Browsing by Author "Erkip, A."
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The CamassaHolm approximation to the double dispersion equation for arbitrarily long times
Erbay, Saadet; Erkip, A.; Kuruk, G. (Springer, 202209)In the present paper we prove the validity of the CamassaHolm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an ... 
The CamassaHolm equation as the longwave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (AIMS, 201611)In the present study we prove rigorously that in the longwave 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 onedimensional nonlinear elastic peridynamic model
Erbay, Hüsnü Ata; Erkip, A.; Muslu, G. M. (Elsevier, 20120415)This paper studies the Cauchy problem for a onedimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local wellposedness and smoothness of the solutions ... 
Comparison of nonlocal nonlinear wave equations in the longwave limit
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Taylor & Francis, 20201117)We consider a general class of convolutiontype 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 convolutiontype 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 onedimensional waves occurs because of a convolution integral in space. For ... 
Convergence of a semidiscrete numerical method for a class of nonlocal nonlinear wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (EDP Sciences, 20180913)In this article, we prove the convergence of a semidiscrete 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 CamassaHolm equations from Boussinesqtype equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Informa Group, 2016)In this paper we derive generalized forms of the CamassaHolm (CH) equation from a Boussinesqtype equation using a twoparameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive ... 
Derivation of the CamassaHolm equations for elastic waves
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 20150605)In this paper we provide a formal derivation of both the Camassa–Holm equation and the fractional Camassa–Holm equation for the propagation of smallbutfinite amplitude long waves in a nonlocally and nonlinearly elastic ... 
Existence and stability of traveling waves for a class of nonlocal nonlinear equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 20150501)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(±up−1u)xx, p>1. The main characteristic of this class of ... 
Global existence and blowup of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations
Babaoglu, C.; Erbay, Hüsnü Ata; Erkip, A. (Elsevier, 201301)This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bidirectional 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, 201603)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 initialvalue problem for onedimensional strainlimiting viscoelasticity
Erbay, Hüsnü Ata; Erkip, A.; Şengül, Y. (Elsevier, 20201115)In this work we prove local existence of strong solutions to the initialvalue problem arising in onedimensional strainlimiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized ... 
Longtime existence of solutions to nonlocal nonlinear bidirectional wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (American Institute of Mathematical Sciences, 201905)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, 202011)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, 202112)We consider a general class of convolutiontype 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 CamassaHolm equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (American Institute of Mathematical Sciences, 201706)We rigorously establish that, in the longwave 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, 202209)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 semidiscrete numerical method for convolutiontype unidirectional wave equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 20210515)Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolutiontype nonlocality in space is considered. A semidiscrete numerical method based on both a uniform space discretization ... 
A semidiscrete numerical scheme for nonlocally regularized KdVtype equations
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Elsevier, 202205)A general class of KdVtype wave equations regularized with a convolutiontype nonlocality in space is considered. The class differs from the class of the nonlinear nonlocal unidirectional wave equations previously studied ... 
Some remarks on the stability and instability properties of solitary waves for the double dispersion equation
Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. (Estonian Academy of Sciences, 2015)In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(up−1u)xx for p > 1, a ≥ b ...
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