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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.