Browsing by Author "Atay, Mehmet Tarik"

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Analysis of the motion of a rigid rod on a circular surface using interpolated variational iteration method
(YILDIZ TECHNICAL UNIV, 2022) Coskun, Safa Bozkurt; Senturk, Erman; Atay, Mehmet Tarik; AGÜ, Mühendislik Fakültesi, Mühendislik Bilimleri Bölümü; Atay, Mehmet Tarik
In this paper, interpolated variational iteration method (IVIM) is applied to investigate the vibration period and steady-state response for the motion of rigid rod rocking back and forth on a circular surface without slipping. The problem can be considered as a strongly nonlinear oscillator. In this solution procedure, analytical variational iteration technique is utilized by evaluating the integrals numerically. The approximate analytical results produced by the presented method are compared with the other existing solutions available in the literature. The advantage of using numerical evaluation of integrals, the method becomes fast convergent and a highly accurate solution can be obtained within seconds. The authors believe that the presented technique has potentially wide application in the other nonlinear oscillation problems.
Haar wavelet collocation method for linear first order stiff differential equations
(E D P SCIENCES17 AVE DU HOGGAR PARC D ACTIVITES COUTABOEUF BP 112, F-91944 CEDEX A, FRANCE, 2020) Atay, Mehmet Tarik; Mertaslan, Onur Metin; Agca, Musa Kasim; Yilmaz, Abdulkadir; Toker, Batuhan; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü; Atay, Mehmet Tarık; Mertaslan, Onur Metin; Ağca, Musa Kasım; Yılmaz, Abdulkadir
In general, there are countless types of problems encountered from different disciplines that can be represented by differential equations. These problems can be solved analytically in simpler cases; however, computational procedures are required for more complicated cases. Right at this point, the wavelet-based methods have been using to compute these kinds of equations in a more effective way. The Haar Wavelet is one of the appropriate methods that belongs to the wavelet family using to solve stiff ordinary differential equations (ODEs). In this study, The Haar Wavelet method is applied to stiff differential problems in order to demonstrate the accuracy and efficacy of this method by comparing the exact solutions. In comparison, similar to the exact solutions, the Haar wavelet method gives adequate results to stiff differential problems.
Interpolated variational iteration method for solving the jamming transition problem
(ELSEVIER, RADARWEG 29, 1043 NX AMSTERDAM, NETHERLANDS, 2019) Coskun, Safa Bozkurt; Atay, Mehmet Tarik; Senturk, Erman; 0000-0002-0833-7113; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü
The purpose of this study is to present an analytical based numerical solution for Jamming Transition Problem (JTP) using Interpolated Variational Iteration Method (IVIM). The method eliminates the difficulties on analytical integration of expressions in analytical variational iteration technique and provides numerical results with analytical accuracy. JTP may be transformed into a nonlinear non-conservative oscillator by Lorenz system in which jamming transition is presented as spontaneous deviations of headway and velocity caused by the acceleration/breaking rate to be higher than the critical value. The resulting governing equation of JTP has no exact solution due to existing nonlinearities in the equation. The problem was previously attempted to be solved semi-analytically via analytical approximation methods including analytical variational iteration technique. The results of this study show that IVIM solutions agree very well with the numerical solution provided by the mathematical software. IVIM with two different formulation according to governing equation is introduced. Required order of the solution and number of time steps for a good agreement is determined according to the analyses performed using IVIM. (C) 2019 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
The Numerical Solutions for Stiff Ordinary Differential Equations by Using Interpolated Variational Iteration Method with Comparison to Exact Solutions
(AMER INST PHYSICS, 2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA, 2018) Ciftci, Cihan; Cayci, Hatice Sinem Sas; Atay, Mehmet Tarik; Toker, Batuhan; Guncan, Berkay; Yildirim, Afsin Talha; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü
Recently proposed Interpolated Variational Iteration Method (IVIM) is used to find numerical solutions of stiff ordinary differential equations for both linear and nonlinear problems. The examples are given to illustrate the accuracy and effectiveness of IVIM method and IVIM results are compared with exact results. In recent analytical approximate methods based studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method, Homotopy Analysis Method etc. In this study comparisons with exact solutions reveal that the Interpolated Variational Iteration Method (IVIM) is easy to implement. In fact, this method is promising methods for various systems of linear and nonlinear stiff ordinary differential equations as an initial value problem. Furthermore, IVIM is giving very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.
