Solve Non-Periodic Signal Equation: (1-\nabla^2)B

[Heimdall]

Hi,

I'm looking for an efficient algorithm to solve this kind of equation :

$$S = (1-\nabla^2)B$$

where both S(x,y) and B(x,y) can both be non-periodic functions. We know S and want to find out what is B.

I was wondering if there was a 'well known' method to solve this kind or problem in the case where both S and B are non-periodic functions...

I've started to write something...

Let $$S=S_0+S_*$$ where S0 and S* are non periodic and periodic functions respectively. I take S0 such that I have $$S_0 = S$$ on the boundaries of my domain, so S* is null there.

you have : $$S_* = (1-\nabla^2)(B+S_1)$$

with $$-(1-\nabla^2)S_1 = S_0$$

You can Fourier transform and obtain :

$$\mathcal{F}\left(S_*\right)= \mathcal{F}\left((1-\nabla^2)(B+S_1)\right) = (1+k^2) \mathcal{F}(B+S_1)$$

so that you can find :

$$B = \mathcal{F}^{-1}\left(\frac{\tilde{S_*}}{1+k^2}\right) - S_1$$

you can then have a solution of the problem by finding the analytical easy-to-integrate function S0.

In 1D it seems ok, but in 2D S0 must have the correct values on all borders which seems a bit complicated...

 

[fresh_42]

Looks like $B=S+c_1\exp(B) +c_2\exp(-B)$.