AbstractWe solve the Milne, constant-source, and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener–Hopf method. The Flatland H-function is derived a...
In the study of energy-dependent neutron transport in three-dimensional volumes with plane symmetry, @cite_0 presented a general family of solutions to the Milne problem using the method of singular eigenfunctions. When their variable- @math factor @math takes on the specific quantity (in our notation) @math , their energy-dependent 3D solution becomes equivalent to our monoenergetic Flatland solution. Thus, the Flatland @math -function has, if only inadvertently, been presented long ago.
@cite_0
1968399379 2964349548 2227626537 2035369868
Abstract Mathematical equivalence is demonstrated between problems for a separable scattering kernel in energy-dependent neutron transport and for the uniform line-blanketing model of nongrey radiative transfer. As an example, the solution of the Milne problem is rederived by the method of singular eigenfunctions, with the application of orthogonality relations. AbstractWe solve the Milne, constant-source, and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener–Hopf method. The Flatland H-function is derived a... Abstract Two classic problems of radiative transfer and neutron transport are solved for a spatially-uniform semi-infinite medium with isotropic scattering. General analytical equations are derived for (1) angular moments of the outward current and (2) spatial moments of the total flux within the half-space. Such moments, for example, can provide analytically explicit equations for determining the surface albedo of the medium as well as the mean depth and mean square distance of travel within the medium. The analysis is done with the Case-style eigenmode method as expressed in terms of the Chandrasekhar H -function and its moments. We study the flatland (two dimensional) linear transport equation, under an angular2@pperiodicity assumption both on particle density function @j(x,y,@q) and on the differential scattering @s"s(@q). We consider the beam problem, with a forward peaked source on phase-space, and derive P"1 approximation with a diffusion coefficient of 1 2@s"t"r, (versus 1 3@s"t"r of the three dimensional problem), where @s"t"r is the transport cross section. Further assumptions as peaked @s"s(@q) near @q=0 (small angle of scattering), and small angle of flight (@q 0) yield Fokker-Planck and Fermi approximations with the diffusion coefficients @s"t"r (rather than @s"t"r 2 of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples.