On the convergence of the nonlocal nonlinear model to the classical elasticity equation

Erbay, H. A. and Erbay, S. and Erkip, Albert (2021) On the convergence of the nonlocal nonlinear model to the classical elasticity equation. Physica D: Nonlinear Phenomena, 427 . ISSN 0167-2789 (Print) 1872-8022 (Online)

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Abstract

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional propagation of nonlinear waves in a continuous medium. In the limit of vanishing nonlocality we study the behavior of solutions to the Cauchy problem. We prove that, as the kernel functions of the convolution integral approach the Dirac delta function, the solutions converge strongly to the corresponding solutions of the classical elasticity equation. An energy estimate with no loss of derivative plays a critical role in proving the convergence result. As a typical example, we consider the continuous limit of the discrete lattice dynamic model (the Fermi–Pasta–Ulam–Tsingou model) and show that, as the lattice spacing approaches zero, solutions to the discrete lattice equation converge to the corresponding solutions of the classical elasticity equation.
Item Type: Article
Uncontrolled Keywords: Discrete-to-continuum convergence; Lattice dynamics; Long wave limit; Nonlocal elasticity
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Albert Erkip
Date Deposited: 28 Aug 2022 20:57
Last Modified: 28 Aug 2022 20:57
URI: https://research.sabanciuniv.edu/id/eprint/43774

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