WoS İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8653
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Article Citation - WoS: 5Citation - Scopus: 5On Some Fractional Operators Generated From Abel's Formula(Tubitak Scientific & Technological Research Council Turkey, 2022) Ugurlu, EkinThis work aims to share some fractional integrals and derivatives containing three real parameters. The main tool to introduce such operators is the corresponding Abel's equation. Solvability conditions for the Abel's equations are shared. Semigroup property for fractional integrals are introduced. Integration by parts rule is given. Moreover, mean value theorems and related results are shared. At the end of the paper, some directions for some fractional operators are given.Article Citation - WoS: 11Citation - Scopus: 13Fractional Hamilton's Equations of Motion in Fractional Time(de Gruyter Poland Sp Z O O, 2007) Baleanu, Dumitru; Rabei, Eqab M.; Muslih, Sami I.The Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton's equations are obtained and two examples are investigated in detail. (C) Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.Article Citation - WoS: 11Citation - Scopus: 11Steady Periodic Response for a Vibration System With Distributed Order Derivatives To Periodic Excitation(Sage Publications Ltd, 2018) Baleanu, Dumitru; Duan, Jun-ShengSteady-state periodic responses for a vibration system with distributed order derivatives are investigated, where the fractional derivative operator -infinity D-t(beta) is utilized. The response to complex harmonic excitation is derived and the amplitude-frequency and phase-frequency relations are obtained. For a periodic excitation, we decompose it into the Fourier series, and then make use of the principle of superposition and the results of harmonic excitations to obtain the response. Finally, we examine three numerical examples by using the proposed method.Article Citation - WoS: 15Citation - Scopus: 17Fractional Dynamics of an Erbium-Doped Fiber Laser Model(Springer, 2019) Saad, K. M.; Baleanu, D.; Gomez-Aguilar, J. F.In this paper we investigate the model of the time-fractional dynamics of an erbium-doped fiber laser model (TFDEFL) with Liouville-Caputo (LC), Caputo-Fabrizio-Caputo (CFC) and Atangana-Baleanu-Caputo (ABC) time-fractional derivatives. We employ the homotopy analysis transform method (HATM) to calculate approximate solutions for the TFDEFL model. This method gives the solution in the form of a rapidly convergent series that can ensure the convergence in solving the resultant series. We study the convergence analysis of HATM by computing the interval of convergence through the h-curves, the residual error function and the average residual error, respectively. We also show the effectiveness and accuracy of this method by comparing the approximate solutions based upon the LC, CFC and ABC time-fractional derivatives.Article Citation - WoS: 74Citation - Scopus: 86Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations(Mdpi, 2019) Baleanu, Dumitru; Waheed, Asif; Khan, Mansoor Shaukat; Affan, Hira; Javeed, ShumailaThe analysis of Homotopy PerturbationMethod (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for alpha = 1, is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.Article Citation - WoS: 44Citation - Scopus: 53Some New Fractional-Calculus Connections Between Mittag-Leffler Functions(Mdpi, 2019) Fernandez, Arran; Baleanu, Dumitru; Srivastava, Hari M.We consider the well-known Mittag-Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag-Leffler function as a fractional derivative of the two-parameter Mittag-Leffler function, which is in turn a fractional integral of the one-parameter Mittag-Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag-Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.Article Citation - WoS: 14Hamiltonian Structure of Fractional First Order Lagrangian(Springer/plenum Publishers, 2010) Golmankhaneh, Alireza Khalili; Baleanu, Dumitru; Baleanu, Mihaela Cristina; Golmankhaneh, Ali KhaliliIn this paper, we show that the fractional constraint Hamiltonian formulation, using Dirac brackets, leads to the same equations as those obtained from fractional Euler-Lagrange equations. Furthermore, the fractional Faddeev-Jackiw formalism was constructed.Article Citation - WoS: 62Citation - Scopus: 70Fractional Newtonian Mechanics(de Gruyter Poland Sp Z O O, 2010) Golmankhaneh, Alireza K.; Nigmatullin, Raoul; Golmankhaneh, Ali K.; Baleanu, DumitruIn the present paper, we have introduced the generalized Newtonian law and fractional Langevin equation. We have derived potentials corresponding to different kinds of forces involving both the right and the left fractional derivatives. Illustrative examples have worked out to explain the formalism.Conference Object Citation - WoS: 7Citation - Scopus: 7Nonconservative Systems Within Fractional Generalized Derivatives(Sage Publications Ltd, 2008) Baleanu, Dumitru; Muslih, Sami I.A fractional derivative generalizes an ordinary derivative, and therefore the derivative of the product of two functions differs from that for the classical ( integer) case ; the integration by parts for Riemann-Liouville fractional derivatives involves both the left and right fractional derivatives. Despite these restrictions, fractional calculus models are good candidates for description of nonconservative systems. In this article, nonconservative Lagrangian mechanics are investigated within the fractional generalized derivative approach. The fractional Euler-Lagrange equations based on the Riemann-Liouville fractional derivatives are briefly presented. Using generalized fractional derivatives, we give a meaning for the term which appears in fractional Euler-Lagrange equations and contains the second order fractional derivative. The fractional Lagrangians and Hamiltonians of two illustrative nonconservative mechanical systems are investigated in detail.Article Citation - WoS: 168Citation - Scopus: 192The Hamilton Formalism With Fractional Derivatives(Academic Press inc Elsevier Science, 2007) Nawafleh, Khaled I.; Hijjawi, Raed S.; Muslih, Sami I.; Baleanu, Dumitru; Rabei, Eqab M.Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism. (c) 2006 Elsevier Inc. All rights reserved.