{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T07:21:24Z","timestamp":1763018484846,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,15]],"date-time":"2022-04-15T00:00:00Z","timestamp":1649980800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020 (CIDMA)"],"award-info":[{"award-number":["UIDB\/04106\/2020 (CIDMA)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann\u2013Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag\u2013Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler\u2013Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.<\/jats:p>","DOI":"10.3390\/axioms11040178","type":"journal-article","created":{"date-parts":[[2022,4,16]],"date-time":"2022-04-16T07:42:41Z","timestamp":1650094961000},"page":"178","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Weighted Generalized Fractional Integration by Parts and the Euler\u2013Lagrange Equation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8412-7783","authenticated-orcid":false,"given":"Houssine","family":"Zine","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3604-3841","authenticated-orcid":false,"given":"El Mehdi","family":"Lotfi","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8435-5981","authenticated-orcid":false,"given":"Noura","family":"Yousfi","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Anastassiou, G.A. (2021). 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