New Method for Numerical Solution of Z-Fractional Differential Equations

Parisa Keshavarz (), Tofigh Allahviranloo, Farajollah M. Yaghoobi () and Ali Barahmand ()
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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

New Mathematics and Natural Computation (NMNC), 2021, vol. 17, issue 01, 45-61

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.

Keywords: Z-number; fractional differential equations; fraction problem with initial value; Hukuhara differences with fuzzy derivatives of the Caputo type; exponential distribution function (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S1793005721500034

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