The Random Steady State Diffusion Problem. II: Random Solutions to Nonlinear, Inhomogeneous, Steady State Diffusion Problems
Georges A. Bécus, Francis A. Cozzarelli · SIAM Journal on Applied Mathematics · 1976
The steady state diffusion problem in a random medium under the influence of a random source term and a random Dirichlet boundary condition is considered. Assuming isotropy and separability of the inhomogeneous and nonlinear effects for the diffusion tensor, the problem is transformed into its normal form and then reformulated as an integral equation. A solution in terms of the random data, and based on our previous results concerning random solutions to Laplace’s equation, is defined. The properties of such a solution are established and indications on the inversion of the solution to the normal form are provided. In the Appendix the results are extended to encompass generalized diffusion problems.