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q-fractional differential equations with uncertainty

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Abstract

In this paper, first we are going to introduce the fuzzy q-derivative and fuzzy q-fractional derivative in Caputo sense by using generalized Hukuhara difference, and then provide the related theorems and properties in detail. Moreover, the characterization theorem between the solutions of fuzzy Caputo q-fractional initial value problem (for short FCqF-IVP), which allows us to translate a FCqF-IVP and system of ordinary Caputo q-fractional differential equations (for short OCqF-DEs), is presented. In detail, the existence and uniqueness theorem is proved for the solution of FCqF-IVP. Finally, we restrict our attention to explain our idea for solving the FCqF-IVP and introducing its numerical solution by means of q-Mittag-Leffler function. The numerical examples demonstrate that the proposed idea is quite reasonable.

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Notes

  1. \([\alpha ]\) denotes the smallest integer greater or equal to \(\alpha \).

  2.  The set of fuzzy-valued function f which are defined on \(_{q}\Omega \), and be fuzzy continuous from the interior points of \(_{q}\Omega \).

  3.  i.e., for any \(\varepsilon >0\) and any \((t, x, y)\in \mathbb {T}_{q}\times \mathbb {R}^2\) we have \(\mid \underline{f}(t, x, y;r)-\underline{f}(t_1, x_1, y_1;r)\mid <\varepsilon \) and \(\mid {\overline{f}}(t, x, y;r)-{\overline{f}}(t_1, x_1, y_1;r)\mid <\varepsilon \),\( \forall r\in [0,1]\) whenever \(\Vert (t_1,x_1, y_1)-(t, x, y)\Vert <\delta .\)

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Acknowledgements

The work of J. J. Nieto has been partially supported by Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and XUNTA de Galicia under Grants GRC2015-004 and R2016-022. The authors wish to thank the editor in chief, the editor and the anonymous reviewers for their constructive comments on this study.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Shahed University, Tehran, Iran

    Z. Noeiaghdam

  2. Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    T. Allahviranloo

  3. Departamento de Estatística, Analise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Juan J. Nieto

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  1. Z. Noeiaghdam
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  2. T. Allahviranloo
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  3. Juan J. Nieto
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Correspondence to Z. Noeiaghdam.

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Noeiaghdam, Z., Allahviranloo, T. & Nieto, J.J. q-fractional differential equations with uncertainty. Soft Comput 23, 9507–9524 (2019). https://doi.org/10.1007/s00500-019-03830-w

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