Analytical solution of the Photon Diffusion Equation

[Stearge]

Homework Statement

Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique.

Homework Equations

Starting equation integrated over t:
int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1)
Result:
sqrt(r^2/(D*c))*exp(-sqrt(a*c)*sqrt(r^2/(D*c)))*D*c^2/(4*r^2*Pi*sqrt(D^3*c^3)) (2)
Integral to find transmission:
int(exp(-sqrt(a*c*(1/(D*c)))*sqrt(x^2+y^2))/(4*Pi*D*sqrt(x^2+y^2)), x = -infinity .. infinity) (3)

The Attempt at a Solution

I have started with homogeneous solution calculating flux at a given point and a given time (1).
First, I integrated it over the time to get the time independent solution as it can be seen above. It gave me the flux at any given point independent of time (2).
The second thing I wanted to obtain from it is transmission, which I think should be obtainable by converting to Cartesian coordinates (r^2 = x^2+y^2) and integrating over x while keeping y constant (slab thickness) (3).
Here I am a bit stuck. I was trying to find table integrals or alternative solutions (I think there is a way to represent it using Bessel function). I would be very grateful if someone could help with this integral or point me towards the textbook with good derivation.

 

[Charles Link]

$\\$ Equation (3) looks incorrect to me. $\\$Why don't you simply integrate equation (2) over $dx dy=2 \pi r \, dr$ to get the final result?

 

[Stearge]

$\\$ Equation (3) looks incorrect to me. $\\$Why don't you simply integrate equation (2) over $dx dy=2 \pi r \, dr$ to get the final result?

Thank you for reply
I am integrating only over dx because y is thickness of the slab and I want to find number of photons on output facet.

 

[Charles Link]

I'll need to study it further. Usually in these Optics problems, "z" is the direction of propagation, and that is evidenced by your equation for the intensity $I(x,y)$ as the number of photons (or energy) per unit time per unit area is symmetric in "x" and "y". Check your equation again and see if the slab thickness isn't in the "z" direction.