MaplePrimes - answers and comments on Question, inverse LaPlace transform

inverse laplace transform

Robert Israel — Wed, 29 Dec 2010 05:02:59 Z

I could be wrong, but I doubt that the inverse laplace transform of exp(x*sqrt((a*s^2+b*s)/(1+c*s^2))) can be expressed in closed form. Do you have any reason to think that it can?

inverse laplace transform

tsunamiBTP — Wed, 29 Dec 2010 06:25:15 Z

Not really, but I have seen people employ the LaPlace transform in wave problems for elasticity to remove the time derivatives & solve for the spatial distribution & then invert back to the time domain. But I think they employ the Bromwich integral. I simply tried to use invlaplace or Cauchy residues to get the job done, but that does not work either,

I did not want to use the Fourier transform because I thought that biases my solutions to be of periodic form.

Is there a better approach?

inverse laplace transform

tsunamiBTP — Wed, 29 Dec 2010 06:52:02 Z

Actually if you inspect the ratio within the sq root sign you see for the extreme cases for the denominator for very small s where the unity term dominates the inverse would exist & vice versa for very large s it does as well. I played with it by simply removing terms from the denom. So there must be some approach to handloe cases for intermediate s values?

inverse laplace transform

tsunamiBTP — Wed, 29 Dec 2010 07:10:47 Z

1 more thought:

The Fourier transform obviously works because you simply substitue e^jwt in for T & that divides out, but again that biases my temporal solution. So I attempted what I have seen for solutions in longitudinal & shear wave problems in elastic media.

I suppose if there is another integral transform that would not bias my solution I could try that. I'll do some experimenting or try to find some literature on what others may have attempted.

When the inverse LaPlace transform does not exist can s decomposed into Re & Im components

tsunamiBTP — Fri, 31 Dec 2010 00:46:17 Z

See attached.

I decomposed s into its Re & Im components & expressed that in polar form & then took the inverse. MAPLE gave an answer but is this VALID?

If I do this decomposition instead of taking the inverse LaPlace should I simply do the inverse Fourier instead & multiply the result by exp(Re(s)*t)?

Does this yield a proper result for the time domain given the original governing eq for F posed in the 1st posting?

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Download inv_laplace_computat.mw

When the inverse LaPlace transform does not exist...

tsunamiBTP — Fri, 31 Dec 2010 00:56:10 Z

Another possibility, but is it valid?

Decompose s-->express it in polar form which gives an exponential with a Re exponent * expontial with an Im exponent. The invlaplace exists for both so then convolve them to get the proper answer?

I have no reference basis for this to test if this approach is valid, but it does give an answer.

appreciate any counsel!!