Question

Is it possible to show, that this http://prntscr.com/2zbunw is a solution to the PDE http://prntscr.com/2zbuze
With fourier transform/Laplace transform.
If it's possible how should i start out.

Answer 1

I have a few observations while I look into this further:
(1) The original PDE does not specify any initial/boundary conditions? If I recall, Laplace transforms make use of that and I imagine the Fourier transform does too.
(2) The given solution seems like a particular solution (it does not have any variables except those from the initial equation.

Anyone more familiar with this can comment better than I can, though!

Answer 2

it's a particular solution for the probability density function P(x,t), with boundry conditions x=0 for small t. If im right, i can't find the exact boundry conditions.

Answer 3

I think you call this PDE, the heat equation btw

Answer 4

Yeah, 1D Heat Equation compares correctly to me.
Wirh Laplace Transform, I can see this:
\( \displaystyle \frac{\partial P}{\partial t} = D \frac{\partial ^2 P}{\partial x^2 } \)
\( \displaystyle \frac{\partial P}{\partial t} e^{-st} \; \text{d}t = D \frac{\partial ^2 P}{\partial x^2} e^{-st} \; \text{d} t \)
\( \displaystyle \mathcal{L} \left[ \frac{\partial P}{\partial t} \right] = s \mathcal{L} \left[ P \right] - P(x, 0) \)
and then \( \displaystyle \mathcal{L} \left[ D \frac{\partial^2 P}{\partial x^2} \right] = D \frac{\text{d}^2}{\text{d}x^2} \left( \mathcal{L} \left[ P \right] \right) \)
the key note to worry of is the appearance of P(x,0). We at least know t goes from 0 to infinity so right-hand limit as t goes to 0. That all seems reasonable to me.
(Am reading through this: http://science.uwaterloo.ca/~suwingspany/earth661/pdfs/6-1_ES661_W2008_06_Chapter_06.pdf)
Again, will say I have not done much with Laplace transform for partial derivatives / PDE so definitely go over anything I write carefully!

Answer 5

http://science.uwaterloo.ca/~suwingspany/earth661/pdfs/6-1_ES661_W2008_06_Chapter_06.pdf
original link has a ) at the end so it breaks.

Answer 6

Thank you, i just have to understand this, i havn't had any training in fourier transform or anything like that yet. I just read somewhere that it'd be easier to prove that P is a solution to the PDE with fourier transform, instead of doing the plug-n-chug method (inserting the solution into the PDE).

Answer 7

Ah, alright. No problem.
Have you had work with Laplace transform in the past, though? Fourier transform is for the most part like a special case of Laplace transform (that is, s = a + wi but a=0 for Fourier). I just have not used Fourier transforms as often as Laplace when I was going over it on my own time!

Answer 8

Got it. Anyways, best of luck with your studies! :)
I personally prefer plug n chug on anything because I just brute force things, but I do know Laplace/Fourier are powerful transforms in ODE/PDE once you know their properties!

Answer 9

Yea, i'll probably end out brute focing my way through this, i just thought it'd be cool to learn a new approach :)

Answer 10

thank you very much for your help :)

Answer 11

You're welcome!