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I have been working on some unique solutions to advection-diffusion type problems.

One inverse Laplace transform that I seem to continue to encounter is the following:

Inverse Laplace[F(s)] where F(s)=[(1/(((s-α)^2)+β)*exp(-x*sqrt(s/D))]

In their classic 1959 text, Carslaw and Jaeger gave an inverse solution to F(s) for the case when β=0. It is an erfc based solution (as the exp(-x*sqrt(s/D)) would indicate).

If anyone has seen anything like this, please let me know. I have checked the following:

1. The Laplace Transform by Widder

2. Laplace Transforms and Applications by Watson

3. CRC Handbook of Tables for Mathematics, 4th edition

4. Table of Laplace Transforms by Roberts and Kaufman

5. Conduction of Heat in Solids by Carslaw and Jaeger

6. Analytical Solution to the one-dimensional advective-dispersive solute transport equation by van Genuchten and Alves

7....and a pile of elementary PDE Books

It may have to go back to first principles on this one..

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# Inverse laplace transform for unique diffusion type problem

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