Browsing by Author "Atay, Mehmet T."

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Axial free vibration analysis of a tapered nanorod using Adomian decomposition method
(TECHNO-PRESS, 2025) Coskun, Safa B.; Kara, Ozge; Atay, Mehmet T.; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü; Atay, Mehmet T.
This study aimed to conduct an analysis of the axial free vibration of tapered nanorods based on nonlocal elasticity theory. The small-scale effect on the free axial vibration of a tapered nanorod was studied employing the Adomian decomposition method (ADM) and the finite difference method (FDM) as a checking tool where a contradiction existed between the results of this study and available results in one highly cited work in the literature, which was used for comparison purposes in this work. Different boundary conditions for the nanorod were considered: fixed-fixed nanorod, fixed-free nanorod, and fixed-linear spring nanorod. The governing equation of the problem is a variable coefficient differential equation for which analytical solutions are strictly limited. For this type of problem, analytical approximate methods are effective, and there are many studies available in the literature on the application of these methods to solve linear/nonlinear ordinary/partial differential equations. ADM is one of the methods and was successfully used in this study to analyze the free vibration of nanorods. The results obtained in this study have shown that the presented technique is so powerful and has potential for applications in nanomechanics based on nonlocal elasticity theory.
Magnus series expansion method for solving nonhomogeneous stiff systems of ordinary differential equations
(ACADEMIC PUBLICATION COUNCILPO BOX 17225, KHALDIYA 72453, KUWAIT, 2016) Atay, Mehmet T.; Eryilmaz, Aytekin; Komez, Sure; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü; Atay, Mehmet T.
In this paper, Magnus Series Expansion Method, which is based on Lie Groups and Lie algebras is proposed with different orders to solve nonhomogeneous stiff systems of ordinary differential equations. Using multivariate Gaussian quadrature, fourth (MG4) and sixth (MG6) order method are presented. Then, it is applied to nonhomogeneous stiff systems using different step sizes and stiffness ratios. In addition, approximate and exact solutions are demonstrated with figures in detail. Moreover, absolute errors are illustrated with detailed tables.