A general analytical solution for the stochastic Milne problem using Karhunen-Loeve (K-L) expansion

- The solution of the stochastic Milne problem is considered.
- We dealt with the random cross-section itself not with the optical transformation of it.
- Pomraning-Eddington method together with the (K-L) expansion were implemented.
- The solution process is obtained as a functional of a set of uncorrelated random variables.
- Good results are obtained for different distributions of these variables.

This paper considers the solution of the stochastic integro-differential equation of Milne problem with random operator. The Pomraning-Eddington method is implemented to get a closed form solution deterministically. Relying on the spectral properties of the covariance function, the Karhunen-Loeve (K-L) expansion is used to represent the input stochastic process in the deterministic solution. This leads to an explicit expression for the solution process as a multivariate functional of a set of uncorrelated random variables. By using different distributions for these variables, the work is realized through computing the mean and the variance of the solution. The numerical results are found in agreement with those obtained in the literature.