Browsing Faculty of Engineering by OzU Authors "Erbay, Hüsnü Ata"
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The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (AIMS, 2016-11)In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions ... -
The cauchy problem for a one-dimensional nonlinear elastic peridynamic model
Erbay, Hüsnü Ata; Erkip, A.; Muslu, G. M. (Elsevier, 2012-04-15)This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions ... -
Comparison of nonlocal nonlinear wave equations in the long-wave limit
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Taylor & Francis, 2020-11-17)We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear ... -
A comparison of solutions of two convolution-type unidirectional wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Taylor and Francis, 2023)In this work, we prove a comparison result for a general class of nonlinear dispersive unidirectional wave equations. The dispersive nature of one-dimensional waves occurs because of a convolution integral in space. For ... -
Convergence of a linearly regularized nonlinear wave equation to the p-system
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. K. (TÜBİTAK, 2023)We consider a second-order nonlinear wave equation with a linear convolution term. When the convolution operator is taken as the identity operator, our equation reduces to the classical elasticity equation which can be ... -
Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (EDP Sciences, 2018-09-13)In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. ... -
Derivation of generalized Camassa-Holm equations from Boussinesq-type equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Informa Group, 2016)In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive ... -
Derivation of the Camassa-Holm equations for elastic waves
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2015-06-05)In this paper we provide a formal derivation of both the Camassa–Holm equation and the fractional Camassa–Holm equation for the propagation of small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic ... -
Dispersive transverse waves for a strain-limiting continuum model
Erbay, Hüsnü Ata; Rajagopal, K. R.; Saccomandi, G.; Şengül, Y. (Sage, 2023-10)It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model ... -
Existence and stability of traveling waves for a class of nonlocal nonlinear equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2015-05-01)In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt−Luxx=B(±|u|p−1u)xx, p>1. The main characteristic of this class of ... -
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations
Babaoglu, C.; Erbay, Hüsnü Ata; Erkip, A. (Elsevier, 2013-01)This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem ... -
Instability and stability properties of traveling waves for the double dispersion equation
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2016-03)In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation View the MathML source for View the MathML source, View the MathML source. The ... -
Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity
Erbay, Hüsnü Ata; Erkip, A.; Şengül, Y. (Elsevier, 2020-11-15)In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized ... -
Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (American Institute of Mathematical Sciences, 2019-05)We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution ... -
Numerical computation of solitary wave solutions of the Rosenau equation
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2020-11)We construct numerically solitary wave solutions of the Rosenau equation using the Petviashvili iteration method. We first summarize the theoretical results available in the literature for the existence of solitary wave ... -
On the convergence of the nonlocal nonlinear model to the classical elasticity equation
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2021-12)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 ... -
On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (American Institute of Mathematical Sciences, 2017-06)We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional solutions of the improved Boussinesq equation tend to associated solutions of two ... -
On the full dispersion Kadomtsev–Petviashvili equations for dispersive elastic waves
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2022-09)Full dispersive models of water waves, such as the Whitham equation and the full dispersion Kadomtsev–Petviashvili (KP) equation, are interesting from both the physical and mathematical points of view. This paper studies ... -
A semi-discrete numerical method for convolution-type unidirectional wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2021-05-15)Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization ... -
A semi-discrete numerical scheme for nonlocally regularized KdV-type equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 2022-05)A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The class differs from the class of the nonlinear nonlocal unidirectional wave equations previously studied ...
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