{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T17:49:26Z","timestamp":1759340966861},"reference-count":18,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["New Math. and Nat. Computation"],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:p> In this paper, at first, we introduce fractional differential equations with [Formula: see text]-valuation. Then, we propose a numerical method to approximate the solution. The proposed method is a hybrid method based on the corrected fractional Euler\u2019s method and the probability distribution function. Moreover, the corrected fractional Euler\u2019s method based on the generalized Taylor formula and the modified trapezoidal rule is proposed that this method can be used in the problems\u2019 limitation section of the [Formula: see text]-fractional Initial value problem of order [Formula: see text] 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 [Formula: see text]-valuation. <\/jats:p>","DOI":"10.1142\/s1793005721500034","type":"journal-article","created":{"date-parts":[[2020,8,12]],"date-time":"2020-08-12T13:50:24Z","timestamp":1597240224000},"page":"45-61","source":"Crossref","is-referenced-by-count":2,"title":["New Method for Numerical Solution of Z-Fractional Differential Equations"],"prefix":"10.1142","volume":"17","author":[{"given":"Parisa","family":"Keshavarz","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hamedan Branch, Islamic Azad University, Prof. Mousivand Blvd, Imam Khomeini Blvd, Hamedan 6518115743, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tofigh","family":"Allahviranloo","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Natural Sciences, Bahcesehir University, \u00c7\u0131ra\u011fan Cd., Beshiktash, 34353 Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Farajollah M.","family":"Yaghoobi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hamedan Branch, Islamic Azad University, Prof. Mousivand Blvd, Imam Khomeini Blvd, Hamedan 6518115743, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ali","family":"Barahmand","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hamedan Branch, Islamic Azad University, Prof. Mousivand Blvd, Imam Khomeini Blvd, Hamedan 6518115743, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,3,29]]},"reference":[{"key":"S1793005721500034BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2009.06.106"},{"key":"S1793005721500034BIB002","first-page":"12","volume-title":"Fractional Calculus and Its Applications","author":"Bagley R. 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