Initial-Boundary Value Problem for the Non-Stationary Radiative Transfer Equation with Fresnel Reflection and Refraction Conditions

Initial-boundary value problem for a radiative transfer equation with generalized matching conditions

“…As relation (33) is also valid for the function f ∈ D, repeating the proof it is easy to verify the existence of the limit functions f + ± ∈ L ∞ (X + ). Passing to the limit in equality (33), we conclude that the functions f ± satisfy equation (31) with Φ = Φ ± . Hence, we have shown the existence of the solution f of equation (31) satisfying estimate (32).…”

“…Solvability of the stationary boundary-value problems with the generalized conjugation conditions describing the Fresnel or diffuse reflection and refraction at the interfaces of the media have been investigated in [19][20][21][22][23][24][25][26]. Problems for the non-stationary radiation transfer equations with the generalized matching conjugation conditions in a threedimensional bounded media and in a plane-parallel layered media have been considered in [27][28][29][30][31][32][33][34][35].…”

“…Consider the following integro-differential equation [29][30][31][32]34] (1) 1 v(r) ∂ ∂t + ω · ∇ r + µ(r) I(r, ω, t) = = σ(r) Ω p(r, ω · ω )I(r, ω , t)dω + J(r, ω, t).…”

“…where B f is the Fresnel matching operator describing the mirror reflection and refraction according to Snell's law on the media interfaces [20,22,29,31,34,35]:…”

“…and B d is the operator describing diffuse reflection and refraction [30,32,35] at the interfaces according to Lambert's law:…”