Fractional physical models based on falling body problem

Bahar Acay; Ramazan Ozarslan; Erdal Bas; Bahar Acay; Ramazan Ozarslan; Erdal Bas

doi:10.3934/math.2020170

AIMS Mathematics

2020, Volume 5, Issue 3: 2608-2628. doi: 10.3934/math.2020170

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Research article

Fractional physical models based on falling body problem

Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey

Received: 19 December 2019 Accepted: 26 February 2020 Published: 12 March 2020
MSC : 26A33, 97M50, 70E17

This article is devoted to investigate the fractional falling body problem relied on Newton's second law. We analyze this physical model by means of Atangana-Baleanu fractional derivative in the sense of Caputo (ABC), generalized fractional derivative introduced by Katugampola and generalized ABC containing the Mittag-Leffler function with three parameters $\mathbb{E}_{\alpha, \mu}^{\gamma}(.)$. For that purpose, the Laplace transform (LT) is utilized to obtain fractional solutions. In order to maintain the dimensionality of the physical parameter in the model, we employ an auxiliary parameter $\sigma$ having a relation with the order of fractional operator. Moreover, simulation analysis is carried out by comparing the underlying fractional derivatives with traditional one to grasp the virtue of the results.
- falling body problem,
- physical model,
- fractional calculus,
- non-local operators,
- fractional model
Citation: Bahar Acay, Ramazan Ozarslan, Erdal Bas. Fractional physical models based on falling body problem[J]. AIMS Mathematics, 2020, 5(3): 2608-2628. doi: 10.3934/math.2020170

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Abstract

This article is devoted to investigate the fractional falling body problem relied on Newton's second law. We analyze this physical model by means of Atangana-Baleanu fractional derivative in the sense of Caputo (ABC), generalized fractional derivative introduced by Katugampola and generalized ABC containing the Mittag-Leffler function with three parameters $\mathbb{E}_{\alpha, \mu}^{\gamma}(.)$. For that purpose, the Laplace transform (LT) is utilized to obtain fractional solutions. In order to maintain the dimensionality of the physical parameter in the model, we employ an auxiliary parameter $\sigma$ having a relation with the order of fractional operator. Moreover, simulation analysis is carried out by comparing the underlying fractional derivatives with traditional one to grasp the virtue of the results.

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