Homogenization and modeling of wave processes in composites with a periodic structure

Abstract

Homogenization and modeling of wave processes with damping in composite materials with a periodic structure, such as photonic crystals, is considered. It is taken into account that the corresponding wave equations depend on additional small parameters characterizing the microscale and permeability of the materials. The asymptotic expansions of solutions and homogenized problems, the solutions of which determine the asymptotic expansions, are given. Such problems are initial boundary value problems for integro-differential equations with convolutions. The presence of convolutions in equations that model processes in continuous media or materials is called the memory effect. Accuracy estimates for the expansions are presented, which allow to simplify the numerical solution and computer modeling for the wave processes.