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Uncertain fractional differential equations on a time scale under Granular differentiability concept

Computational and Applied Mathematics volume 38, Article number: 110 (2019) Cite this article

Abstract

In this work, the Caputo q-fractional initial value problem of order \(\alpha \in (0,1)\) in uncertain environment is investigated. The uncertainties are considered as possibility sets using fuzzy sets. The concepts of horizontal membership function and granular difference are used to give a new definition for fuzzy fractional integral and derivative, called granular Riemann–Liouville q-fractional integral and granular Caputo q-fractional derivative. The existence and uniqueness of solution for the Caputo q-fractional initial value problem under granular differentiability concept is established. Finally, some illustrative examples are also presented.

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Acknowledgements

The authors would like to express deep gratitude to the Associate Editor Professor Rosana Jafelice and anonymous referees for their valuable comments and suggestions which have greatly improved the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2018.311.

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Authors and Affiliations

  1. Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Ho Vu

  2. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ngo Van Hoa

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ngo Van Hoa

Authors
  1. Ho Vu
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  2. Ngo Van Hoa
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Correspondence to Ngo Van Hoa.

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Communicated by Rosana Sueli da Motta Jafelice.

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Vu, H., Hoa, N.V. Uncertain fractional differential equations on a time scale under Granular differentiability concept. Comp. Appl. Math. 38, 110 (2019). https://doi.org/10.1007/s40314-019-0873-x

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  • DOI: https://doi.org/10.1007/s40314-019-0873-x

Keywords

  • Horizontal membership function
  • Granular difference
  • Granular derivative
  • Riemann–Liouville q-fractional integral
  • Caputo q-fractional derivative

Mathematics Subject Classification

  • 03E72
  • 05A30
  • 33D05
  • 39A13
  • 74H20
  • 74H25