Why dummy variables used for the coefficients of a PDE solution?

[AStaunton]

have been solving PDEs by sep of variables, and the solution that comes out is generally a summation the general look of it is something like:

U=SIGMA(n=1 to infinity)E_n(sin(n(pi)x/L)(cos(n)(pi)x/L)t

The above may not be exactly right, I was thinking along the lines of heat equation where U=temperature x=position and t=time..in most of the basic probs I am doing, a boundary condition is given something like U=f(t) at x=0 or something like that...and this BC is used to solve for E_n...

My question is why when we solve for what the E_n's are do we then use dummy variables to express it...in this case the dummy variable would by t^bar...

The details of what I've written above aren't correct I know, but I hope that that doesn't matter in terms of the question I am asking..

 

[LCKurtz]

have been solving PDEs by sep of variables, and the solution that comes out is generally a summation the general look of it is something like:

U=SIGMA(n=1 to infinity)E_n(sin(n(pi)x/L)(cos(n)(pi)x/L)t

The above may not be exactly right, I was thinking along the lines of heat equation where U=temperature x=position and t=time..in most of the basic probs I am doing, a boundary condition is given something like U=f(t) at x=0 or something like that...and this BC is used to solve for E_n...

More likely U(x,0) = f(x), which I will use...

My question is why when we solve for what the E_n's are do we then use dummy variables to express it...in this case the dummy variable would by t^bar...

The details of what I've written above aren't correct I know, but I hope that that doesn't matter in terms of the question I am asking..

So you would likely have something like

$$f(x) = \sum_{n=0}^{\infty}E_n\sin(n\pi x/L) \cdot 1$$

I'm not sure what dummy variable you refer to. At this point you have a simple Fourier series expansion for f(x) and the E_n are the Fourier coefficients. Are you referring to the dummy variable inside the integral for the coefficient E_n? E_n is just a constant...

 

[AStaunton]

yes, sorry I was unclear.

when we solve for E_n and plug into the final eigenfunction, we use x_bar (or some other variable) instead of x...
it has not been clear to my why this has to be done:

$$E_{n}=\frac{2}{L}\int_{0}^{L}f(\bar{x})\sin(\frac{n\pi\bar{x}}{L})d\bar{x}$$

 

[lurflurf]

The dummy variable does not matter. It helps to use a different one for two reasons. It reminds us not to use it ourside the integral and avoids confusion with the our main variable.