Homogenization of hyperbolic equations with periodic coefficients

2016/06/19 by Mark Dorodnyi, Dorodnyi, Mark, T. A. Suslina +1
Computer Science · Engineering · Mathematics · #35L99 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1606.05868

openalex publication_date 2016/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In L₂(ℝ^d;ℂⁿ) we consider selfadjoint strongly elliptic second order differential operators \mathcal A_ε with periodic coefficients depending on \mathbf x/ ε, ε>0. We study the behavior of the operator cosine cos( \mathcal A^1/2_ε τ), τ∈ ℝ, for small ε. Approximations for this operator in the (H^s→ L₂)-operator norm with a suitable s are obtained. The results are used to study the behavior of the solution \mathbf v_ε of the Cauchy problem for the hyperbolic equation ∂²_τ\mathbf v_ε = - A_ε \mathbf v_ε +F. General results are applied to the acoustics equation and the system of elasticity theory.

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