On Aspects of Continuous Approximation of Diatomic Lattice

Igor V. Andrianov (), Lelya A. Khajiyeva, Askar K. Kudaibergenov and Galina A. Starushenko
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Igor V. Andrianov: Chair and Institute of General Mechanics, RWTH Aachen University, Eilfschornsteinstrasse 18, D-52062 Aachen, Germany
Lelya A. Khajiyeva: Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, 71 Al-Farabi Ave., 050040 Almaty, Kazakhstan
Askar K. Kudaibergenov: Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, 71 Al-Farabi Ave., 050040 Almaty, Kazakhstan
Galina A. Starushenko: Department of Information Technology and Information Systems, Dnipro University of Technology, 19 Dmytra Yavornytskoho Ave., 49005 Dnipro, Ukraine

Mathematics, 2024, vol. 12, issue 10, 1-13

Abstract: This paper is devoted to the continualization of a diatomic lattice, taking into account natural intervals of wavenumber changes. Continualization refers to the replacement of the original pseudo-differential equations by a system of PDEs that provides a good approximation of the dispersion relations. In this regard, the Padé approximants based on the conditions for matching the values of the dispersion relations of the discrete and continuous models at several characteristic points are utilized. As a result, a sixth-order unconditionally stable system with modified inertia is obtained. Appropriate boundary conditions are formulated. The obtained continuous approximation accurately describes the amplitude ratios of neighboring masses. It is also shown that the resulting continuous system provides a good approximation for the natural frequencies.

Keywords: diatomic lattice; continualization; Padé approximants; model with modified inertia; normal dispersion; anomalous dispersion (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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