Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media

Gómez-Aguilar, J. F.; Escobar-Jiménez, R.F.; López, G.; Alvarado-Martínez, V.M.

doi:10.1080/09205071.2016.1225521

Cited by 78 publications

(29 citation statements)

References 33 publications

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“…The other definition, which is defined by Yang et al [7] in 2017, is based on the normalized sinc function (NSFDO). Especially in recent years, many important theoretical results and applications have been obtained using an operator which has non-local and non-singular kernel such as the Caputo-Fabrizio, NSFDO and ABO [8][9][10][11][12][13][14][15][16][17][18]. Both of these operators have some illustrative advantages according to the operators defined previously such as Caputo [19] and conformable [20].…”

Section: Yavuzmentioning

confidence: 99%

Characterizations of two different fractional operators without singular kernel

Yavuz

2019

Math. Model. Nat. Phenom.

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In this paper, we analyze the behaviours of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a non-singular kernel. The fractional derivative operator defined with the MLF is developed by Atangana and Baleanu (ABO) in 2016 and the other operator defined with the normalized sinc function (NSFDO) is created by Yang et al. in 2017. These mentioned operators have some advantages to model the real life problems and to solve them. On the other hand, since the Laplace transform (LT) of the ABO can be calculated more easily, it can be preferred to solve linear/nonlinear problems. In this study, we use the perturbation method with coupled the LTs of these operators to analyze their performance in solving some fractional differential equations. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

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Section: Yavuzmentioning

confidence: 99%

Characterizations of two different fractional operators without singular kernel

Yavuz

2019

Math. Model. Nat. Phenom.

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“…Removing the singularity of kernels at t = 0 allows to highlight memory effects better [21]. In this paper, we consider the following bounded kernels: [21,22]; [23,24]. Here, E β and E β,β are one-parametric and two-parametric Mittag-Leffler functions, respectively, given by the formulas:…”

Section: Generalized Fractional Derivativesmentioning

confidence: 99%

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

Kinash

Janno

2019

Mathematics

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In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.

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“…Go´mez-Aguilar et al 27 presented some electrical circuits in terms of Caputo-Fabrizio fractional operator; they obtained numerical simulations of these circuits by applying the numerical Laplace transform algorithm, and more works related to this fractional operator are given by Atangana and Alkahtani, 28 Atangana and Nieto, 29 Atangana and Alkahtani, 30 Go´mez-Aguilar et al, 31 and Alsaedi et al 32,33 Recently, Atangana and Baleanu proposed a new definition with non-local and nonsingular kernel based on the Mittag-Leffler function, this definition has all the benefits of the fractional derivatives of Liouville-Caputo and Caputo-Fabrizio types. [34][35][36][37][38] The fractional operators mentioned above have been widely used. However, none of these fractional operators is capable of handling the concept of heterogeneity with great success.…”

Section: Introductionmentioning

confidence: 99%

Electrical circuits RC and RL involving fractional operators with bi-order

Gómez-Aguilar

Escobar-Jiménez

Olivares-Peregrino

et al. 2017

Advances in Mechanical Engineering

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This article describes electrical series circuits RC and RL using the concept of derivative with two fractional orders a and b in Liouville-Caputo sense. The fractional equations consider derivatives in the range of a, b 2 (0; 1. Numerical solutions are presented considering different source terms introduced in the fractional equation. This new approach considers electrical elements with two different properties. In addition, we prove that if a = 0, the fractional derivative with Mittag-Leffler kernel in Liouville-Caputo sense is recovered, and when b = 0, the Liouville-Caputo fractional derivative is recovered.

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Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media

Cited by 78 publications

References 33 publications

Characterizations of two different fractional operators without singular kernel

Characterizations of two different fractional operators without singular kernel

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

Electrical circuits RC and RL involving fractional operators with bi-order

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