Scopus İndeksli Yayınlar Koleksiyonu

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

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End date: 2019Subject: search.filters.subject.Fractional Calculus

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Now showing 1 - 10 of 80
  • Article
    Citation - WoS: 180
    Citation - Scopus: 192
    On Fractional Calculus with General Analytic Kernels
    (Elsevier Science Inc, 2019) Fernandez, Arran; Ozarslan, Mehmet Ali; Baleanu, Dumitru
    Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators. (C) 2019 Elsevier Inc. All rights reserved.
  • Article
    Citation - Scopus: 2
    Non-Integer Variable Order Dynamic Equations on Time Scales Involving Caputo-Fabrizio Type Differential Operator
    (Eudoxus Press, LLC, 2018) Baleanu, D.; Baleanu, Dumitru; Nategh, M.; Matematik
    This work deals with the conecept of a Caputo-Fabrizio type non-integer variable order differential opertor on time scales that involves a non-singular kernel. A measure theoretic discussion on the limit cases for the order of differentiation is presented. Then, corresponding to the fractional derivative, we discuss on an integral for constant and variable orders. Beside the obtaining solutions to some dynamic problems on time scales involving the proposed derivative, a fractional folrmulation for the viscoelastic oscillation problem is studied and its conversion into a third order dynamic equation is presented. © 2018 by Eudoxus Press, LLC. All rights reserved.
  • Article
    Citation - Scopus: 3
    On Some Impulsive Fractional Neutral Differential Systems With Nonlocal Condition Through Fractional Operators
    (Cambridge Scientific Publishers, 2017) Anuradha, A.; Baleanu, Dumitru; Baleanu, D.; Suganya, S.; Arjunan, M.M.; Matematik
    According to semigroup theories, fractional calculus, Banach contraction principle and Schaefer's fixed point theorem, this paper is fundamentally involved with the existence of mild solutions for an impulsive fractional neutral differential systems (abbreviated, IFNDS) with nonlocal conditions (abbreviated, NLC) in Banach space X. At last, an illustration is also presented to exhibit the use of our theoretical results. © CSP - Cambridge, UK; I & S - Florida, USA, 2017.
  • Article
    Citation - WoS: 59
    Citation - Scopus: 66
    New Aspects of the Motion of a Particle in a Circular Cavity
    (Editura Acad Romane, 2018) Baleanu, Dumitru; Baleanu, Dumitru; Asad, Jihad H.; Jajarmi, Amin; Matematik
    In this work, we consider the free motion of a particle in a circular cavity. For this model, we obtain the classical and fractional Lagrangian as well as the fractional Hamilton's equations (FHEs) of motion. The fractional equations are formulated in the sense of Caputo and a new fractional derivative with Mittag-Leffler nonsingular kernel. Numerical simulations of the FHEs within these two 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. Simulation results show that the fractional calculus provides more flexible models demonstrating new aspects of the real-world phenomena.
  • Article
    Citation - WoS: 43
    Citation - Scopus: 40
    Motion of a Particle in a Resisting Medium Using Fractional Calculus Approach
    (Editura Acad Romane, 2013) Rosales Garcia, J. Juan; Baleanu, Dumitru; Guia Calderon, M.; Martinez Ortiz, Juan; Baleanu, Dumitru; Garcia, J. Juan Rosales; Calderon, M. Guia; Ortiz, Juan Martinez; Matematik
    In this manuscript we propose a fractional differential equation to describe the vertical motion of a body through the air. The order of the derivative was considered to be 0 < gamma <= 1. To keep the dimensionality of the physical parameter in the system, an auxiliary parameter sigma is introduced. This parameter characterizes the existence of fractional components in the given system. We prove that there is a relation between gamma and sigma through the physical parameter of the system and that, due to this relation the analytical solutions are given in terms of the Mittag-Leffler function depending on the order of the fractional differential equation.
  • Article
    Citation - WoS: 20
    Citation - Scopus: 22
    Analytical Treatment of System of Abel Integral Equations by Homotopy Analysis Method
    (Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; Matematik
