MHB Equilibrium Soln for $u_{t}=K u_{xx}+\gamma$

[Markov2]

Well I don't know if is the correct term for this for here goes:

Let

$\begin{align}
& {{u}_{t}}=K{{u}_{xx}}+\gamma ,\text{ }0<x<L,\text{ }t>0, \\
& u(0,t)=\alpha ,\text{ }u(L,t)=\beta ,\text{ }t>0, \\
& u(x,0)=0,
\end{align}
$

where $\alpha,\beta,\gamma$ are constant, then find the equilibrium solution. I don't know what I need to do.

 

[Dustinsfl]

Well I don't know if is the correct term for this for here goes:

Let

$\begin{align}
& {{u}_{t}}=K{{u}_{xx}}+\gamma ,\text{ }0<x<L,\text{ }t>0, \\
& u(0,t)=\alpha ,\text{ }u(L,t)=\beta ,\text{ }t>0, \\
& u(x,0)=0,
\end{align}
$

where $\alpha,\beta,\gamma$ are constant, then find the equilibrium solution. I don't know what I need to do.

To find the equilibrium solution $U(x)$, set

$0=Ku_{xx}+\gamma$
$U(0)=\alpha$
$U(L)=\beta$

 

[Markov2]

So I get $u(x)=-\dfrac\gamma{2K} x^2,$ does this make sense? Why the boundary conditions aren't something like $u(0,t)$ ?

 

[Dustinsfl]

Why the boundary conditions aren't something like $u(0,t)$ ?

We removed the t parameter.

---------- Post added at 02:45 PM ---------- Previous post was at 02:43 PM ----------

So I get $u(x)=-\dfrac\gamma{2K} x^2,$ does this make sense?

Shouldn't you have a constant of integration?

 

[Markov2]

Shouldn't you have a constant of integration?

Oh yes, yes, so then I just put the initial conditions and that's it?