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Review Citation - WoS: 15Citation - Scopus: 21A freely damped oscillating fractional dynamic system modeled by fractional Euler-Lagrange equations(Sage Publications Ltd, 2018) Agila, Adel; Baleanu, Dumitru; Eid, Rajeh; Irfanoglu, BulentThe behaviors of some vibrating dynamic systems cannot be modeled precisely by means of integer representation models. Fractional representation looks like it is more accurate to model such systems. In this study, the fractional Euler-Lagrange equations model is introduced to model a fractional damped oscillating system. In this model, the fractional inertia force and the fractional damping force are proportional to the fractional derivative of the displacement. The fractional derivative orders in both forces are considered to be variable fractional orders. A numerical approximation technique is utilized to obtain the system responses. The discretization of the Coimbra fractional derivative and the finite difference technique are used to accomplish this approximation. The response of the system is verified by a comparison to a classical integer representation and is obtained based on different values of system parameters.Article Citation - WoS: 30Citation - Scopus: 34Fractional Unit-Root Tests Allowing for a Fractional Frequency Flexible Fourier Form Trend: Predictability of Covid-19(Springer, 2021) Omay, Tolga; Baleanu, DumitruIn this study we propose a fractional frequency flexible Fourier form fractionally integrated ADF unit-root test, which combines the fractional integration and nonlinear trend as a form of the Fourier function. We provide the asymptotics of the newly proposed test and investigate its small-sample properties. Moreover, we show the best estimators for both fractional frequency and fractional difference operator for our newly proposed test. Finally, an empirical study demonstrates that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the Covid-19 pandemic.Article A Computationally Efficient Approximation for Fractional Differencing: First-Order Operators(Pergamon-Elsevier Science Ltd, 2026) Omay, Tolga; Baleanu, DumitruThis paper introduces the First-Order Fractional Differencing (FOFD) operator that substantially reduces the computational burden of fractional differencing for large-scale applications. While the standard Gr & uuml;nwald-Letnikov (GL) operator requires O(T2) operations for a series of length T, and recent FFT-based methods achieve O(T log T), our FOFD operator requires only O(T) operations through a simple two-point recursion. We develop an optimal weight calibration framework that ensures this computational efficiency does not compromise statistical accuracy, deriving a general formula wopt = d & sdot; (1-0.9 rho)beta(p) that adapts to the persistence structure of autoregressive processes. Empirical applications demonstrate substantial improvements: for the Chicago Fed National Financial Conditions Index with extreme persistence (rho= 0.992), optimal weight calibration reduces approximation error by 93% while preserving the autocorrelation structure of the GL operator. For a series of 10,000 observations, our method requires 20,000 operations compared to 530,000 for FFT-based methods and 50 million for standard implementations-enabling fractional differencing in real-time and high-frequency contexts previously infeasible due to computational constraints. The method's simplicity, requiring no specialized libraries and providing direct implementation through our calibration formula, makes it immediately accessible to practitioners while maintaining the long-memory properties essential for financial time series modeling.Article Citation - WoS: 53Citation - Scopus: 65On Fractional schrodinger Equation in Α-Dimensional Fractional Space(Pergamon-elsevier Science Ltd, 2009) Eid, Rajeh; Muslih, Sami I.; Baleanu, Dumitru; Rabei, E.The Schrodinger equation is solved in a-dimensional fractional space with a Coulomb potential proportional to 1/r(beta-2), 2 <= beta <= 4. The wave functions are studied in terms of spatial dimensionality alpha and beta and the results for beta = 3 are compared with those obtained in the literature. (C) 2008 Elsevier Ltd. All rights reserved.Article Citation - WoS: 17Citation - Scopus: 22Fractional Dimensional Harmonic Oscillator(Editura Acad Romane, 2011) Eid, R.; Muslih, Sami I.; Baleanu, Dumitru; Rabei, E.; MathematicsThe fractional Schrodinger equation corresponding to the fractional oscillator was investigated. The regular singular points and the exact solutions of the corresponding radial Schrodinger equation were reported.Article Citation - WoS: 48APPLICATIONS OF THE EXTENDED FRACTIONAL EULER-LAGRANGE EQUATIONS MODEL TO FREELY OSCILLATING DYNAMICAL SYSTEMS(Editura Acad Romane, 2016) Agila, Adel; Baleanu, Dumitru; Eid, Rajeh; Irfanoglu, BulentThe fractional calculus and the calculus of variations are utilized to model and control complex dynamical systems. Those systems are presented more accurately by means of fractional models. In this study, an extended version of the fractional Euler-Lagrange equations is introduced. In these equations the damping force term is extended to be proportional to the fractional derivative of the displacement with variable fractional order. The finite difference methods and the Coimbra fractional derivative are used to approximate the solution of the introduced fractional Euler-Lagrange equations model. The free oscillating single pendulum system is investigated.
