{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T15:13:40Z","timestamp":1771946020765,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,20]],"date-time":"2022-06-20T00:00:00Z","timestamp":1655683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Hebei Education Department","award":["ZD2020172"],"award-info":[{"award-number":["ZD2020172"]}]},{"name":"Hebei Education Department","award":["QN2020124"],"award-info":[{"award-number":["QN2020124"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of \u03b1 and \u03bb.<\/jats:p>","DOI":"10.3390\/sym14061279","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T23:11:19Z","timestamp":1655939479000},"page":"1279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Stability of Euler Methods for Fuzzy Differential Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Cuilian","family":"You","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Hebei University, Baoding 071002, China"},{"name":"Hebei Key Lab of Machine Learning and Computational Intelligence, Hebei University, Baoding 071002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Cheng","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Hebei University, Baoding 071002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongyan","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Hebei University, Baoding 071002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.ins.2015.03.029","article-title":"The initial value problem for interval-valued second-order differential equations under generalized H-differentiability","volume":"311","author":"Ngo","year":"2015","journal-title":"Inf. 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[2nd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1279\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:36:01Z","timestamp":1760139361000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1279"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,20]]},"references-count":23,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061279"],"URL":"https:\/\/doi.org\/10.3390\/sym14061279","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,20]]}}}