Asymptotic analysis of the one-dimensional diffusion-absorption equation with rapidly and strongly oscillating absorption coefficient (Q2902751)

The paper studies the one-dimensional Helmholtz equation with rapidly oscillating large potential with a periodic support having a small measure of its intersection with the period NEWLINE\[NEWLINE- u'' +q \left(\frac{x}{\varepsilon}\right) u(x)=f(x)NEWLINE\]NEWLINE \(q(\xi)=\omega,\) if \(0<\xi<\frac{\delta}{2},\) or \(1-\frac{\delta}{2}<\xi< 1\) and \(q(\xi)=0,\) if \(\frac{\delta}{2}<\xi<1-\frac{\delta}{2}, \) \(q(\xi)=q(\xi+1)\) for all real \(\xi\). The dimensionless absorption coefficient \(q\) depends on three parameters: \(\varepsilon\) is the ratio of the period of the potential and the characteristic microscopic size, \(\delta\) is the ratio between the measure of the intersection of the support of \(q\) and the period, and \(\omega\) corresponds to the intensity of absorption. Here, the case when \(q\) takes only two values, \(\omega\) and 0, is considered.NEWLINENEWLINEThe equation under consideration 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. 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 \(\omega\). Both parameters \(\varepsilon\) and \(\delta\) are small. The classical high-order homogenization method is applicable only in the case \(\varepsilon^2\omega\delta\rightarrow 0\), \(\omega \delta\rightarrow \infty\), while, if \( \varepsilon^2\omega\delta\rightarrow\text{const}\) or \( \varepsilon^2\omega\delta\rightarrow \infty\), it does not work and the construction of an asymptotic approximation was an open problem. Here, an asymptotic expansion of the solution is constructed in all three cases. Three settings are considered, namely \({\mathbb R}\), periodic boundary conditions, and the Dirichlet boundary value problem.