A developed solution of the stochastic Milne problem using probabilistic transformations |
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Authors: | A Hussein MM Selim |
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Institution: | a Basic Sciences Department, Higher Technological Institute, 10th Ramadan City 228, Egypt b Physics Department, Faculty of Science, Mansoura University, New Damietta 34517, Damietta, Egypt |
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Abstract: | This paper considers the solution of the Milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium. Properties of the medium are assumed to be continuous random functions of the spatial dimensions. The available solutions - in literature - for this stochastic integro-differential equation (SIDE) are represented only by the ensemble average of the radiant energy density. In this paper, a developed algorithm, based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties, is introduced. A complete stochastic solution represented by the probability-density function (p.d.f) of the radiant energy density is obtained. Using the closed form of the p.d.f, the nth moment of the stochastic solution is evaluated. In realization of this work, Exponential and Gaussian statistics for the medium properties are assumed. Results are physically acceptable and found to be compatible with those in the literature. |
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Keywords: | Milne problem Continuous stochastic medium Stochastic integro-differential equation (SIDE) Integral transformation Random variable transformation (RVT) Extrapolation distance Gaussian variations Exponential variations |
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