New Method for Numerical Solution of Z-Fractional Differential Equations

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.