{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:05:57Z","timestamp":1760148357731,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, many scientists have studied a wide range of strategies for solving characteristic types of symmetric differential equations, including symmetric fractional differential equations (FDEs). In our manuscript, we obtained sufficient conditions to prove the existence and uniqueness of solutions (EUS) for FDEs in the sense \u03c8-Caputo fractional derivative (\u03c8-CFD) in the second-order 1&lt;\u03b1&lt;2. We know that \u03c8-CFD is a generalization of previously familiar fractional derivatives: Riemann-Liouville and Caputo. By applying the Banach fixed-point theorem (BFPT) and the Schauder fixed-point theorem (SFPT), we obtained the desired results, and to embody the theoretical results obtained, we provided two examples that illustrate the theoretical proofs.<\/jats:p>","DOI":"10.3390\/sym15040955","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T10:11:25Z","timestamp":1682071885000},"page":"955","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Processing Fractional Differential Equations Using \u03c8-Caputo Derivative"],"prefix":"10.3390","volume":"15","author":[{"given":"Mahrouz","family":"Tayeb","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, University of Ibn Khladoun, BP P 78 Zaaroura, Tiaret 14000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hamid","family":"Boulares","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis and Control of Differential Equations \u201cACED\u201d, Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1080-3686","authenticated-orcid":false,"given":"Abdelkader","family":"Moumen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 55425, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7684-2616","authenticated-orcid":false,"given":"Moheddine","family":"Imsatfia","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ali, M.A., Kara, H., Tariboon, J., Asawasamrit, S., Budak, H., and Hezenci, F. (2021). 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