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Yayın Contributions of higher order terms to nonlinear waves in fluid-filled elastic tubes: strongly dispersive case(Pergamon-Elsevier Science, 2003-07) Demiray, HilmiIn the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the modified reductive perturbation technique presented by us [15] the amplitude modulation of weakly nonlinear waves is examined. It is shown that the first order term in the perturbation expansion is governed by a nonlinear Schrodinger equation and the second order term is governed by the linearized Schrodinger equation with a nonhomogeneous term. In the longwave limit a travelling wave type of solution to these equations are also given.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 A note on the amplitude modulation of symmetric regularized long-wave equation with quartic nonlinearity(Springer, 2012-12) Demiray, HilmiWe study the amplitude modulation of a symmetric regularized long-wave equation with quartic nonlinearity through the use of the reductive perturbation method by introducing a new set of slow variables. The nonlinear Schrodinger (NLS) equation with seventh order nonlinearity is obtained as the evolution equation for the lowest order term in the perturbation expansion. It is also shown that the NLS equation with seventh order nonlinearity assumes an envelope type of solitary wave solution.Yayın Head-on collision of the solitary waves in fluid-filled elastic tubes(Işık University Press, 2018-04-12) Özden, Ali Erinç; Demiray, HilmiIn the present work, by employing the field equations given in [15] and the extended PLK method derived in [9], we have studied the head-on collision of solitary waves in arteries. Introducing a set of stretched coordinates which include some unknown functions characterizing the higher order dispersive effects and the trajectory functions to be determined from the removal of possible secularities that might occur in the solution. Expanding these unknown functions and the field variables into power series of the smallness parameter epsilon and introducing the resulting expansions into the field equations we obtained the sets of partial differential equations. By solving these differential equations and imposing the requirements for the removal of possible secularities we obtained the speed correction terms and the trajectory functions. The results of our calculation show that both the evolution equations and the phase shifts resulting from the head-on collision of solitary waves are quite different from those of Xue [15], who employed the incorrect formulation of Su and Mirie [4]. As opposed to the result of previous works on the same subject, in the present work the phase shifts depend on the amplitudes of both colliding waves.Yayın Waves in an elastic tube filled with a heterogeneous fluid of variable viscosity(Pergamon-Elsevier Science Ltd, 2009-07) Demiray, HilmiBy treating the artery as a prestressed thin elastic tube and the blood as an incompressible heterogeneous fluid with variable viscosity. we studied the propagation of weakly non-linear waves in such a composite medium through the use of reductive perturbation method. By assuming a variable density and a variable viscosity for blood in the radial direction we obtained the perturbed Korteweg-deVries equation as the evolution equation when the viscosity is of order of epsilon(3/2). We observed that the perturbed character is the combined result of the viscosity and the heterogeneity of the blood. A progressive wave type of solution is presented for the evolution equation and the result is discussed. The numerical results indicate that for a certain value of the density parameter sigma, the wave equation loses its dispersive character and the evolution equation degenerates. It is further shown that, for the perturbed KdV equation both the amplitude and the wave speed decay in the time parameter tau.Yayın Weakly nonlinear waves in water of variable depth: Variable-coefficient Korteweg-de Vries equation(Pergamon-Elsevier Science Ltd, 2010-09) 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 non-linear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as a variable-coefficient Korteweg-de Vries (KdV) equation. A progressive wave type of solution, which satisfies the evolution equation in the integral sense but not point by point, is presented. The resulting solution is numerically evaluated for two selected bottom profile functions, and it is observed that the wave amplitude increases but the band width of the solitary wave decreases with increasing undulation of the bottom profile.Yayın Modulational instability of acoustic waves in a dusty plasma with nonthermal electrons and trapped ions(Pergamon-Elsevier Science Ltd, 2019-04) Demiray, Hilmi; Abdikian, AlirezaIn the present work, employing the nonlinear field equations of a hot dusty plasma in the presence of nonthermal electrons and trapped ions, we studied the amplitude modulation of nonlinear waves in such a plasma medium by use of the reductive perturbation method and obtained the modified nonlinear Schrodinger equation. The modulational instability (MI) was investigated and the effects of the proportion of the fast electrons (alpha), the trapping parameter (b) and the plasma parameters such as the dust-ion temperature ratio (sigma(d)), the partial unperturbed electron to dust density (delta), and the ion-electron temperature ratio (sigma(i)) on it was discussed. For the investigation of modulational instability problems three parameters P/Q,K-max and Gamma(max) play the central role. The variations of these parameters with the wave number k and the other physical parameters are discussed and the possibility of occurence of modulational instability is indicated.Yayın Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube(Pergamon-Elsevier Science, 2002-10) Demiray, HilmiIn the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schrodinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed.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 Nonlinear wave modulation in a prestressed thin elastic tube filled with an inviscid fluid(Wit Press, 2002) Bakırtaş, İlkay; Demiray, HilmiIn the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid and then utilizing the reductive perturbation technique, the amplitude modulation of weakly nonlinear waves is examined. It is shown that, the amplitude modulation of these waves is governed by a nonlinear Schrödinger (NLS) equation. The result is compared with some previous works on the same subject. The modulational instability of the monochromatic wave solution is discussed for some elastic materials and initial deformations. It is shown that the amplitude modulation of weakly nonlinear waves near the marginal state is governed by the Generalized Nonlinear Schrödinger equation (GNLS).
