Person:
ERBAY, Hüsnü Ata

Profile Picture

Email Address

Birth Date

WoSScopusGoogle ScholarORCID

Name

Job Title

First Name

Hüsnü Ata

Last Name

ERBAY
OrgUnit Logo
Organizational Unit

Filters

2012 - 2019152020 - 202311
Reset filters

Settings

Publication Search Results

Now showing 1 - 10 of 26
  • Thumbnail Image
    ArticlePublicationOpen Access
    Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity
    (Elsevier, 2015-12) Erbay, Hüsnü Ata; Şengül, Yasemin; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ŞENGÜL, Yasemin
    In this paper we investigate traveling wave solutions of a non-linear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal. To describe the response of viscoelastic solids we assume a non-linear relationship among the linearized strain, the strain rate and the Cauchy stress. We then concentrate on traveling wave solutions that correspond to the heteroclinic connections between the two constant states. We establish conditions for the existence of such solutions, and find those solutions, explicitly, implicitly or numerically, for various forms of the non-linear constitutive relation.
  • Placeholder
    ArticlePublicationMetadata only
    Derivation of generalized Camassa-Holm equations from Boussinesq-type equations
    (Informa Group, 2016) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.
  • Placeholder
    ArticlePublicationMetadata only
    A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity
    (Springer Nature, 2020-05-28) Erbay, Hüsnü Ata; Sengul, Y.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata
    We introduce a one-dimensional stress-rate type nonlinear viscoelastic model for solids that obey the assumptions of the strain-limiting theory. Unlike the classical viscoelasticity theory, the critical hypothesis in the present strain-limiting theory is that the linearized strain depends nonlinearly on the stress and the stress rate. We show the thermodynamic consistency of the model using the complementary free energy and then using the Gibbs free energy. This allows us to take the stress and the stress rate as primitive variables instead of kinematical quantities such as deformation or strain. We also show that the non-dissipative part of the materials in consideration has a stored energy. We compare the new stress-rate type model with the strain-rate type viscoelastic model due to Rajagopal from the points of view of energy decay, the nonlinear differential equations of motion and Fourier analysis of the corresponding linear models.
  • Placeholder
    ArticlePublicationMetadata only
    Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
    (EDP Sciences, 2018-09-13) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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. The most important characteristic of the numerical method is that it is directly applied to the nonlocal equation by introducing the discrete convolution operator. Starting from the continuous Cauchy problem defined on the real line, we first construct the discrete Cauchy problem on a uniform grid of the real line. Thus the semi-discretization in space of the continuous problem gives rise to an infinite system of ordinary differential equations in time. We show that the initial-value problem for this system is well-posed. We prove that solutions of the discrete problem converge uniformly to those of the continuous one as the mesh size goes to zero and that they are second-order convergent in space. We then consider a truncation of the infinite domain to a finite one. We prove that the solution of the truncated problem approximates the solution of the continuous problem when the truncated domain is sufficiently large. Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions for various convolution kernels.
  • Placeholder
    ArticlePublicationMetadata only
    A comparison of solutions of two convolution-type unidirectional wave equations
    (Taylor and Francis, 2023) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 two specific choices of the kernel function, the Benjamin–Bona–Mahony equation and the Rosenau equation that are particularly suitable to model water waves and elastic waves, respectively, are two members of the class. We first prove an energy estimate for the Cauchy problem of the non-local unidirectional wave equation. Then, for the same initial data, we consider two distinct solutions corresponding to two different kernel functions. Our main result is that the difference between the solutions remains small in a suitable Sobolev norm if the two kernel functions have similar dispersive characteristics in the long-wave limit. As a sample case of this comparison result, we provide the approximations of the hyperbolic conservation law.
  • Placeholder
    ArticlePublicationMetadata only
    Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
    (American Institute of Mathematical Sciences, 2019-05) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 integral. We prove a long-time existence result for the nonlocal wave equations with a power-type nonlinearity and a small parameter. As the energy estimates involve a loss of derivatives, we follow the Nash-Moser approach proposed by Alvarez-Samaniego and Lannes. As an application to the long-time existence theorem, we consider the limiting case in which the kernel function is the Dirac measure and the nonlocal equation reduces to the governing equation of one-dimensional classical elasticity theory. The present study also extends our earlier result concerning local well-posedness for smooth kernels to nonsmooth kernels.
  • Placeholder
    ArticlePublicationMetadata only
    The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
    (AIMS, 2016-11) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters \epsilon and \delta measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.
  • Placeholder
    ArticlePublicationMetadata only
    On the full dispersion Kadomtsev–Petviashvili equations for dispersive elastic waves
    (Elsevier, 2022-09) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 analogous full dispersive KP models of nonlinear elastic waves propagating in a nonlocal elastic medium. In particular we consider anti-plane shear elastic waves which are assumed to be small-amplitude long waves. We propose two different full dispersive extensions of the KP equation in the case of cubic nonlinearity and ”negative dispersion”. One of them is called the Whitham-type full dispersion KP equation and the other one is called the BBM-type full dispersion KP equation. Most of the existing KP-type equations in the literature are particular cases of our full dispersion KP equations. We also introduce the simplified models of the new proposed full dispersion KP equations by approximating the operators in the equations. We show that the line solitary wave solution of a simplified form of the Whitham-type full dispersion KP equation is linearly unstable to long-wavelength transverse disturbances if the propagation speed of the line solitary wave is greater than a certain value. A similar analysis for a simplified form of the BBM-type full dispersion KP equation does not provide a linear instability assessment.
  • Placeholder
    ArticlePublicationMetadata only
    A semi-discrete numerical method for convolution-type unidirectional wave equations
    (Elsevier, 2021-05-15) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a finite domain. Restricting the problem to a finite domain introduces a localization error and it is proved that this localization error stays below a given threshold if the finite domain is large enough. For two particular kernel functions, the numerical examples concerning solitary wave solutions illustrate the expected accuracy of the method. Our class of nonlocal wave equations includes the Benjamin–Bona–Mahony equation as a special case and the present work is inspired by the previous work of Bona, Pritchard and Scott on numerical solution of the Benjamin–Bona–Mahony equation.
  • Placeholder
    ArticlePublicationMetadata only
    Derivation of the Camassa-Holm equations for elastic waves
    (Elsevier, 2015-06-05) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 medium. We first show that the equation of motion for the nonlocally and nonlinearly elastic medium reduces to the improved. Boussinesq equation for a particular choice of the kernel function appearing in the integral-type constitutive relation. We then derive the Camassa–Holm equation from the improved Boussinesq equation using an asymptotic expansion valid as nonlinearity and dispersion parameters that tend to zero independently. Our approach follows mainly the standard techniques used widely in the literature to derive the Camassa–Holm equation for shallow-water waves. The case where the Fourier transform of the kernel function has fractional powers is also considered and the fractional Camassa–Holm equation is derived using the asymptotic expansion technique.