Asymptotic Analysis of the One-Dimensional Diffusion-Absorption Equation with Rapidly and Strongly Oscillating Absorption Coefficient
Alexander Elbert, Grigory P. Panasenko · SIAM Journal on Mathematical Analysis · 2012
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 $\omega$. Two other parameters appear in the problem: $\varepsilon$ is the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and $\delta$, which is the ratio of the thickness of thin vessels and the period. Both parameters $\varepsilon$ and $\delta$ are small. The one-dimensional setting is considered. The classical high order homogenization method is applicable only in the case $\varepsilon^2\omega\delta\to 0, \omega\delta\to \infty$, while if $\varepsilon^2\omega\delta\to const$ or $\varepsilon^2\omega\delta\to \infty$, 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 $\mathbb{R}$, the periodic boundary conditions, and the Dirichlet boundary value problem.