Natural and Mathematical Sciences

Permanent URI for this collectionhttps://hdl.handle.net/10679/313

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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
(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.
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Comparison of nonlocal nonlinear wave equations in the long-wave limit
(Taylor & Francis, 2020-11-17) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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 and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular, our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations.
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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.
Open Access
Convergence of a linearly regularized nonlinear wave equation to the p-system
(TÜBİTAK, 2023) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A. K.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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 written as a p-system of first-order differential equations. We first establish the local well-posedness of the Cauchy problem. We then investigate the behavior of solutions to the Cauchy problem in the limit as the kernel function of the convolution integral approaches to the Dirac delta function, that is, in the vanishing dispersion limit. We consider two different types of the vanishing dispersion limit behaviors for the convolution operator depending on the form of the kernel function. In both cases, we show that the solutions converge strongly to the corresponding solutions of the classical elasticity equation.
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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.
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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.
Open Access
Dispersive transverse waves for a strain-limiting continuum model
(Sage, 2023-10) Erbay, Hüsnü Ata; Rajagopal, K. R.; Saccomandi, G.; Şengül, Y.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata
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 for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation.
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Instability and stability properties of traveling waves for the double dispersion equation
(Elsevier, 2016-03) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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 main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms uxxxx and uxxtt. We obtain an explicit condition in terms of a, b and p on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation (b=0), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide analytical as well as numerical results on the variation of the stability region of wave velocities with a, b and p and then state explicitly the conditions under which the traveling waves are orbitally stable.
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Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity
(Elsevier, 2020-11-15) Erbay, Hüsnü Ata; Erkip, A.; Şengül, Y.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata
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 strain, the rate of change of the linearized strain and the stress. The model is a generalization of the nonlinear Kelvin-Voigt viscoelastic solid under the assumption that the strain and the strain rate are small. We define an initial-value problem for the stress variable and then, under the assumption that the nonlinear constitutive function is strictly increasing, we convert the problem to a new form for the sum of the strain and the strain rate. Using the theory of variable coefficient heat equation together with a fixed point argument we prove local existence of solutions. Finally, for several constitutive functions widely used in the literature we show that the assumption on which the proof of existence is based is not violated.
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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.
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Numerical computation of solitary wave solutions of the Rosenau equation
(Elsevier, 2020-11) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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 solutions. We then apply two numerical algorithms based on the Petviashvili method for solving the Rosenau equation with single or double power law nonlinearity. Numerical calculations rely on a uniform discretization of a finite computational domain. Through some numerical experiments we observe that the algorithm converges rapidly and it is robust to very general forms of the initial guess.
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On the convergence of the nonlocal nonlinear model to the classical elasticity equation
(Elsevier, 2021-12) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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.
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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.
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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.
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A semi-discrete numerical scheme for nonlocally regularized KdV-type equations
(Elsevier, 2022-05) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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 by the addition of a linear convolution term involving third-order derivative. To solve the Cauchy problem we propose a semi-discrete numerical method based on a uniform spatial discretization, that is an extension of a previously published work of the present authors. We prove uniform convergence of the numerical method as the mesh size goes to zero. We also prove that the localization error resulting from localization to a finite domain is significantly less than a given threshold if the finite domain is large enough. To illustrate the theoretical results, some numerical experiments are carried out for the Rosenau-KdV equation, the Rosenau-BBM-KdV equation and a convolution-type integro-differential equation. The experiments conducted for three particular choices of the kernel function confirm the error estimates that we provide.
Open Access
Some remarks on the stability and instability properties of solitary waves for the double dispersion equation
(Estonian Academy of Sciences, 2015) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
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 = −(|u|p−1u)xx for p > 1, a ≥ b > 0. After a brief reminder of the general class of nonlocal wave equations to which the double dispersion equation belongs, we summarize our findings for both the existence and orbital stability/instability of travelling wave solutions to the general class of nonlocal wave equations. We then state (i) the conditions under which travelling wave solutions of the double dispersion equation are unstable by blow-up and (ii) the conditions under which the travelling waves are orbitally stable. We plot the instability/stability regions in the plane defined by wave velocity and the quotient b/a for various values of p.
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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.
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Transverse linear instability of solitary waves for coupled long-wave-short-wave interaction equations
(Elsevier, 2012-12) Erbay, Hüsnü Ata; Erbay, Saadet; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.
Open 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.
Open Access
Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: The KdV, BBM and CH equations
(Estonian Academy of Sciences, 2015) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
We consider unidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral with a suitable kernel function. We first give a brief review of asymptotic wave models describing the unidirectional propagation of small-but-finite amplitude long waves. When the kernel function is the well-known exponential kernel, the asymptotic description is provided by the Korteweg–de Vries (KdV) equation, the Benjamin–Bona–Mahony (BBM) equation, or the Camassa–Holm (CH) equation. When the Fourier transform of the kernel function has fractional powers, it turns out that fractional forms of these equations describe unidirectional propagation of the waves. We then compare the exact solutions of the KdV equation and the BBM equation with the numerical solutions of the nonlocal model. We observe that the solution of the nonlocal model is well approximated by associated solutions of the KdV equation and the BBM equation over the time interval considered.

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