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Yayın Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations(Academic Press Inc Elsevier Science, 2011-02-01) Duruk, Nilay; Erbay, Hüsnü Ata; Erkip, AlbertWe study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite-time blow-up and as well as global existence of solutions of the problem.Yayın Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity(IOP Publishing, 2010-01) Duruk, Nilay; Erbay, Hüsnü Ata; Erkip, AlbertWe study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided.Yayın A Higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity(Oxford Univ Press, 2009-02) Duruk, Nilay; Erkip, Albert; Erbay, Hüsnü AtaIn one space dimension, a non-local elastic model is based on a single integral law, giving the stress when the strain is known at all spatial points. In this study, we first derive a higher-order Boussinesq equation using locally non-linear theory of 1D non-local elasticity and then we are able to show that under certain conditions the Cauchy problem is globally well-posed.Yayın The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials(IOP Publishing Ltd, 2011-04) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, AlbertThis paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is reflected by a convolution integral in the space variables. The Fourier transform of the convolution kernel is nonnegative and satisfies a certain growth condition at infinity. For initial data in L-2 Sobolev spaces, conditions for global existence or finite time blow-up of the solutions of the Cauchy problem are established.Yayın Numerical simulation of blow-up solutions for the generalized Davey-Stewartson system(Taylor & Francis Ltd, 2011-03) Muslu, Gülçin Mihriye; Erbay, Hüsnü AtaBlow-up solutions for the generalized Davey-Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed.Yayın A numerical study of the long wave-short wave interaction equations(Elsevier B.V., 2007-03-07) Borluk, Handan; Muslu, Gülçin Mihriye; Erbay, Hüsnü AtaTwo numerical methods are presented for the periodic initial-value problem of the long wave-short wave interaction equations describing the interaction between one long longitudinal wave and two short transverse waves propagating in a generalized elastic medium. The first one is the relaxation method, which is implicit with second-order accuracy in both space and time. The second one is the split-step Fourier method, which is of spectral-order accuracy in space. We consider the first-, second- and fourth-order versions of the split-step method, which are first-, second- and fourth-order accurate in time, respectively. The present split-step method profits from the existence of a simple analytical solution for the nonlinear subproblem. We numerically test both the relaxation method and the split-step schemes for a problem concerning the motion of a single solitary wave. We compare the accuracies of the split-step schemes with that of the relaxation method. Assessments of the efficiency of the schemes show that the fourth-order split-step Fourier scheme is the most efficient among the numerical schemes considered.Yayın Dynamic extension of a compressible nonlinearly elastic membrane tube(Oxford Univ Press, 2005-02) Erbay, Hüsnü Ata; Tüzel, Vasfiye HandeThe dynamic response of an isotropic compressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. The equations governing dynamic axially symmetric deformations of the membrane tube are presented for a general form of compressible isotropic elastic strain-energy function. Numerical results, obtained using a Godunov-type finite volume method and valid up to the time at which reflections occur at the fixed end of the tube, are given for two specific forms of the strain-energy function that characterizes a class of compressible elastomers (the Blatz-Ko model). The question of how the numerical results are related to the exact solution obtained for a limiting case is discussed.Yayın Two remarks on a generalized Davey-Stewartson system(Elsevier Ltd, 2006-03-01) Eden, Osman Alp; Erbay, Hüsnü Ata; Muslu, Gülçin MihriyeWe present two results on a generalized Davey-Stewartson system, both following from the pseudo-conformal invariance of its solutions. In the hyperbolic-elliptic-elliptic case, under some conditions on the physical parameters, we establish a blow-up profile. These conditions turn out to be necessary conditions for the existence of a special '' radial '' solution. In the elliptic-elliptic-elliptic case, under milder conditions, we show the L-P-norms of the solutions decay to zero algebraically in time for 2 < p < infinity.Yayın Closing the gap in the purely elliptic generalized Davey-Stewartson system(Pergamon-Elsevier Science Ltd, 2008-10-15) Eden, Osman Alp; Erbay, Hüsnü Ata; Muslu, Gülçin MihriyeIn this note we improve the results presented previously on global existence and global nonexistence for the Solutions of the purely elliptic generalized Davey-Stewartson system. These results left a gap in the parameter range where neither a global existence result nor a global nonexistence result could be established. Here we are able to show that when the coupling parameter is negative there is no gap. Moreover, in the case where the coupling parameter is positive we reduce the size of the gap.Yayın The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension(Pergamon-Elsevier Science Ltd, 2004-06) Tüzel, Vasfiye Hande; Erbay, Hüsnü AtaThe dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.
