{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T18:58:48Z","timestamp":1770836328105,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,3,25]],"date-time":"2021-03-25T00:00:00Z","timestamp":1616630400000},"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>In this research study, we establish some necessary conditions to check the uniqueness-existence of solutions for a general multi-term \u03c8-fractional differential equation via generalized \u03c8-integral boundary conditions with respect to the generalized asymmetric operators. To arrive at such purpose, we utilize a procedure based on the fixed-point theory. We follow our study by suggesting two numerical algorithms called the Dafterdar-Gejji and Jafari method (DGJIM) and the Adomian decomposition method (ADM) techniques in which a series of approximate solutions converge to the exact ones of the given \u03c8-RLFBVP and the equivalent \u03c8-integral equation. To emphasize for the compatibility and the effectiveness of these numerical algorithms, we end this investigation by providing some examples showing the behavior of the exact solution of the existing \u03c8-RLFBVP compared with the approximate ones caused by DGJIM and ADM techniques graphically.<\/jats:p>","DOI":"10.3390\/sym13040532","type":"journal-article","created":{"date-parts":[[2021,3,25]],"date-time":"2021-03-25T02:54:03Z","timestamp":1616640843000},"page":"532","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["Numerical Solutions Caused by DGJIM and ADM Methods for Multi-Term Fractional BVP Involving the Generalized \u03c8-RL-Operators"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3463-2607","authenticated-orcid":false,"given":"Shahram","family":"Rezapour","sequence":"first","affiliation":[{"name":"Department of Medical Research, China Medical University Hospital, Taichung 404, Taiwan"},{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1574-1800","authenticated-orcid":false,"given":"Sina","family":"Etemad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brahim","family":"Tellab","sequence":"additional","affiliation":[{"name":"Laboratory of Applied Mathematics, Kasdi Merbah University, B.P. 511, Ouargla 30000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7556-8942","authenticated-orcid":false,"given":"Praveen","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab Emirates"},{"name":"International Center for Basic and Applied Sciences, Jaipur 302029, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan Luis","family":"Garcia Guirao","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada y Estad\u00edstica, Universidad Polit\u00e9cnica de Cartagena, Campus de la Muralla, 30203 Cartagena, Murcia, Spain"},{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,25]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies."},{"key":"ref_2","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_3","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.2478\/amns.2020.2.00011","article-title":"Fractional calculus involving (p,q)-Mathieu type series","volume":"5","author":"Kaur","year":"2020","journal-title":"Appl. Math. 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