Scopus İndeksli Yayınlar Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651

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Publisher: search.filters.publisher.Sage Publications LtdSubject: search.filters.subject.Fractional Calculus

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Now showing 1 - 9 of 9
  • Article
    Citation - WoS: 4
    Citation - Scopus: 3
    On Generalized Asymmetric Harmonic Oscillator With Quadratic Nonlinearity Within Fractional Variational Principles
    (Sage Publications Ltd, 2024) Baleanu, Dumitru; Jajarmi, Amin; Defterli, Ozlem; Mohammad, Noorhan F. AlShaikh; Asad, Jihad; AlShaikh Mohammad, Noorhan F
    This work studies the nonlinear fractional dynamics of asymmetric harmonic oscillators. The classical description of the physical system is generalized using the principles of fractional variational analysis. As a system of two-coupled fractional differential equations with a quadratic nonlinear component, the fractional Euler-Lagrange equations of the motion of the corresponding system are obtained. The Adams-Bashforth predictor-corrector numerical approach is used to approximate the system's outcomes, which are then simulated comparatively with respect to various model parameter values, including mass, linear and quadratic nonlinear stiffness, and the order of the fractional derivative. The simulations provided the possibility of investigating various dynamical behaviours within the same physical model that is generalized by the use of fractional operators.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 8
    On the New Hadamard Fractional Optimal Control Problems
    (Sage Publications Ltd, 2023) Tajani, Asmae; Jajarmi, Amin; Baleanu, Dumitru; Zguaid, Khalid
    The main goal of this manuscript is to investigate a fractional optimal control problem subject to a dynamical system involving Hadamard fractional derivatives. Necessary conditions for the optimality of the considered problem are derived in terms of the corresponding Euler-Lagrange equations. An iterative method is also proposed to numerically solve the obtained equations from the necessary optimality conditions. Two illustrative examples are considered and simulated in order to show the applicability and efficiency of the proposed method. Numerical simulations show that the used method presents some satisfying results regarding the absolute error values.
  • Conference Object
    Citation - WoS: 233
    Citation - Scopus: 268
    A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems
    (Sage Publications Ltd, 2007) Baleanu, Dumitru; Agrawal, Om P.
    This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann-Liouville Fractional Derivatives (RLFDs). It is demonstrated that right RLFDs automatically arise in the formulation even when the dynamics of the system is described using left RLFDs only. For numerical computation, the FDs are approximated using the Grunwald-Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. Two examples, one time-invariant and the other time-variant, are considered to demonstrate the effectiveness of the formulation. Results show that as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system. The approach requires dividing of the entire time domain into several sub-domains. Further, as the sizes of the sub-domains are reduced, the solutions converge to unique solutions. However, the convergence is slow. A scheme that improves the convergence rate will be considered in a future paper. Other issues to be considered in the future include formulations using other types of derivatives, nonlinear and stochastic fractional optimal controls, existence and uniqueness of the solutions, and the error analysis.
  • Article
    Citation - WoS: 85
    Citation - Scopus: 97
    Suboptimal Control of Fractional-Order Dynamic Systems With Delay Argument
    (Sage Publications Ltd, 2018) Baleanu, Dumitru; Jajarmi, Amin
    In this paper, an efficient linear programming formulation is proposed for a class of fractional-order optimal control problems with delay argument. By means of the Lagrange multiplier in the calculus of variations and using the formula for fractional integration by parts, the Euler-Lagrange equations are derived in terms of a two-point fractional boundary value problem including an advance term as well as the delay argument. The derived equations are then reduced into a linear programming problem by using a Grunwald-Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations. The new scheme is also effective for the delay fractional optimal control problems influenced by the external persistent disturbances. Numerical simulations and comparative results verify that the proposed approach is efficient and easy to implement.
  • Article
    Citation - WoS: 111
    Citation - Scopus: 122
    A Hybrid Functions Numerical Scheme for Fractional Optimal Control Problems: Application To Nonanalytic Dynamic Systems
    (Sage Publications Ltd, 2018) Moradi, L.; Baleanu, D.; Jajarmi, A.; Mohammadi, F.
