Eur. Phys. J. Plus (2017) 132: 505
https://doi.org/10.1140/epjp/i2017-11763-6

Regular Article

Solution of the finite Milne problem in stochastic media with RVT Technique

¹ Theoretical Physics Group, Physics Department, Faculty of Science, Damietta University, 34517, New Damietta City, Egypt
² Mathematics Department, Faculty of Science, Damietta University, 34517, New Damietta City, Egypt

^* e-mail: mselim2000@yahoo.com

Received: 20 August 2017
Accepted: 27 October 2017
Published online: 5 December 2017

Abstract

This paper presents the solution to the Milne problem in the steady state with isotropic scattering phase function. The properties of the medium are considered as stochastic ones with Gaussian or exponential distributions and hence the problem treated as a stochastic integro-differential equation. To get an explicit form for the radiant energy density, the linear extrapolation distance, reflectivity and transmissivity in the deterministic case the problem is solved using the Pomraning-Eddington method. The obtained solution is found to be dependent on the optical space variable and thickness of the medium which are considered as random variables. The random variable transformation (RVT) technique is used to find the first probability density function (1-PDF) of the solution process. Then the stochastic linear extrapolation distance, reflectivity and transmissivity are calculated. For illustration, numerical results with conclusions are provided.