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  • MDPI Publishing  (18)
  • 11
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    MDPI Publishing
    In: Entropy
    Publication Date: 2016-04-20
    Description: In this paper, we consider the local fractional decomposition method, variational iteration method, and differential transform method for analytic treatment of linear and nonlinear local fractional differential equations, homogeneous or nonhomogeneous. The operators are taken in the local fractional sense. Some examples are given to demonstrate the simplicity and the efficiency of the presented methods.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
    Published by MDPI Publishing
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  • 12
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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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  • 13
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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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  • 14
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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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  • 15
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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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  • 16
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    MDPI Publishing
    In: Entropy
    Publication Date: 2015-09-09
    Description: The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
    Published by MDPI Publishing
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  • 17
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    MDPI Publishing
    In: Entropy
    Publication Date: 2016-02-06
    Description: In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE) by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
    Published by MDPI Publishing
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  • 18
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    MDPI Publishing
    In: Entropy
    Publication Date: 2016-01-30
    Description: In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on the fractals subset of real-line lies in the fact that they are better at modeling processes with memory effect.
    Electronic ISSN: 1099-4300
    Topics: Chemistry and Pharmacology , Physics
    Published by MDPI Publishing
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