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Entropy 2015, 17(2), 885-902; doi:10.3390/e17020885

On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem

1
Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran
2
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400UPM, Serdang, Selangor, Malaysia
3
Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, Balgat 0630, Ankara, Turkey
4
Institute of Space Sciences, P.O.Box MG-23, Magurele-Bucharest, Romania
5
Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India
*
Author to whom correspondence should be addressed.
Received: 24 November 2014 / Revised: 15 January 2015 / Accepted: 15 January 2015 / Published: 16 February 2015
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
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Abstract

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing length of their support (Hukuhara difference). Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method. View Full-Text
Keywords: fuzzy fractional differential equation; fuzzy Laplace transform; Caputo differentiability; dynamical systems; Basset problem fuzzy fractional differential equation; fuzzy Laplace transform; Caputo differentiability; dynamical systems; Basset problem
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Salahshour, S.; Ahmadian, A.; Senu, N.; Baleanu, D.; Agarwal, P. On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem. Entropy 2015, 17, 885-902.

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