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Yayın Interactions of nonlinear acoustic waves in a fluid-filled elastic tube(Pergamon-Elsevier Science, 2001-03) Akgün, Güler; Demiray, HilmiIn the present work, the nonlinear interactions of two acoustical waves propagating in a fluid-filled elastic tube with different wave numbers, frequencies and group velocities are examined. Employing the multiple-scale expansion method, expanding the field quantities into asymptotic series of the smallness parameter and solving the resulting differential equations of various orders of the same parameter, we obtained two coupled nonlinear Schrodinger equations. The nonlinear plane wave solutions to these equations are also given for some special cases.Yayın Higher order perturbation expansion of waves in water of variable depth(Elsevier Ltd, 2010-01) Demiray, HilmiIn this work, we extended the application of "the modified reductive perturbation method" to long waves in water of variable depth and obtained a set of KdV equations as the governing equations. Seeking a localized travelling wave solution to these evolution equations we determine the scale function c(1)(tau) so as to remove the possible secularities that might occur. We showed that for waves in water of variable depth, the phase function is not linear anymore in the variables x and t. It is further shown that, due to the variable depth of the water, the speed of the propagation is also variable in the x coordinateYayın Head-on-collision of nonlinear waves in a fluid of variable viscosity contained in an elastic tube(Pergamon-Elsevier Science Ltd, 2009-08-30) Demiray, HilmiIn this work, treating the arteries as a thin walled, prestressed elastic tube and the blood as an incompressible viscous fluid of variable viscosity, we have studied the interactions of two nonlinear waves, in the long wave approximation, through the use of extended PLK perturbation method, and the evolution equations are shown to be the Korteweg-deVries-Burgers equation. The results show that, Up to O(is an element of(3/2)), the head-on-collision of two nonlinear progressive waves is elastic and the nonlinear progressive waves preserve their original properties after the collision. The phase functions for each wave are derived explicitly and it is shown that they are not straight lines anymore, they are rather some curves.Yayın Forced Korteweg-de Vries-Burgers equation in an elastic tube filled with a variable viscosity fluid(Pergamon-Elsevier Science Ltd, 2008-11) Gaik, Tay Kim; Demiray, HilmiIn the present work, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as a Newtonian fluid with variable viscosity, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method [Jeffrey A, Kawahara T. Asymptotic methods in nonlinear wave theory. Boston: Pitman; 1981]. We obtained the forced Korteweg-de Vries-Burgers (FKdVB) equation with variable coefficients as the evolution equation. By use of the coordinate transformation, it is shown that this type of evolution equation admits a progressive wave solution with variable wave speed. As might be expected from physical consideration, the wave speed reaches its maximum value at the center of stenosis and gets smaller and smaller as we go away from the center of the stenosis. The variations of radial displacement and the fluid pressure with the distance parameter are also examined numerically. The results seem to be consistent with physical intuition.Yayın Variable coefficient modified KdV equation in fluid-filled elastic tubes with stenosis: Solitary waves(Pergamon-Elsevier Science Ltd, 2009-10-15) Demiray, HilmiIn the present work, treating the arteries as a thin walled prestressed elastic tube with variable radius, 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 non-viscous fluid, the evolution equation is obtained as variable coefficients Korteweg-de Vries equation. Noticing that for a set of initial deformations, the coefficient characterizing the nonlinearity vanish, by re-scaling the stretched coordinates we obtained the variable coefficient modified KdV equation. Progressive wave solution is sought for this evolution equation and it is found that the speed of the wave is variable along the tube axis.Yayın Nonlinear waves in an elastic tube with variable prestretch filled with a fluid of variable viscosity(Pergamon-Elsevier Science Ltd, 2008-10) Demiray, HilmiIn the present work, by employing the reductive perturbation method to the nonlinear equations of an incompressible, prestressed, homogeneous and isotropic thin elastic tube and to the exact equations of an incompressible Newtonian fluid of variable viscosity, we have studied weakly nonlinear waves in such a medium and obtained the variable coefficient Korteweg-deVries-Burgers (KdV-B) equation as the evolution equation. For this purpose, we treated the artery as an incompressible, homogeneous and isotropic elastic material subjected to variable stretches both in the axial and circumferential directions initially, and the blood as an incompressible Newtonian fluid whose viscosity changes with the radial coordinate. By seeking a travelling wave solution to this evolution equation, we observed that the wave front is not a plane anymore, it is rather a curved surface. This is the result of the variable radius of the tube. The numerical calculations indicate that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations, like Bernoulli's law. We further observed that, the amplitude of the Burgers shock gets smaller and smaller with increasing time parameter along the tube axis. This is again due to the variable radius, but the effect of it is quite small.Yayın Multiple time scale formalism and its application to long water waves(Elsevier Science Inc, 2010-05) Demiray, HilmiIn the present work, by employing the multiple time scaling method, we studied the non-linear waves in shallow-water problem and obtained a set of Korteweg-deVries equations governing the various order terms in the perturbation expansion. By seeking a travelling wave type of solutions for the evolution equations, we have obtained a set of wave speeds associated with each time parameter. It is shown that the speed corresponding to the lowest order time parameter given the wave speed of the conventional reductive perturbation method, whereas the wave speeds corresponding to the higher order time parameters give the speed correction terms. The result obtained here is exactly the same with that of Demiray [H. Demiray, Modified reductive perturbation method as applied to long water waves: Korteweg-deVries hierarchy, Int. J. Nonlinear Sci. 6 (2008) 11-20] who employed the modified reductive perturbation method.Yayın The modified reductive perturbation method as applied to Boussinesq equation: strongly dispersive case(Elsevier Science Inc, 2005-05-05) Demiray, HilmiIn this work, we extended the application of "the modified reductive perturbation method" to Boussinesq equation for strongly dispersive case and tried to obtain the contribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schrodinger equation and the second order term is governed by the linearized Schrodinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given.Yayın Modulation of nonlinear waves near the marginal state of instability in fluid-filled elastic tubes(Elsevier Inc, 2004-02-05) Bakırtaş, İlkay; Demiray, HilmiUsing the nonlinear differential equations governing the motion of a fluid-filled and prestressed long thin elastic tube, the propagation of nonlinear waves-near the marginal state is examined through the use of reductive perturbation method. It is shown that the amplitude modulation near the marginal state is governed by a generalized nonlinear Schrodinger (GNLS) equation. Some exact solutions, including oscillatory and solitary waves of the GNLS equation are presented.
