Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation

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Abstract
We consider initial−boundary value and boundary value problems for transport equations in inhomogeneous media. We consider the case when the mean free path is small compared to typical lengths in the domain (e.g., the size of a reactor). Employing the boundary layer technique of matched asymptotic expansions, we derive a uniform asymptotic expansion of the solution of the problem. In so doing we find that in the interior of the domain, i.e., away from boundaries and away from the initial line, the leading term of the expansion satisfies a diffusion equation which is the basis of most computational work in reactor design. We also derive boundary conditions appropriate to the diffusion equation. Comparisons with existing results such as the asymptotic and P1 diffusion theories, the PN approximation, and the extrapolated end point condition for these approximations, are made. Finally the uniform validity of our expansions is proved, thus yielding the desired error estimates.