On Critical Buckling Loads of Columns under End Load Dependent on Direction
(Hindawi Publishing Corporation, 2014) Başbük, Musa; Eryılmaz, Aytekin; Atay, Mehmet Tarik; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü; Atay, M.Tarık
Most of the phenomena of various fields of applied sciences are nonlinear problems. Recently, various types of analytical approximate solution techniques were introduced and successfully applied to the nonlinear differential equations. One of the aforementioned techniques is the Homotopy analysis method (HAM). In this study, we applied HAM to find critical buckling load of a column under end load dependent on direction. We obtained the critical buckling loads and compared them with the exact analytic solutions in the literature.
Preface of the Symposium on: "advances in Analytical, Analytical Approximate, Numerical and Asymptotic Analysis of Sturm-Liouville Problems with the Applications in Engineering"
(American Institute of Physics Inc., 2017) Atay, Mehmet Tarik; Erdogan, Hakan; AGÜ, Mühendislik Fakültesi, Mühendislik Bilimleri Bölümü; Atay, Mehmet Tarik
Preface of the Symposium on: "advances in Analytical, Analytical Approximate, Numerical and Asymptotic Analysis of Sturm-Liouville Problems with the Applications in Engineering"
Solution of jamming transition problem using adomian decomposition method
(EMERALD GROUP PUBLISHING LTD, HOWARD HOUSE, WAGON LANE, BINGLEY BD16 1WA, W YORKSHIRE, ENGLAND, 2018) Senturk, Erman; Coskun, Safa Bozkurt; Atay, Mehmet Tarik; 0000-0002-0833-7113; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü
Purpose The purpose of the study is to obtain an analytical approximate solution for jamming transition problem (JTP) using Adomian decomposition method (ADM). Design/methodology/approach In this study, the jamming transition is presented as a result of spontaneous deviations of headway and velocity that is caused by the acceleration/breaking rate to be higher than the critical value. Dissipative dynamics of traffic flow can be represented within the framework of the Lorenz scheme based on the car-following model in the one-lane highway. Through this paper, an analytical approximation for the solution is calculated via ADM that leads to a solution for headway deviation as a function of time. Findings A highly nonlinear differential equation having no exact solution due to JTP is considered and headway deviation is obtained implementing a number of different initial conditions. The results are discussed and compared with the available data in the literature and numerical solutions obtained from a built-in numerical function of the mathematical software used in the study. The advantage of using ADM for the problem is presented in the study and discussed on the basis of the results produced by the applied method. Originality/value This is the first study to apply ADM to JTP.
Twist-Bend Instability of a Cantilever Beam Subjected to an End Load via Homotopy Perturbation Method
(AMER INST PHYSICS, 2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA, 2018) Yucesoy, Ahmet; Coskun, Safa Bozkurt; Atay, Mehmet Tarik; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü
In this article, twist-bend buckling analysis of a cantilever beam subjected to a concentrated end load is conducted using Homotopy Perturbation Method (HPM). Even in the linear stability analysis, obtaining an exact solution for some cases is not an easy task. However, by the use of HPM this difficulty can be overcome easily. This issue is presented with a case study and the results show that HPM can be used successfully in the analysis of twist-bend buckling of beams.
A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions
(SPRINGER INTERNATIONAL PUBLISHING AG, GEWERBESTRASSE 11, CHAM, CH-6330, SWITZERLAND, 2016) Cengizci, Suleyman; Atay, Mehmet Tarik; Eryilmaz, Aytekin; AGÜ, Mühendislik Fakültesi, Makine Mühendisliği Bölümü;
This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.