find the steady state temp for the following Partial Diff. Eqn:

    Ut = (alpha)^2 * Uxx - (beta)U

    IC: 0<x<1

    BCs: U(0,t) = 1 and U(1,t) = 1.

    I know how to approach the problem, but I am getting stuck on the Integrating U. Here is what I have:

    1.) SETUP OF PROBLEM
    steady-state => Ut = 0, so
    (alpha)^2 * Uxx - (beta)U = 0
    (alpha)^2 * Uxx = (beta)U

    2.) INTEGRATION/SOLVE
    Integral((alpha)^2 * Uxx)dx = Integral ((beta)U)dx
    ((alpha)^2 * Ux = (beta) * How do I do [Integral(U) dx] ????

    This is where I get stuck, so if you could please show me how to Integrate(U) dx or what I need to do to fix this? Thanks for any and all help.
    In the steady state you have:

    \alpha^2 u''(x) =\beta u(x)

    If you are familiar with ODE's you should know that the general solution of this second order constant coefficients homogeneous ODE is:

    u(x)=A e^{\sqrt{\beta}/\alpha x}+Be^{-\sqrt{\beta}/\alpha x}

    Now use the boundary conditions to deduce A and B

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Because is U_{t} =0 , the U(*,*) is a function of the only x and the PDE becomes the following ordinary second order initial values problem...

    \alpha^{2}\cdot \frac{d^{2}U}{dx^{2}} - \beta\cdot U= 0 , U(0)=U(1)=1

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    Thanks a bunch CaptainBlack and Chisigma. I had to review some ODE notes from a while back, but I am able to work out the problem now. Thanks again.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Steady State Numerical ODEs
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: September 10th 2010, 04:46 AM
  2. Help with steady-state unemployment
    Posted in the Advanced Applied Math Forum
    Replies: 6
    Last Post: March 31st 2010, 04:13 PM
  3. steady state
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: March 20th 2010, 01:38 AM
  4. Steady State Vector
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 4th 2009, 06:47 PM
  5. finding a steady state matrix
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 28th 2009, 11:59 AM

Search Tags


/mathhelpforum @mathhelpforum