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A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion

Author

Listed:
  • Tofigh Allahviranloo

    (Faculty of Engineering and Natural Sciences, Bahcesehir University, 34353 Istanbul, Turkey)

  • Zahra Noeiaghdam

    (Department of Mathematics, Shahed University, Tehran 3319118651, Iran)

  • Samad Noeiaghdam

    (Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia
    Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia)

  • Juan J. Nieto

    (Instituto de Matemáticas, Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain)

Abstract

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.

Suggested Citation

  • Tofigh Allahviranloo & Zahra Noeiaghdam & Samad Noeiaghdam & Juan J. Nieto, 2020. "A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion," Mathematics, MDPI, vol. 8(12), pages 1-24, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2166-:d:457121
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    References listed on IDEAS

    as
    1. Barnab?s Bede & Luciano Stefanini, 2012. "Generalized Differentiability of Fuzzy-valued Functions," Working Papers 1209, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2012.
    2. Luciano Stefanini & Barnabas Bede, 2008. "Generalized Hukuhara Differentiability of Interval-valued Functions and Interval Differential Equations," Working Papers 0803, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2008.
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