Hi NFuller, thank you for your answer!
I have to admit that it has nothing to do with it. :)
However, when I got to the point of defining the ##C(\omega)## as ##c_1(\omega)## and ##c_2(\omega)## I questioned myself about how these two different constants could be rearranged to get the situation...
Hi everyone, I'm reading this paper about the solution of the heat equation inside an infinite domain: https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/fourtran.pdf
1) Please let me know if the following discussion is correct.
The...
Hi Orodruin,
I got what you were saying and I realize that I made a mistake in my last reply. I supposed that the solution of the heat equation was already written with a single exponential, while I think it should be written as $$\Psi (x,t)= \int_{0}^{+\infty} [c_{1}(\omega) e^{i\omega x}+...
Wonderful, now I got what you meant! Your solution is in the general case where I start from a complete Fourier transform and I don't have constraint on the spectrum.
In the case discussed inside the paper, we agreed that the solution, before applying the boundary conditions and keeping the...
I'm sorry but I don't understand, but I admit I'm not very used "thinking" and working with the sine and cosine transforms, my bad. Even in the discussions before I wasn't exactly following you when you introduced the ##\omega## sign change for the coefficients...
I think that boundary conditions are the key to my problem. From your reply and from what I read, the separation of variable technique starts by separating variables and then analyze AS FIRST EQUATION the one that has homogeneous boundary conditions. The wave equation for ##\Psi (x,t)## where...
I'm really sorry Orodruin, but I'm stucked with this idea that a singular SL problem solution is written as a Fourier transform, hence using a superposition of single positive exponentials (with the integral going from ##-\infty## to ##+\infty##). That's what I read on many books. And since both...
I think I expressed myself badly. My doubt is about the difference between the solution that I would write if I'd follow the rule written in Snider (i.e. using the Fourier transform as a solution of every separated equation) - so obtaining $$\Psi(x,t)=\int_{-\infty}^{+\infty} A(\omega) e^{i...
This now confuses me. From what I read in Snider, the solution to ##\frac{X''}{X}=\lambda## over an infinite domain is given by the superposition of single exponentials (that is the Fourier transform), as in this picture: https://ibb.co/ekFq7G
Thank you for your answer.
But how come (in the 3D case) it is ok to keep just ##\int_{-\infty}^{+\infty}e^{i\omega x}## instead of ##\int_{-\infty}^{+\infty} (e^{i\omega x}+e^{-i\omega x})## for the spatial variables and not for the time variable? And why in the 1D case I get the two...
Hi everyone,
I'm reading about the solution of the wave equation in free space on Stratton - Electromagnetic Theory and Snider - PDE and I got a little confused. The wave equation in 3D (plus time) is the following $$\frac{\partial^{2} \Psi (x,y,z,t)} {\partial t^{2}}=\nabla ^{2}\Psi...
Hi everyone,
I have some knowledge of Hilbert spaces and Functional Analysis and I have the following question.
I ofter have read that "Fourier transform diagonalize the convolution operator". So, we can say that for LTI systems (that can always be described with a convolution and "live" in...
Ok, so from what I understand I can get different solutions of an ODE/PDE depending on the hypothesis of the space I'm working with.
If I suppose to work with functions in L^2(-\infty, +\infty), then the Fourier transform is valid and I can exploit the fact that the unknown function x(t) in a...
Hi Jason, thank you very much for your answer.
I believe I found an answer to my doubt in this thread: https://dsp.stackexchange.com/questions/31011/why-are-fourier-analysis-and-transform-only-applicable-for-lti-systems and I'll try to summary the answer.
LTI systems are a special case of linear...