    Abel equation has important applications in describing the least time for an object which is sliding on surface without friction in uniform gravity, and the classical theory of elasticity of materials is modeled by a system of Abel integral equations. In this manuscript, the homotopy analysis method is presented for obtaining analytical solutions of a system of Abel integral equations as fractional equations. The applied method has lessened the size of calculation and improved the accuracy of solution in the case of the singular Abel integral equation. The illustrated examples and numerical results have proved the assertion.
  • Article
    Citation - WoS: 19
    Citation - Scopus: 20
    Fractional Calculus Analysis of the Cosmic Microwave Background
    (Editura Acad Romane, 2013) Tenreiro Machado, J. A.; Baleanu, Dumitru; Stefanescu, Petruta; Tintareanu, Ovidiu; Baleanu, Dumitru; Matematik
    Cosmic microwave background (CMB) radiation is the imprint from an early stage of the Universe and investigation of its properties is crucial for understanding the fundamental laws governing the structure and evolution of the Universe. Measurements of the CMB anisotropies are decisive to cosmology, since any cosmological model must explain it. The brightness, strongest at the microwave frequencies, is almost uniform in all directions, but tiny variations reveal a spatial pattern of small anisotropies. Active research is being developed seeking better interpretations of the phenomenon. This paper analyses the recent data in the perspective of fractional calculus. By taking advantage of the inherent memory of fractional operators some hidden properties are captured and described.
  • Article
    Citation - WoS: 5
    Citation - Scopus: 7
    About Fractional Calculus of Singular Lagrangians
    (Fuji Technology Press Ltd, 2005) Baleanu, Dumitru
    In this paper the solutions of the fractional Euler-Lagrange equations corresponding to singular fractional Lagrangians were examined. We observed that if a Lagrangian is singular in the classical sense, it remains singular after being fractionally generalized. The fractional Lagrangian is non-local but its gauge symmetry was preserved despite complexity of equations in fractional cases. We generalized four examples of singular Lagrangians admitting gauge symmetry in fractional case and found solutions to corresponding Euler-Lagrange equations.
  • Book Part
    Citation - WoS: 25
    Citation - Scopus: 1
    New Treatise in Fractional Dynamics
    (Springer-verlag Berlin, 2012) Baleanu, Dumitru; Baleanu, Dumitru
    Fractional calculus becomes a powerful tool used to investigate complex phenomena from various fields of science and engineering. In this context, the researchers paid a lot of attention for the fractional dynamics. However, the fractional modeling is still at the beginning of its developing. The aim of this chapter is to present some new results in the area of fractional dynamics and its applications.
  • Article
    Citation - WoS: 26
    Citation - Scopus: 32
    On a More General Fractional Integration by Parts Formulae and Applications
    (Elsevier, 2019) Gomez-Aguilar, J. F.; Jarad, Fahd; Abdeljawad, Thabet; Atangana, Abdon
    The integration by part comes from the product rule of classical differentiation and integration. The concept was adapted in fractional differential and integration and has several applications in control theory. However, the formulation in fractional calculus is the classical integral of a fractional derivative of a product of a fractional derivative of a given function f and a function g. We argue that, this formulation could be done using only fractional operators: thus, we develop fractional integration by parts for fractional integrals, Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. We allow the left and right fractional integrals of order alpha > 0 to act on the integrated terms instead of the usual integral and then make use of the fractional type Leibniz rules to formulate the integration by parts by means of new generalized type fractional operators with binomial coefficients defined for analytic functions. In the case alpha = 1, our formulae of fractional integration by parts results in previously obtained integration by parts in fractional calculus. The two disciplines or branches of mathematics are built differently, while classical differentiation is built with the concept of rate of change of a given function, a fractional differential operator is a convolution. (C) 2019 Elsevier B.V. All rights reserved.