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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 An application of the modified reductive perturbation method to a generalized boussinesq equation(Walter De Gruyter GMBH, 2013-02) Demiray, HilmiIn this work, we apply "the modified reductive perturbation method" to the generalized Boussinesq equation and obtain various form of generalized KdV equations as the evolution equations. Seeking a localized travelling wave solutions for these evolution equations we determine the scale parameters g(1) and g(2), which corresponds to the correction terms in the wave speed, so as to remove the possible secularities that might occur. Depending on the sign and the values of certain parameters the resulting solutions are shown to be a solitary wave or a periodic solution. The suitability of the method is also shown by comparing the results with the exact travelling wave solution for the generalized Boussinesq equation.Yayın Head-on collisions of solitary waves(Işık Üniversitesi, 2015-12-11) Özden, Ali Erinç; Demiray, Hilmi; Işık Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Doktora ProgramıThe interaction of solitary waves in various physical media is a long time studied subject in nonlinear wave theory. For overtaking collision between solitary waves, one can use the inverse scattering transform method to obtain the overtaking colliding effect of solitary waves. However, for the head-on collision between solitary waves, one must employ some kind of asymptotic expansion to solve the original field equations. This thesis addresses head-on collision problem between two solitary waves. The head-on collision of solitary waves in shallow water is re-examined upon discovering the wrongness of the statement about the secular terms in the pioneering work of Su and Mirie (J. Fluid Mech., 98:509-525, 1980). In the first part, based on the above argument, the head-on collision of two solitary waves propagating in shallow water is studied by introducing a set of stretched coordinates that includes some unknown trajectory functions which are to be determined so as to remove secularities that might occur in the solution. Expanding the field variables and trajectory functions into power series, a set of differential equations governing various terms in the perturbation expansion is obtained. By solving them under non-secularity condition, the evolution equations and also the expressions for phase shifts are determined. As opposed to the result of previous studies our calculation shows that the phase shifts depend on amplitudes of both colliding waves. In the second part, the head-on-collision of solitary waves in shallow water theory is examined through the use of extended Poincaré-Lighthill-Kuo(PLK) method. Following a similar procedure with the previous part, the speed correction terms and the trajectory functions are determined. The result obtained here is exactly same with that found in the first part. In the third part, the head-on collision of the solitary waves in fluid-filled elastic tubes is studied by employing the extended PLK method. Pursuing the procedure in the previous part, the speed correction terms and the trajectory functions are obtained. The results of our calculation show that both the evolution equations and the phase shifts are quite different from those of Xue (Phys. Lett. A, 331:409-413, 2004). As opposed to the result of previous works on the same subject, the phase shifts depend on the amplitudes of both colliding waves.
