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

Title	On the convergence of the nonlocal nonlinear model to the classical elasticity equation
Author	Erbay, Hüsnü Ata, Erbay, Saadet, Erkip, A.
Publication Date:	2021-12
Publication Place	- Elsevier
Subject	Nonlocal elasticity, Long wave limit, Discrete-to-continuum convergence, Lattice dynamics
Type	Periodical
Language	English
Digital	Yes
Manuscript	No
Library:	Özyeğin University
Library Asset ID	0167-2789
Record ID	c5a47cce-92db-4eed-bfee-578e31762b71
Library Location	Natural and Mathematical Sciences
Date	2021-12
Sample Text	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.
DOI	10.1016/j.physd.2021.133010
Cilt	427

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Özyeğin University

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

Author Erbay, Hüsnü Ata, Erbay, Saadet, Erkip, A.

Publication Date 2021-12

Publication Place - Elsevier

Subject Nonlocal elasticity, Long wave limit, Discrete-to-continuum convergence, Lattice dynamics

Type Periodical

Language English

Digital Yes

Manuscript No

Library Özyeğin University

Library Asset ID 0167-2789

Record ID c5a47cce-92db-4eed-bfee-578e31762b71

Library Location Natural and Mathematical Sciences

Date 2021-12

Sample Text 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.

DOI 10.1016/j.physd.2021.133010

Cilt 427