Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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Book Part Citation - Scopus: 1Fractional Lagrangian and Hamiltonian Mechanics With Memory(De Gruyter, 2019) Muslih, S.I.; Baleanu, D.Fractional variational principles are very important for science and engineering. Within this field of study, the fractional Lagrangian and Hamiltonian equations are challenging ones from the viewpoint of mathematics. During the last fifteen years, the field of fractional variational principles was continuously improved and developed. In this chapter, the fractional variational principles-with and without delay-will be briefly reviewed. Several illustrative examples from mechanics are presented. © 2019 Walter de Gruyter GmbH, Berlin/Boston.Article Citation - WoS: 20Citation - Scopus: 23A Numerical Scheme for Two-Dimensional Optimal Control Problems With Memory Effect(Pergamon-elsevier Science Ltd, 2010) Defterli, OzlemA new formulation for multi-dimensional fractional optimal control problems is presented in this article. The fractional derivatives which are coming from the formulation of the problem are defined in the Riemann-Liouville sense. Some terminal conditions are imposed on the state and control variables whose dimensions need not be the same. A numerical scheme is described by using the Grunwald-Letnikov definition to approximate the Riemann-Liouville Fractional Derivatives. The set of fractional differential equations, which are obtained after the discretization of the time domain, are solved within the Grunwald-Letnikov approximation to obtain the state and the control variable numerically. A two-dimensional fractional optimal control problem is studied as an example to demonstrate the performance of the scheme. (C) 2009 Elsevier Ltd. All rights reserved.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.