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Yayın A note on the exact travelling wave solution to the KdV-Burgers equation(Elsevier Science, 2003-10) Demiray, HilmiIn the present note, by use of the hyperbolic tangent method, a progressive wave solution to the Korteweg-de Vries-Burgers (KdVB) equation is presented. The solution we introduced here is less restrictive and comprises some solutions existing in the current literature (see [Wave Motion 11 (1989) 559; Wave Motion 14 (1991) 369]).Yayın On the realization of optical mappings and transformation of amplitudes by means of an aspherical "thick" lens(Gustav Fischer Verlag, 2000) Hasanoğlu, Elman; Polat, Burak DenizThe constraints for the realization of a given optical mapping by means of an aspherical ''thick" lens are investigated by using the laws of geometrical optics. The analysis yields us a partial differential equation which the optical mapping functions must satisfy as a necessary and sufficient condition. It is shown that thick lenses, which convert plane waves to plane waves, can be considered as a pure amplitude element, An interesting feature of this equation is that it does not involve the lens profiles. The problem of realization is later discussed for some special mappings and graphical illustrations of the aspherical lens profiles for a linear mapping are presented.Yayın On travelling wave solutions of a generalized Davey-Stewartson system(Oxford Univ Press, 2005-02) Eden, Osman Alp; Erbay, SaadetThe generalized Davey-Stewartson (GDS) equations, as derived by Babaoglu & Erbay (2004, Int. J. Non-Linear Mech., 39, 941-949), is a system of three coupled equations in (2 + 1) dimensions modelling wave propagation in an infinite elastic medium. The physical parameters (gamma, m(1), m(2), lambda and n) of the system allow one to classify the equations as elliptic-elliptic-elliptic (EEE), elliptic-elliptic-hyperbolic (EEH), elliptic-hyperbolic-hyperbolic (EHH), hyperbolic-elliptic-elliptic (HEE), hyperbolic-hyperbolic-hyperbolic (HHH) and hyperbolic-elliptic-hyperbolic (HEH) (Babaoglu et al., 2004, preprint). In this note, we only consider the EEE and HEE cases and seek travelling wave solutions to GDS systems. By deriving Pohozaev-type identities we establish some necessary conditions on the parameters for the existence of travelling waves, when solutions satisfy some integrability conditions. Using the explicit solutions given in Babaoglu & Erbay (2004) we also show that the parameter constraints must be weaker in the absence of such integrability conditions.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 A note on the wave propagation in water of variable depth(Elsevier Science Inc, 2011-11-01) Demiray, HilmiIn the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called "the method of integrating factor" is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.Yayın Influence of the velocity on the energy patterns of moving scatterers(Taylor & Francis, 2004) İdemen, Mehmet Mithat; Alkumru, AliParallel to the developments in the communication through space vehicles achieved during the last two decades, the scattering problems connected with moving objects became more and more important from both theoretical and practical points of view. Same problems are also arisen in point of space science, radio astronomy, radar techniques and particle physics. The earlier investigations available in the open literature concern the analysis of the scattered field pattern and, hence, treat the polarization, frequency shift (Doppler effect), aberration, etc, which are all important from both pure scientific and technological points of view. But, another issue which is also important in regard to the communication, antennas and particle physics is the influence of the motion on the scattered energy patterns which involves the radar cross-section and scattering coefficient. This paper is devoted to this purpose and aims to study the influence of the velocity on the received and scattered energies. Notice that the scattered wave is not time-harmonic even though the incident wave is so because the Lorentz transformation formulas interrelate the space coordinates and time, which makes impossible to extend the notion of radar cross-section to moving bodies. For the sake of simplicity of the mathematical manipulations, only two-dimensional case is taken into account but the method can be adapted by straightforward extensions to other types of scatterer.Yayın Weakly non-linear waves in a tapered elastic tube filled with an inviscid fluid(Pergamon-Elsevier Science Ltd, 2005-07) Bakırtaş, İlkay; Demiray, HilmiIn the present work, treating the artery as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly non-linear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equation admits a solitary wave-type solution with variable wave speed. It is observed that, the wave speed decreases with distance for positive tapering while it increases for negative tapering. It is further observed that, the progressive wave profile for expanding tubes (a > 0) becomes more steepened whereas for narrowing tubes (a < 0) it becomes more flattened.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 Head-on collision of solitary waves in fluid-filled elastic tubes(Pergamon-Elsevier Science Ltd, 2005-08) Demiray, HilmiIn this work, treating the arteries as a thin walled, prestressed thin elastic tube and the blood as an inviscid fluid, we have studied the propagation of nonlinear waves, in the longwave approximation, through the use of extended PLK perturbation method. The results show that, up to O(epsilon(2)), the head-on collision of two solitary waves is elastic and the solitary waves preserve their original properties after the collision. The leading-order analytical phase shifts and the trajectories of two solitons after the collision are derived explicitly.Yayın The effect of a bump on wave propagation in a fluid-filled elastic tube(Pergamon-Elsevier Science Ltd., 2004-01) Demiray, HilmiIn the present work, treating the arteries as a thin walled prestressed elastic tube with variable cross-section, and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible viscous fluid the evolution equation is obtained as the perturbed Korteweg-de Vries equation with variable coefficients. It is shown that this type of equations admit a solitary wave type of solution with variable wave speed. It is observed that, the wave speed gets smaller and smaller as we go away from the center of the bump. The wave speed reaches to its maximum value at the center of the bump.
