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New Method for Numerical Solution of Z-Fractional Differential Equations

Author

Listed:
  • Parisa Keshavarz

    (Department of Mathematics, Hamedan Branch, Islamic Azad University, Prof. Mousivand Blvd, Imam Khomeini Blvd, Hamedan 6518115743, Iran)

  • Tofigh Allahviranloo

    (Faculty of Engineering and Natural Sciences, Bahcesehir University, Çırağan Cd., Beshiktash, 34353 Istanbul, Turkey)

  • Farajollah M. Yaghoobi

    (Department of Mathematics, Hamedan Branch, Islamic Azad University, Prof. Mousivand Blvd, Imam Khomeini Blvd, Hamedan 6518115743, Iran)

  • Ali Barahmand

    (Department of Mathematics, Hamedan Branch, Islamic Azad University, Prof. Mousivand Blvd, Imam Khomeini Blvd, Hamedan 6518115743, Iran)

Abstract

In this paper, at first, we introduce fractional differential equations with Z-valuation. Then, we propose a numerical method to approximate the solution. The proposed method is a hybrid method based on the corrected fractional Euler’s method and the probability distribution function. Moreover, the corrected fractional Euler’s method based on the generalized Taylor formula and the modified trapezoidal rule is proposed that this method can be used in the problems’ limitation section of the Z-fractional Initial value problem of order α ∈ (0, 1) with the fuzzy Caputo fractional differential (fractional derivatives are defined on the basis of the Hukuhara differences and the generalized fuzzy derivatives). The probability function is based on exponential distribution function and used to represent the reliability of the problem limitation part. Finally, by two examples, we show that the proposed method can arbitrarily approximate the fractional differential equations with Z-valuation.

Suggested Citation

  • Parisa Keshavarz & Tofigh Allahviranloo & Farajollah M. Yaghoobi & Ali Barahmand, 2021. "New Method for Numerical Solution of Z-Fractional Differential Equations," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 45-61, March.
    • Handle: RePEc:wsi:nmncxx:v:17:y:2021:i:01:n:s1793005721500034
    DOI: 10.1142/S1793005721500034
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