Thread: 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.

Originally Posted by spearfish

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:



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



Now use the boundary conditions to deduce and

CB

Because is , the is a function of the only and the PDE becomes the following ordinary second order initial values problem...

,

Kind regards

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.