{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:34:02Z","timestamp":1760240042311,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,3,16]],"date-time":"2019-03-16T00:00:00Z","timestamp":1552694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["753750 \u2013 RaSiR"],"award-info":[{"award-number":["753750 \u2013 RaSiR"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.<\/jats:p>","DOI":"10.3390\/axioms8010035","type":"journal-article","created":{"date-parts":[[2019,3,18]],"date-time":"2019-03-18T12:18:53Z","timestamp":1552911533000},"page":"35","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Operator Ordering and Solution of Pseudo-Evolutionary Equations"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8738-5040","authenticated-orcid":false,"given":"Nicolas","family":"Behr","sequence":"first","affiliation":[{"name":"Institut de Recherche en Informatique Fondamentale (IRIF), Universit\u00e9 Paris-Diderot, F-75013 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Dattoli","sequence":"additional","affiliation":[{"name":"ENEA\u2014Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6520-3077","authenticated-orcid":false,"given":"Ambra","family":"Lattanzi","sequence":"additional","affiliation":[{"name":"ENEA\u2014Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy"},{"name":"H. Niewodnicza\u0144ski Institute of Nuclear Physics, Polish Academy of Science, 31-342 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,16]]},"reference":[{"key":"ref_1","unstructured":"Babusci, D., and Dattoli, G. (arXiv, 2011). Umbral methods and operator ordering, arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3427","DOI":"10.1140\/epjst\/e2018-00073-1","article-title":"Comments on the properties of Mittag-Leffler function","volume":"226","author":"Dattoli","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_3","unstructured":"Dattoli, G. (1999). Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle. Proceedings of the Workshop on Advanced Special Functions and Applications, Aracne."},{"key":"ref_4","unstructured":"Weyl, H. (1950). 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