# Inverse laplace transform for unique diffusion type problem

1. Feb 20, 2012

### Groundwater

Hi all

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..

2. Feb 20, 2012

### kai_sikorski

The most difficult part if you try to do the inverse transform with integration in the complex plane will be this integral along the branch cut

$$\int_0^{\infty } \frac{e^{-t \rho } \left(e^{-i x \sqrt{\frac{\rho }{d}}}-e^{i x \sqrt{\frac{\rho }{d}}}\right)}{\beta +(-\alpha -\rho )^2} \, d\rho$$

Maybe you can find that in a table somewhere. I couldn't even find/figure out the case where $\beta = 0$. If you can't find the one with non-zero beta, but perhaps can find

$$\int_0^{\infty } \frac{e^{-t \rho } \left(e^{-i x \sqrt{\frac{\rho }{d}}}-e^{i x \sqrt{\frac{\rho }{d}}}\right)}{(-\alpha -\rho )^4} \, d\rho$$

Then if beta is small maybe you could expand as a series in beta?

3. Feb 21, 2012

### Groundwater

Thanks...I think I am a bit out of my league here. I once took an introductory course in ODE that introduced me to Laplace transforms. I have slowly figured out how to apply the technique in the solution of PDE. However, at my level, I am doing well if I can solve a solution using tabulated inverse Laplace transforms.

I did try to solve the inverse Laplace, but I am sure that I didn't even get as far as you. I may have to adjust my boundary condition.

4. Feb 21, 2012

### JJacquelin

Hi !

May be, you could try this :
Split the function into two terms, thanks to
1/(((s-α)^2)+B²) = c/(s-a -i B) - c/(s-a +i B)
c = 1/(2 i B)
The inverse Laplace of (1/(s-A))*exp(-C*sqrt(s)) can be found in tables.
But, there is a major difficulty, since A is complex : A=-a-i B or A=-a+i B
The formal result will involves Erfc functions in the complex range. The theoretical proof of validity should be rather ardous. Instead of doing it directly, I suggest to compute the Laplace transform of the result and check if it is consistent with the initial function F(s)

I suppose that beta>0 , so beta=B². If beta<0, then beta=-B² and there is no difficulty since A=-a-B or A=-a+B which are reals.

Last edited: Feb 21, 2012