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  • SpringerOpen  (26)
  • MDPI Publishing  (18)
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  • 1
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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
    Published by MDPI Publishing
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  • 2
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    SpringerOpen
    Publication Date: 2016-09-07
    Description: In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and th...
    Print ISSN: 1687-1839
    Electronic ISSN: 1687-1847
    Topics: Mathematics
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  • 3
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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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  • 4
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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
    Published by MDPI Publishing
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  • 5
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    SpringerOpen
    Publication Date: 2015-07-17
    Description: In this work, we study the fractional order Lane-Emden differential equations by using the reproducing kernel method. The exact solution is shown in the form of a series in the reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of the present method. The results obtained from the method are compared with the exact solutions and another method. The obtained numerical results are better than the ones provided by the collocation method. Results of numerical examples show that the presented method is simple, effective, and easy to use.
    Print ISSN: 1687-1839
    Electronic ISSN: 1687-1847
    Topics: Mathematics
    Published by SpringerOpen
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  • 6
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    SpringerOpen
    Publication Date: 2015-08-07
    Description: In this paper, we discussed the existence of solutions for the fractional finite difference inclusion Δ ν x ( t ) ∈ F ( t , x ( t ) , Δ x ( t ) , Δ 2 x ( t ) ) via the boundary value conditions ξ x ( ν − 3 ) + β Δ x ( ν − 3 ) = 0 , x ( η ) = 0 , and γ x ( b + ν ) + δ Δ x ( b + ν ) = 0 , where η ∈ N ν − 2 b + ν − 1 , 2 〈 ν 〈 3 , and F : N ν − 3 b + ν + 1 × R × R × R → 2 R is a compact valued multifunction.
    Print ISSN: 1687-1839
    Electronic ISSN: 1687-1847
    Topics: Mathematics
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  • 7
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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
    Published by MDPI Publishing
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  • 8
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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
    Published by MDPI Publishing
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  • 9
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    SpringerOpen
    Publication Date: 2018-03-12
    Description: We establish analogues of the mean value theorem and Taylor’s theorem for fractional differential operators defined using a Mittag–Leffler kernel. We formulate a new model for the fractional Boussinesq equatio...
    Print ISSN: 1687-1839
    Electronic ISSN: 1687-1847
    Topics: Mathematics
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  • 10
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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
    Published by MDPI Publishing
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