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In the last decade, theoretical and applied studies were done in order to provide a suitable definition of fractional derivative, which meets all the requirement of a derivative in its primary sense. It was concluded by some eminent researchers that the Riemann-Liouville version was the most suitable. However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann-Liouville definition, from power-law kernel to generalized Mittag-Leffler-law via exponential-decay-law. K E Y W O R D S exponential-decay-law, Mittag-Leffler-law, numerical approximation, power-law, Riemann-Liouville definition Numer Methods Partial Differential Eq. 2017;1-22.wileyonlinelibrary.com/journal/num

In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo-Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag-Leffler function; for the Caputo-Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one.

Summary In this paper, the analytical solutions for the electrical series circuits RC, LC, and RL using novel fractional derivatives of type Atangana–Baleanu with non‐singular and nonlocal kernel in Liouville–Caputo and Riemann–Liouville sense were obtained. The fractional equations in the time domain are considered derivatives in the range α∈(0;1]; analytical solutions are presented considering different source terms introduced in the fractional equation. We solved analytically the fractional equation using the properties of Laplace transform operator together with the convolution theorem. On the basis of the Mittag–Leffler function, new behaviors for the voltage and current were obtained; the classical cases are recovered when α=1. Copyright © 2017 John Wiley & Sons, Ltd.

Abstract:Surface deformation in Morocco derived from five years of GPS survey observations of a 22-station network, four continuously recording GPS stations, and four IGS stations in Iberia indicate roughly southward motion (~3 mm/yr) of the Rif Mountains, Morocco relative to stable Africa. Motion of the Rif is approximately normal to the direction of Africa-Eurasia relative motion, which is predominantly strike slip, and results in shortening of the Rif and subsequent crustal extension of the adjacent Alboran Sea region. The sense, and the N-S asymmetry of the observed deformation (i.e., no evidence for north-directed shortening in the Betic Mountains north of the Alboran Sea) cannot be easily explained in terms of crustal plate interactions suggesting that dynamic processes below the crust are driving the recent geologic evolution of the western Mediterranean. The model that best fits the observations involves delamination and southward roll back of the African lithospheric mantle under the Alboran and Rif domains.

Abstract:In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R L and C an auxiliary parameter σ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between γ and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order γ of the fractional differential equation.PACS (2008)

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