tsunamiBTP

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node1.html

Why MAPLE does not seem to cooperate I will need to experiment further.

I experimented with known functions with known transforms & what I proposed does not seem to work.

What I do not understand is the practice of employing the LaPlace transform to remove the time deriv's from anything other that an ORDINARY diff eq.  I have seen it done at least in practice of elasticity.  In electromagnetism they primarily stick with the fourier transform since they presume the signal is periodic in time.

If anyone...

Examination of the Bromwich integral vs invfourier they appear identical except the Bromwich integral contains a Re component in its integration bounds; whereas, the Re part for the invFourier = 0.

The logic seems to support the approach I mentioned earlier.

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!!

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?

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.

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?

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.

I guess I need to familiarize myself with pdsolve & its output format if I am going to continue working problems like this!!!!!

Your output I can interpret.

I sometimes have difficulty with interpreting MAPLE output, but my result does not appear anything like yours.  Maybe it is telling me the same thing, but am I in some ODD display mode that causes MAPLE to display the answer as it is below or is it indeed giving me something different from yours?

Look at pdsolve(Q2), I think I put in the correct equation since I copied it from your posting?

Robert,

OK, I agree the eq is nonlinear, but what if I need to consider that term of the gradient magnitude.  Would it be more appropriate to move the LaPlacian & the zeroth order term of X to the other side of the eq & square it to get rid of the sq root?  Then you have a Helmholtz eq that is squared & you pick a value for the gradient that is considered a nominal value for the gradient magnitude for a defined region.

Then the Helmholtz...