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

In summary, the conversation discusses an efficient algorithm for solving a non-periodic equation involving functions S and B. The suggested method involves using a Fourier transform and finding a solution for B using an analytical function S0. The challenge lies in finding the correct values for S0 on all borders in 2D.
  • #1
Heimdall
42
0
Hi,

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

[tex]S = (1-\nabla^2)B[/tex]

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

you have : [tex]S_* = (1-\nabla^2)(B+S_1)[/tex]

with [tex] -(1-\nabla^2)S_1 = S_0 [/tex]

You can Fourier transform and obtain :

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

so that you can find :

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

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...
 
Last edited:
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  • #2
Looks like ##B=S+c_1\exp(B) +c_2\exp(-B)##.
 

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

1. What is the meaning of the "non-periodic signal equation"?

The non-periodic signal equation refers to a mathematical expression used to describe a signal that does not repeat itself at regular intervals. This type of signal is common in real-world situations, such as in audio or video signals, where the signal values change continuously over time.

2. What does (1-∇2)B represent in the equation?

In this equation, (1-∇2)B represents the Laplace operator applied to the signal B. The Laplace operator is a mathematical operator used to describe the change in a function over space or time.

3. How is the non-periodic signal equation solved?

The non-periodic signal equation is typically solved using numerical methods, such as finite difference methods or spectral methods. These methods involve discretizing the signal and solving the resulting system of equations using algorithms.

4. What are some applications of solving non-periodic signal equations?

Solving non-periodic signal equations is important in understanding and analyzing various physical phenomena, such as heat transfer, fluid dynamics, and electromagnetic fields. It is also used in various engineering fields, such as signal processing and image analysis.

5. Are there any limitations to solving non-periodic signal equations?

One limitation of solving non-periodic signal equations is that it can be computationally expensive, especially for complex signals or systems. Additionally, these equations may not accurately capture all aspects of a real-world system, leading to potential errors in the solution.

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