Scopus İndeksli Yayınlar Koleksiyonu

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

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Subject: search.filters.subject.Euler-Lagrange Equation

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Now showing 1 - 7 of 7
  • Article
    Citation - WoS: 128
    Citation - Scopus: 138
    Fractional Differences and Integration by Parts
    (Eudoxus Press, Llc, 2011) Abdeljawad, Thabet; Abdeljawad, Thabet; Baleanu, Dumitru; Baleanu, Dumitru; Matematik
    In this paper we define the right fractional sum and difference following the delta time scale calculus and obtain results on them analogous to those obtained for the left ones studied in [6], [7], [8]. In addition of that a formula for the integration by parts was obtained. The obtained formula is used to obtain a discrete Euler-Lagrange equation in fractional calculus.
  • Article
    Citation - WoS: 69
    Citation - Scopus: 73
    Classical and Fractional Aspects of Two Coupled Pendulums
    (Editura Acad Romane, 2019) Baleanu, D.; Baleanu, Dumitru; Jajarmi, A.; Asad, J. H.; Matematik
    In this study, we consider two coupled pendulums (attached together with a spring) having the same length while the same masses are attached at their ends. After setting the system in motion we construct the classical Lagrangian, and as a result, we obtain the classical Euler-Lagrange equation. Then, we generalize the classical Lagrangian in order to derive the fractional Euler-Lagrange equation in the sense of two different fractional operators. Finally, we provide the numerical solution of the latter equation for some fractional orders and initial conditions. The method we used is based on the Euler method to discretize the convolution integral. Numerical simulations show that the proposed approach is efficient and demonstrate new aspects of the real-world phenomena.
  • Article
    Citation - WoS: 67
    Citation - Scopus: 69
    The Fractional Model of Spring Pendulum: New Features Within Different Kernels
    (Editura Acad Romane, 2018) Baleanu, Dumitru; Baleanu, Dumitru; Asad, Jihad H.; Jajarmi, Amin; Matematik
    In this work, new aspects of the fractional calculus (FC) are examined for a model of spring pendulum in fractional sense. First, we obtain the classical Lagrangian of the model, and as a result, we derive the classical Euler-Lagrange equations of the motion. Second, we generalize the classical Lagrangian to fractional case and derive the fractional Euler-Lagrange equations in terms of fractional derivatives with singular and nonsingular kernels, respectively. Finally, we provide the numerical solution of these equations within two fractional operators for some fractional orders and initial conditions. Numerical simulations verify that taking into account the recently features of the FC provides more realistic models demonstrating hidden aspects of the real-world phenomena.
  • Article
    Citation - WoS: 86
    Citation - Scopus: 97
    On Fractional Derivatives With Generalized Mittag-Leffler Kernels
    (Pushpa Publishing House, 2018) Abdeljawad, Thabet; Baleanu, Dumitru
    Fractional derivatives with three parameter generalized Mittag-Leffler kernels and their properties are studied. The corresponding integral operators are obtained with the help of Laplace transforms. The action of the presented fractional integrals on the Caputo and Riemann type derivatives with three parameter Mittag-Leffler kernels is analyzed. Integration by parts formulas in the sense of Riemann and Caputo are proved and then used to formulate the fractional Euler-Lagrange equations with an illustrative example. Certain nonconstant functions whose fractional derivatives are zero are determined as well.
  • Article
    Citation - WoS: 8
    Citation - Scopus: 8
    A New Type of Equation of Motion and Numerical Method for a Harmonic Oscillator With Left and Right Fractional Derivatives
    (Elsevier, 2020) Baleanu, Dumitru; Ullah, Malik Zaka
    The aim of this research is to propose a new fractional Euler-Lagrange equation for a harmonic oscillator. The theoretical analysis is given in order to derive the equation of motion in a fractional framework. The new equation has a complicated structure involving the left and right fractional derivatives of Caputo-Fabrizio type, so a new numerical method is developed in order to solve the above-mentioned equation effectively. As a result, we can see different asymptotic behaviors according to the flexibility provided by the use of the fractional calculus approach, a fact which may be invisible when we use the classical Lagrangian technique. This capability helps us to better understand the complex dynamics associated with natural phenomena.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 7
    On the Motion of a Heavy Bead Sliding on a Rotating Wire - Fractional Treatment
    (Elsevier, 2018) Asad, Jihad H.; Alipour, Mohsen; Baleanu, Dumitru
    In this work, we consider the motion of a heavy particle sliding on a rotating wire. The first step carried for this model is writing the classical and fractional Lagrangian. Secondly, the fractional Hamilton's equations (FHEs) of motion of the system is derived. The fractional equations are formulated in the sense of Caputo. Thirdly, numerical simulations of the FHEs within the fractional operators are presented and discussed for some fractional derivative orders. Numerical results are based on a discretization scheme using the Euler convolution quadrature rule for the discretization of the convolution integral. Finally, simulation results verify that, taking into account the fractional calculus provides more flexible models demonstrating new aspects of the real world phenomena.
  • Article
    On fractional derivatives with generalized Mittag-Leffler kernels
    (Pushpa Publishing House, 2018) Abdeljawad, Thabet; Baleanu, Dumitru
    Fractional derivatives with three parameter generalized Mittag-Leffler kernels and their properties are studied. The corresponding integral operators are obtained with the help of Laplace transforms. The action of the presented fractional integrals on the Caputo and Riemann type derivatives with three parameter Mittag-Leffler kernels is analyzed. Integration by parts formulas in the sense of Riemann and Caputo are proved and then used to formulate the fractional Euler-Lagrange equations with an illustrative example. Certain nonconstant functions whose fractional derivatives are zero are determined as well.