Included below is vector partial diff eq I am working with. To get rid of the time deriv's I took the LaPlace transform & the remaining spatial eq in the s-domain is listed. To make matters simpler I set beta = 0 to get rid of the curl of the field. What remains is essentially the Helmholtz eq. To simplify further I just found the homogeneous soln for the x direction only.
As can be seen the eigenfunctions are exponentials with s beneath a square root sign. I have not had any success at inverting back into the time domain. I also tried the residue theorem without success. Should I be employing a different transform or is there some trick to get the inversion to work with this solution form?
As far as I can tell I executed the forward transform correctly. The dimensions beneath the sq root sign is 1/length^2 which then multiplied by the lenght dimension x gives dimensionless units for the exponential. So I am reasonable confident there.
Suggestions???
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Download inv_laplace_computat.mw
