Separation of variables - Getting the Fourier coefficients

[NicolaiTheDane]

TL;DR

Using separation of variables I have found a Fourier series to solve a PDE. How do I find the coefficients?

Hey there!

I am current taking an introductory course on PDE's, and our professor hasn't really emphasized last part of solutions from separation of variables. Now its not strictly going to be on the exam, however I remember doing this with ease a few years back, but for some reason now I simply cannot recall, or translate my books explanation into action. This below is the solution

$$u(x,y,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}C_{m,n}\cdot e^{-(m^2+n^2)\cdot t} \cdot \sin(m \cdot x) \cdot \cos(n \cdot y)$$

with $(x,y,t) \in (0,\pi) \times (0,\pi) \times (0,\infty)$. I need an expression for $C_{m,n}$, which seems pretty clear to me is the Fourier Coefficients. My attempt is to use the Fourier Transform:
$$C_{m,n}=\frac{1}{\pi} \cdot \int_{0}^{\pi} \frac{1}{\pi} \cdot \int_{0}^{\pi}u(x,y,t) \cdot e^{-i \cdot m \cdot x} \cdot e^{-i \cdot n \cdot y} dydx$$
However this seems odd. Can someone enlighten me?

 

[Orodruin]

Your coefficients $C_{n,m}$ cannot depend on time. You need to use the initial condition to fix them.

 

[NicolaiTheDane]

Interesting, as the assignment I'm doing atm, askes for en expression of for $C_{m,n}$, before going on to ask for the use of initial conditions in a later sub assignment. If I had taking time into account. Then what do I do?

 

[Infrared]

I think there's a typo in your expression for $u$ since $y$ does not appear in the RSH. Could you post the problem that you're working on?

It's possible that you're looking for a solution of the form $\sum_{n,m} c_{nm}(t)X_n(x)Y_m(y)$ but in the expression that you have, it looks likely that the $c_{nm}$ are constants since a typical function of $t$ already appears.

 

[NicolaiTheDane]

I think there's a typo in your expression for $u$ since $y$ does not appear in the RSH. Could you post the problem that you're working on?

It's possible that you're looking for a solution of the form $\sum_{n,m} c_{nm}(t)X_n(x)Y_m(y)$ but in the expression that you have, it looks likely that the $c_{nm}$ are constants since a typical function of $t$ already appears.

Yes I have made a correction. There should be a y in the cos function. $C_{m,n}$ shouldn't just look like it, they are constants, depending entirely on n and m. Also that is the problem I'm working on.

EDIT: If you wanted the original, problem here it is: