Spectral approach to homogenization of hyperbolic equations with\n periodic coefficients

2017/08/02 by Mark Dorodnyi, Dorodnyi, Mark, T. A. Suslina +1
Computer Science · Engineering · Mathematics · #35B27 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1708.00859

openalex publication_date 2017/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In L₂(\ℝ^d;\ℂⁿ), we consider selfadjoint strongly\nelliptic second order differential operators mathcal A_\ε with\nperiodic coefficients depending on mathbf x/ \ε, \ε>0.\nWe study the behavior of the operators \cos( mathcal A^1/2_\ε\n\τ) and mathcal A^-1/2_\ε \sin( mathcal\nA^1/2_\ε \τ), \τ \∈ \ℝ, for small \ε.\nApproximations for these operators in the (H^s\→ L₂)-operator norm with a\nsuitable s are obtained. The results are used to study the behavior of the\nsolution mathbf v_\ε of the Cauchy problem for the hyperbolic\nequation \∂²_\τ mathbf v_\ε = - \A_\ε\n mathbf v_\ε +\F. General results are applied to the\nacoustics equation and the system of elasticity theory.\n

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