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Article Citation - WoS: 11Citation - Scopus: 20The Taylor Series Method and Trapezoidal Rule on Time Scales(Elsevier Science inc, 2020) Georgiev, Svetlin G.; Erhan, Inci M.The Taylor series method for initial value problems associated with dynamic equations of first order on time scales with delta differentiable graininess function is introduced. The trapezoidal rule for the same types of problems is derived and applied to specific examples. Numerical results are presented and discussed. (c) 2020 Elsevier Inc. All rights reserved.Book Citation - Scopus: 27Lyapunov Inequalities and Applications(Springer International Publishing, 2021) Agarwal,R.P.; Bohner,M.; Özbekler,A.This book provides an extensive survey on Lyapunov-type inequalities. It summarizes and puts order into a vast literature available on the subject, and sketches recent developments in this topic. In an elegant and didactic way, this work presents the concepts underlying Lyapunov-type inequalities, covering how they developed and what kind of problems they address. This survey starts by introducing basic applications of Lyapunov's inequalities. It then advances towards even-order, odd-order, and higher-order boundary value problems; Lyapunov and Hartman-type inequalities; systems of linear, nonlinear, and quasi-linear differential equations; recent developments in Lyapunov-type inequalities; partial differential equations; linear difference equations; and Lyapunov-type inequalities for linear, half-linear, and nonlinear dynamic equations on time scales, as well as linear Hamiltonian dynamic systems. Senior undergraduate students and graduate students of mathematics, engineering, and science will benefit most from this book, as well as researchers in the areas of ordinary differential equations, partial differential equations, difference equations, and dynamic equations. Some background in calculus, ordinary and partial differential equations, and difference equations is recommended for full enjoyment of the content. © Springer Nature Switzerland AG 2021. All rights reserved.Article Citation - Scopus: 5Adomian Polynomials Method for Dynamic Equations on Time Scales(DergiPark, 2021) Georgiev,S.G.; Erhan,I.M.A recent study on solving nonlinear differential equations by a Laplace transform method combined with the Adomian polynomial representation, is extended to the more general class of dynamic equations on arbitrary time scales. The derivation of the method on time scales is presented and applied to particular examples of initial value problems associated with nonlinear dynamic equations of first order. © 2021, DergiPark. All rights reserved.Article Citation - WoS: 3Citation - Scopus: 3Lyapunov type inequalities for second-order forced dynamic equations with mixed nonlinearities on time scales(Springer-verlag Italia Srl, 2017) Agarwal, Ravi P.; Cetin, Erbil; Ozbekler, AbdullahIn this paper, we present some newHartman and Lyapunov inequalities for second-order forced dynamic equations on time scales T with mixed nonlinearities: x(Delta Delta)(t) + Sigma(n)(k=1) qk (t)vertical bar x(sigma) (t)vertical bar (alpha k-1) x(sigma) (t) = f (t); t is an element of [t(0), infinity)(T), where the nonlinearities satisfy 0 < alpha(1) < ... < alpha(m) < 1 < alpha(m+1) < ... < alpha(n) < 2. No sign restrictions are imposed on the potentials qk, k = 1, 2, ... , n, and the forcing term f. The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature.Book Part Approximation of Discontinuous Functions by q-bernstein Polynomials(Springer international Publishing Ag, 2016) Ostrovska, Sofia; Ozban, Ahmet YasarThis chapter presents an overview of the results related to the q-Bernstein polynomials with q > 1 attached to discontinuous functions on [0, 1]. It is emphasized that the singularities of such functions located on the set Jq : = {0} boolean OR {q-l}(l=0, infinity), q > 1 are definitive for the investigation of the convergence properties of their q-Bernstein polynomials.Article Citation - WoS: 1Citation - Scopus: 1Prescribed Asymptotic Behavior of Nonlinear Dynamic Equations Under Impulsive Perturbations(Springer Basel Ag, 2024) Zafer, Agacik; Dogru Akgol, SibelThe asymptotic integration