ASYMPTOTIC ANALYSIS OF THE ONE-DIMENSIONAL DIFFUSION-ABSORPTION EQUATION WITH RAPIDLY AND STRONGLY OSCILLATING ABSORPTION COEFFICIENT.

The Helmholtz equation with rapidly oscillating absorption coefficient and constant scattering (diffusion) coefficient is considered. It models the light absorption in a tissue containing a periodic set of thin blood vessels; it is assumed that the absorption takes place within these vessels only. So, the scattering coefficient is supposed to be constant while the absorption coefficient is equal to zero everywhere except for a periodic set of thin parallel strips simulating the blood vessels, where it is equal to the large parameter ω. Two other parameters appear in the problem: ε is the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and δ, which is the ratio of the thickness of thin vessels and the period. Both parameters ε and δ are small. The one-dimensional setting is considered. The classical high order homogenization method is applicable only in the case ε2ωδ → 0, ωδ →∞, while if ε2ωδ → const or ε2ωδ →∞, it doesn't work and the construction of an asymptotic approximation was an open problem. Here we construct an asymptotic expansion of the solution in all three cases. Three settings are considered: in ℝ, the periodic boundary conditions, and the Dirichlet boundary value problem. [ABSTRACT FROM AUTHOR]