    In this paper, a numerical scheme based on hybrid Chelyshkov functions (HCFs) is presented to solve a class of fractional optimal control problems (FOCPs). To this end, by using the orthogonal Chelyshkov polynomials, the HCFs are constructed and a general formulation for their operational matrix of the fractional integration, in the Riemann-Liouville sense, is derived. This operational matrix together with HCFs are used to reduce the FOCP to a system of algebraic equations, which can be solved by any standard iterative algorithm. Moreover, the application of presented method to the problems with a nonanalytic dynamic system is investigated. Numerical results confirm that the proposed HCFs method can achieve spectral accuracy to approximate the solution of FOCPs.
  • Article
    Citation - WoS: 111
    Citation - Scopus: 138
    Fractional Optimal Control Problems With Several State and Control Variables
    (Sage Publications Ltd, 2010) Baleanu, Dumitru; Agrawal, Om P.; Defterli, Ozlem
    In many applications, fractional derivatives provide better descriptions of the behavior of dynamic systems than other techniques. For this reason, fractional calculus has been used to analyze systems having noninteger order dynamics and to solve fractional optimal control problems. In this study, we describe a formulation for fractional optimal control problems defined in multi-dimensions. We consider the case where the dimensions of the state and control variables are different from each other. Riemann-Liouville fractional derivatives are used to formulate the problem. The fractional differential equations involving the state and control variables are solved using Grunwald-Letnikov approximation. The performance of the formulation is shown using an example.
  • Conference Object
    Citation - WoS: 4
    Citation - Scopus: 4
    Fractional Wavelet Analysis of the Composite Signals of Two-Component Mixture by Multivariate Spectral Calibration
    (Sage Publications Ltd, 2007) Baleanu, Dumitru; Tas, Kenan; Dinc, Erdal
    The fractional wavelet transform is a new mathematical tool for signal and image analysis. In this study, fractional wavelet analysis of the composite signals of the components in a binary mixture was investigated. In the quantitative evaluation of hydrochlorothiazide and cilazapril, partial least squares calibration was applied to the fractional wavelet coefficients after optimization of the experimental conditions. The validation of the partial least squares approach was tested by analyzing various synthetic mixtures consisting of active compounds of interest. Hydrochlorothiazide and cilazapril in real samples were analyzed using partial least squares calibration based on the fractional wavelet coefficients without a prior chemical separation procedure. This combined method is a potential new chemometric approach and is suitable for quantitative analysis of the components in samples.
  • Conference Object
    Citation - WoS: 33
    Citation - Scopus: 35
    Fractional Euler-Lagrange Equations of Motion in Fractional Space
    (Sage Publications Ltd, 2007) Baleanu, Dumitru; Muslih, Sami I.
    Fractional variational principles have gained considerable importance during the last decade due to their various applications in several areas of science and engineering. In this study, the fractional Euler-Lagrange equations corresponding to a prescribed fractional space are obtained. These equations are obtained using the traditional method of calculus of variations adapted to the case of fractional space. The most general fractional Lagrangian is considered and the limit case when the parameters involved in fractional derivatives are equal to one, is obtained. Two examples are investigated in this study, namely the free particle on fractional space and the fractional simple pendulum, and their corresponding fractional Euler-Lagrange equations ar obtained.
  • Conference Object
    Citation - WoS: 20
    Citation - Scopus: 20
    Heisenberg's Equations of Motion With Fractional Derivatives
    (Sage Publications Ltd, 2007) Tarawneh, Derar M.; Muslih, Sami I.; Baleanu, Dumitru; Rabei, Eqab M.
    Fractional variational principles is a new topic in the field of fractional calculus and it has been subject to intense debate during the last few years. One of the important applications of fractional variational principles is fractional quantization. In this present study, fractional calculus is applied to obtain the Hamiltonian formalism of non-conservative systems. The definition of Poisson bracket is used to obtain the equations of motion in terms of these brackets. The commutation relations and the Heisenberg equations of motion are also obtained. The proposed approach was tested on two examples and good agreements with the classical fractional are reported.