problem has a rich historical background and has been extensively studied in the context of ordinary differential equations, delay differential equations, dynamic equations, and impulsive differential equations. However, the problem has not been explored for impulsive dynamic equations due to the lack of essential tools such as principal and nonprincipal solutions, as well as certain compactness results. In this work, by making use of the principal and nonprincipal solutions of the associated linear dynamic equation, recently obtained in [Acta Appl. Math. 188, 2 (2023)], we investigate the asymptotic integration problem for a specific class of nonlinear impulsive dynamic equations. Under certain conditions, we prove that the given impulsive dynamic equation possesses solutions with a prescribed asymptotic behavior at infinity. These solutions can be expressed in terms of principal and nonprincipal solutions as in differential equations. In addition, the necessary compactness results are also established. Our findings are particularly valuable for better understanding the long-time behavior of solutions, modeling real-world problems, and analyzing the solutions of boundary value problems on semi-infinite intervals.Article Citation - WoS: 112Citation - Scopus: 124Oscillation of Second-Order Delay Differential Equations on Time Scales(Pergamon-elsevier Science Ltd, 2005) Sahiner, Y.By means of Riccati transformation technique, we establish some new oscillation criteria for a second-order delay differential equation on time scales in terms of the coefficients. (C) 2005 Elsevier Ltd. All rights reserved.Article Citation - WoS: 17Citation - Scopus: 19Weak Solutions for the Dynamic Cauchy Problem in Banach Spaces(Pergamon-elsevier Science Ltd, 2009) Cichon, Mieczyslaw; Kubiaczyk, Ireneusz; Sikorska-Nowak, Aneta; Yantir, AhmetThis paper is devoted to unify and extend the results of the existence of the weak solutions of continuous and discrete Cauchy problem in Banach spaces. We offer the existence of the weak solution of dynamic Cauchy problem on an infinite time scale. The measure of weak noncompactness and the fixed point theorem of Kubiaczyk are used to prove the main result. (C) 2009 Elsevier Ltd. All rights reserved.Master Thesis Zaman Skalalarında Yüksek Mertebeden Çok Noktalı İmpalsif Sınır Değer Problemlerinin Çözümlerinin Varlığı(2022) Kuş, Murat Eymen; Akgöl, Sibel Doğru; Georgıev, Svetlin G.Bu tezde, çok noktalı yüksek mertebeden impalsif sınır değer problemlerinin zaman skalalarında çözümlerinin bulunması için yeterli koşulları araştırdık. Özellikle, üçüncü mertebeden impalsif sınır değer problemlerinin bir sınıfı ve 2n + 1, n ≥ 1 mertebeden bir impalsif sınır değer problemi sınıfı incelenmiştir. Bölüm 1'de zaman skalası ve bazı ilgili kavramların tanımları ile birlikte örnekler verilmiştir. Sonrasında tezde kullanılan sabit nokta teoremleri verilmiştir. Bölüm 2, üçüncü mertebeden çok noktalı dinamik impalsif sınır değer problemlerinin çözümlerinin varlığına ayrılmıştır. Bölüm 3'de tek sayı mertebeli çok noktalı dinamik impalsif sınır değer problemlerinin çözümlerinin varlığına odaklanılmıştır. Son olarak, Bölüm 4'te kısa bir sonuc¸ verilmiştir. Bu tezdeki sonuçların bir kısmı Georgian Mathematical Journal dergisinde basılmış, bir kısmı da Miskolc Mathematical Notes dergisinde basılmak üzere kabul edilmiştir.Master Thesis Zaman skalasında interpolasyon(2022) Jaddoa, Najlaa Abd Zaıd Jaddoa; Adıgüzel, Rezan Sevinik; Erhan, İnciBu tezde, zaman skalasında interpolasyon konusunu inceledik. Keyfi bir zaman skalası üzerinde, Lagrange, sigma-Lagrange, Hermite, sigma-Hermite, Newton ve sigma-Newton polinomlarını tanımladık. Bölünen ve sigma-bölünen farkları tanımlayarak, verilen bir veri kümesi için, Hermite polinomunu kolay yoldan elde etmek amacıyla bölünen farklar tablosu oluşturduk. Verilen bir veri kümesini, zaman skalasının yapısına bağlı olarak polinom olmayabilen fonksiyonlar olan sigma-polinomları ile temsil etmek (interpole etmek) alışılmadık bir yöntemdir. Bu şekilde, zaman skalasında interpolasyon için farklı bir bakış açısı sunmaktayız. Çeşitli zaman skalalarında birçok örnek inceledik. Bu örnekler Matlab ile elde edilen sayısal hesaplamalar ve ilgili grafikler ile desteklenmiştir.
