{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:16:54Z","timestamp":1760059014117,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T00:00:00Z","timestamp":1747180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12050001","11672150"],"award-info":[{"award-number":["12050001","11672150"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by space\u2013time symmetry breakage. Classical mathematical equations can be transformed into differential equations of undetermined operators. We confirmed that the presence of fractional-order operator solutions in operator differential equations is contingent upon the mismatch of differential orders of space and time, which can induce both fractional operators in the time domain and fractional operators in the space domain. The introduction of symmetry breakage and operators of space and time offers novel insights into understanding nonlocal phenomena within the space\u2013time continuum.<\/jats:p>","DOI":"10.3390\/sym17050762","type":"journal-article","created":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T12:22:17Z","timestamp":1747225337000},"page":"762","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Fractional-Order Effect Induced by Space\u2013Time Order Mismatch in Operator Differential Equations"],"prefix":"10.3390","volume":"17","author":[{"given":"Yajun","family":"Yin","sequence":"first","affiliation":[{"name":"Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianyi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruiheng","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaobin","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-2370-157X","authenticated-orcid":false,"given":"Chaoqian","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,14]]},"reference":[{"key":"ref_1","unstructured":"Oldham, K.B. 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