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  • MDPI Publishing  (18)
  • 1
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    MDPI Publishing
    In: Entropy
    Publication Date: 2017-01-29
    Description: In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based on the Mittag–Leffler kernel with arbitrary order α. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when α is equal to 1.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
    Published by MDPI Publishing
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  • 2
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    MDPI Publishing
    In: Entropy
    Publication Date: 2018-04-10
    Description: Entropy, Vol. 20, Pages 259: An Efficient Computational Technique for Fractal Vehicular Traffic Flow Entropy doi: 10.3390/e20040259 Authors: Devendra Kumar Fairouz Tchier Jagdev Singh Dumitru Baleanu In this work, we examine a fractal vehicular traffic flow problem. The partial differential equations describing a fractal vehicular traffic flow are solved with the aid of the local fractional homotopy perturbation Sumudu transform scheme and the local fractional reduced differential transform method. Some illustrative examples are taken to describe the success of the suggested techniques. The results derived with the aid of the suggested schemes reveal that the present schemes are very efficient for obtaining the non-differentiable solution to fractal vehicular traffic flow problem.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 3
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    MDPI Publishing
    In: Entropy
    Publication Date: 2018-05-21
    Description: Entropy, Vol. 20, Pages 384: Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors Entropy doi: 10.3390/e20050384 Authors: Jesús Emmanuel Solís Pérez José Francisco Gómez-Aguilar Dumitru Baleanu Fairouz Tchier This paper deals with a numerical simulation of fractional conformable attractors of type Rabinovich–Fabrikant, Thomas’ cyclically symmetric attractor and Newton–Leipnik. Fractional conformable and β -conformable derivatives of Liouville–Caputo type are considered to solve the proposed systems. A numerical method based on the Adams–Moulton algorithm is employed to approximate the numerical simulations of the fractional-order conformable attractors. The results of the new type of fractional conformable and β -conformable attractors are provided to illustrate the effectiveness of the proposed method.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 4
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    MDPI Publishing
    In: Entropy
    Publication Date: 2018-07-03
    Description: Entropy, Vol. 20, Pages 504: Diffusion on Middle-ξ Cantor Sets Entropy doi: 10.3390/e20070504 Authors: Alireza Khalili Golmankhaneh Arran Fernandez Ali Khalili Golmankhaneh Dumitru Baleanu In this paper, we study Cζ-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the Cζ-calculus on the generalized Cantor sets known as middle-ξ Cantor sets. We have suggested a calculus on the middle-ξ Cantor sets for different values of ξ with 0<ξ<1. Differential equations on the middle-ξ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 5
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    MDPI Publishing
    In: Symmetry
    Publication Date: 2018-08-16
    Description: Symmetry, Vol. 10, Pages 341: Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation Symmetry doi: 10.3390/sym10080341 Authors: Aliyu Isa Aliyu Mustafa Inc Abdullahi Yusuf Dumitru Baleanu In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
    Published by MDPI Publishing
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  • 6
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    MDPI Publishing
    In: Entropy
    Publication Date: 2018-04-27
    Description: Entropy, Vol. 20, Pages 321: Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative Entropy doi: 10.3390/e20050321 Authors: Sadia Arshad Dumitru Baleanu Jianfei Huang Maysaa Mohamed Al Qurashi Yifa Tang Yue Zhao In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 7
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    MDPI Publishing
    In: Entropy
    Publication Date: 2016-11-24
    Description: In this manuscript, we prove the existence and uniqueness of solutions for local fractional differential equations (LFDEs) with local fractional derivative operators (LFDOs). By using the contracting mapping theorem (CMT) and increasing and decreasing theorem (IDT), existence and uniqueness results are obtained. Some examples are presented to illustrate the validity of our results.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
    Published by MDPI Publishing
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  • 8
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    MDPI Publishing
    In: Entropy
    Publication Date: 2016-08-21
    Description: In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. Different source terms are considered in the fractional differential equations. The classical behaviors are recovered when the fractional order α is equal to 1.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 9
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    MDPI Publishing
    In: Entropy
    Publication Date: 2015-02-17
    Description: In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing length of their support (Hukuhara difference). Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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  • 10
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    MDPI Publishing
    In: Entropy
    Publication Date: 2014-11-29
    Description: In this paper, based on fixed entropy, the adiabatic equation of state in fractal flow is discussed. The local fractional wave equation for the velocity potential is also obtained by using the non-differential perturbations for the pressure and density of fractal hydrodynamics.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
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