I just want to see if I completed this problem right. Here is the problem:
Consider $\frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^2} -\alpha T$ where $k,\alpha >0$ are constants and $\partial_x(0,t) = \partial_x (L,t) = 0$ and $T(x,0) = f(x)$. Find the equilibrium temperature and T(x,t). Find the long time asymptotic limit of T and compare to the equilibrium temperature.
Here is my attempt at the problem:
Assume the solution is in the form $T(x,t) = F(x)G(t)$. Then, By Separation of Variables, I got
$(\lambda +\alpha) G \frac{\partial G}{\partial t} = \frac{k}{F} \frac{\partial^{2} T}{\partial x^2}=\lambda $, which lambda is the separation constant.I have examined the cases for $\lambda$, the cases for $\lambda = 0$, and $\lambda >0$, they would yield the trivial solution for F(x). For $\lambda <0$, I would have $F(x) = A \cos \sqrt{\frac{\lambda}{k} }x +B \sin \sqrt{\frac{\lambda}{k} }x$ which means
$F'(X) = \sqrt{\frac{\lambda}{k}} (-A \sin (\sqrt{\frac{\lambda}{k}}x) +B\cos (\sqrt{\frac{\lambda}{k}}x))$.
When I initialize the initial conditions, I got the general solution to be a sequence of functions, which is $F_n(x) = A_n \sin (\frac{n \pi x}{L}) $. For G(t), I got the function to be $G(t) = C e ^{-(\lambda -\alpha)t} $.
Since $\lambda = (\frac{n \pi x}{L})^2 k$. $G(t)$ is also generalized as a sequence of functions which is
$G_n(t) = C_n e^{((\frac{n \pi x}{L})^2 k - \alpha)t} $
Thus the general solution would be a linear sum of sequence of functions:
$T(x,t) = \sum_{n=1}^{\infty} a_n \sin (\frac{n \pi x}{L})e^{((\frac{n \pi x}{L})^2 k - \alpha)t}$
From the initial condition, $T(x,0) = F(x)$, this would turn out to be
$T(x,0) = \sum_{n=1}^{\infty} a_n \sin (\frac{n \pi x}{L})=F(x) $ which is a Fourier Sine Series representation.
Am I on the right track on this? Also, for the equilibrium temperature, this is where we set $ \frac{\partial T}{\partial t} = 0$ and solve. What I got for the equilibrium temperature function was
$F(x) = A e^\sqrt{\frac{\alpha}{k}x} +B e^{-\sqrt{\frac{\alpha}{k}x}}$. From the boundary conditions, I got the solution was the trivial solution. Did I do this correctly?
Also, for the coefficients, $a_n$, after you are done solving them, do you have to substitute it in into $T(x,t)$?
Thank you for all of your help.