TR Dizin İndeksli Yayınlar Koleksiyonu / TR Dizin Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/20.500.14365/4
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Article The Legendre Matrix-Collocation Approach for Some Nonlinear Differential Equations Arising in Physics and Mechanics(2019-03-31) Kürkçü, Ömür Kıvanç; Dönmez Demir, Duygu; Sezer, Mehmet; Çınardalı, Tuğçe; Demir, Duygu DönmezIn this study, the Legendre operational matrix method based on collocation points is introduced to solve high order ordinary differentialequations with some nonlinear terms arising in physics and mechanics. This technique transforms the nonlinear differential equationinto a matrix equation with unknown Legendre coefficients via mixed conditions. This solution of this matrix equation yields theLegendre coefficients of the solution function. Thus, the approximate solution is obtained in terms of Legendre polynomials. Some testproblems together with residual error estimation are given to show the usefulness and applicability of the method and the numericalresults are compared.Article Lucas Polynomial Approach for Second Order Nonlinear Differential Equations(2020-04-20) Gümgüm, Sevin; Kürkçü, Ömür Kıvanç; Sezer, Mehmet; Bayku S Sava Saner Il, Nurcan; Savaşaneril, Nurcan BaykuşThis paper presents the Lucas polynomial solution of second-order nonlinearordinary differential equations with mixed conditions. Lucas matrix method is based oncollocation points together with truncated Lucas series. The main advantage of the methodis that it has a simple structure to deal with the nonlinear algebraic system obtained frommatrix relations. The method is applied to four problems. In the first two problems, exactsolutions are obtained. The last two problems, Bratu and Duffing equations are solvednumerically; the results are compared with the exact solutions and some other numericalsolutions. It is observed that the application of the method results in either the exact oraccurate numerical solutions.Article A Numerical Technique Based on Lucas Polynomials Together With Standard and Chebyshev-Lobatto Collocation Points for Solving Functional Integro-Differential Equations Involving Variable Delays(2018-12-01) Gümgüm, Sevin; Sezer, Mehmet; Savaşaneril, Nurcan Baykuş; Kürkçü, Ömür KıvançIn this paper, a new numerical matrix-collocation technique is considered to solve functional integrodifferentialequations involving variable delays under the initial conditions. This technique is basedessentially on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points. Somedescriptive examples are performed to observe the practicability of the technique and the residual erroranalysis is employed to improve the obtained solutions. Also, the numerical results obtained by using thesecollocation points are compared in tables and figures.Article Citation - WoS: 11Citation - Scopus: 11A Novel Graph-Operational Matrix Method for Solving Multidelay Fractional Differential Equations With Variable Coefficients and a Numerical Comparative Survey of Fractional Derivative Types(Scientific Technical Research Council Turkey-Tubitak, 2019-01-18) Kürkçü, ÖmÜr Kıvanç; Aslan, Ersin; Sezer, MehmetIn this study, we introduce multidelay fractional differential equations with variable coefficients in a unique formula. A novel graph-operational matrix method based on the fractional Caputo, Riemann-Liouville, Caputo-Fabrizio, and Jumarie derivative types is developed to efficiently solve them. We also make use of the collocation points and matrix relations of the matching polynomial of the complete graph in the method. We determine which of the fractional derivative types is more appropriate for the method. The solutions of model problems are improved via a new residual error analysis technique. We design a general computer program module. Thus, we can explicitly monitor the usefulness of the method. All results are scrutinized in tables and figures. Finally, an illustrative algorithm is presented.Article Citation - WoS: 11Citation - Scopus: 15Legendre Wavelet Solution of Neutral Differential Equations With Proportional Delays(Springer Heidelberg, 2019-04-01) Gumgum, Sevin; Ersoy Özdek, Demet; Özaltun, Gökçe; Bildik, Necdet; Özdek, Demet Ersoy; Kürkçü, Ömür Kıvanç; Sezer, Mehmet; Savaşaneril, Nurcan BaykuşThe aim of this paper is to solve neutral differential equations with proportional delays by using Legendre wavelet method. Using orthonormal polynomials is the main advantage of this method since it enables a decrease in the computational cost and runtime. Some examples are displayed to illustrate the efficiency and accuracy of the proposed method. Numerical results are compared with various numerical methods in literature and show that the present method is very effectual in solving neutral differential equations with proportional delays.