I'm curious to know if anyone knows of the inverse Fourier transform of the function depicted in the attached picture. The transform variable is s and all other variables are constants.
I would be much obliged for any answer.
By using Mathcad software, you can easily get as shown in the attached file
That's not an answer... I do know the definition for an inverse Fourier transformation, but there seems to be no known analytic solution to the integral.
Thank you very much for your efforts.
Hello Luisiana Cundin ,
Can you tell us the problem or type of problem that gave rise to this inverse Fourier transform?
Regards,
Tom Cuff
Thomas Cuff
Sure, the basis or source of the problem is hyperbolic partial differential equations, essentially, wave equations. I'm working on solving non-linear wave equations. The function shown is simply the solution for a wave with both time and spatial dispersion added. If we seek a simpler solution, then e^{2\pi ist} would be the solution, whose inverse Fourier transform is the Dirac Delta function, hence, it describes a wave impulse traveling in time and space. The non-linear portion of the problem is a bit more complicated.
I know complex integration techniques, but I do believe -- looking and playing with this function -- that intractable integrals arise, therefore, I know of no known analytic solution to the inverse Fourier transform for the function shown in my question. I believe there is no known solution. I've scoured the tables for integrals and Fourier transform pairs to no avail!
Cheers!
Hello Luisiana Cundin ,
Thank you for the explanation. Are you aware of the discussion of this class of problems in the book by Truesdell, see below. Perhaps some of his citations, or the citations within these citations, may be of some help to you.
[1] C. Truesdell; An Idiot's Fugitive Essays on Science; Springer-Verlag; 1984; pp. 623-625.
Regards,
Tom Cuff
Thomas Cuff
I totally agree with everything said in the pages given. I am not a proponent of computer assisted calculations. Too many do not understand the accumulation of round-off error, how discretization of the problem causes a totally new phenomena to be considered and whether or not these exotic systems describe anything at all, which destroys any hope for accurate simulation, especially, in the case for non-linear descriptions. Good show!
I am also aware of the silliness of adding another derivative, for the sake of adding it. What I am developing is a method for solving these systems and am not concerned with what they may describe. That may seem disingenuous, even flighty, at first, but my aim is to provide a mathematical method for solutions.
From earlier studies of non-linear PDEs, I know these systems actually do suffer a catastrophe, especially, as the order of non-linearity is raised, in fact, as the order of non-linearity approaches infinity, the system approaches the simpler non-linear system in behavior and solution.
Cheers!
Thomas Cuff
...and, it may be that Airy functions or Meijer-G functions would result from careful inspection of the function's inverse Fourier transform; but, such series solutions are dissatisfying on so many levels, in fact, I consider these types of solutions as "non-solutions," because you are stuck with an analytic representation in series form. The reason it is more than likely a Bessel function -- or something similar -- is that the function must represent a wave spread out over time and space. This would be dissatisfying on so many levels.
If there is a closed-form, compact, analytic solution to the problem proposed, then I would accept such a result, because formulation, calculation would be simple.
