The Heat Equation
Find the Steady State Solution and state the Transient Problem.
a) Boundary Conditions:
b) Boundary Conditions:
I've handled conditions of the first kind, u(x,t) = c. But I have not dealt with either of these before. Any help would be appreciated.
Are you sure you're solving my problem??
I'm supposed to be finding the steady state solution... Which, as far as I've seen, does not have the same form as , given that when you want to solve by splitting u into products you get:
And when you want a Steady State and Transient solution you get:
I'm wanting to solve for v(x), and then produce a new differential equation with w(x,t) as opposed to u(x,t).
Perhaps this is a new way of asking for a rod with constant temperature at the ends...I'll go through a general example, see if it is applicable here.
Go through the normal seperation process
Getting this into the form of the heat equation we get
This is the regular heat equation if
Now fit this with the boundary conditions. So we have..
Now, let us note that
This is true when
Therefore, our new heat equation becomes
I'm not sure if this will help...but again, i'm confused at your original question. Perhaps it's a form I have never seen either but from what you described it doesn't seem to model what I have above (which is when we use what you suggested)
I've already solved a problem like that...
I'm stuck just like you are.... I don't need to solve the equation with a product solution and I don't need the eigenvalues of either the transient solution or the original solution.
I simply need a steady state equation and a statement of the new transient problem.
Since I've never dealt with this before, it could very well be impossible without initial conditions on u itself instead of all initial conditions relying on derivatives...
But I do not know.
Yeah, this isn't a standard condition. So....yeah, i'm fresh outta ideas. I'm not sure what else you can do here...
Originally Posted by Aryth