{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T15:19:24Z","timestamp":1778339964415,"version":"3.51.4"},"reference-count":14,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,25]],"date-time":"2022-07-25T00:00:00Z","timestamp":1658707200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University","award":["TURSP-2020\/320"],"award-info":[{"award-number":["TURSP-2020\/320"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work proposes a valuable and successful strategy for approximating the solutions to the Bagley\u2013Torvik system, which plays an essential role in fractional calculus. The Caputo sense is used to derive the basic conformable fractional. The Bagley\u2013Torvik problem is numerically solved in this study using an effective symmetric projection method. From this symmetry, there are some interesting original results. The proposed approach has two key benefits. We began by converting the connected fractional Bagley\u2013Torvik equations into two fractional-order Bagley\u2013Torvik equations, which we then solved using the current method. Second, two linear equation systems are solved to obtain approximate solutions.<\/jats:p>","DOI":"10.3390\/sym14081514","type":"journal-article","created":{"date-parts":[[2022,7,25]],"date-time":"2022-07-25T04:52:47Z","timestamp":1658724767000},"page":"1514","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["An Effective Projection Method for Solving a Coupled System of Fractional-Order Bagley\u2013Torvik Equations via Fractional Shifted Legendre Polynomials"],"prefix":"10.3390","volume":"14","author":[{"given":"Saeed","family":"Althubiti","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0791-5866","authenticated-orcid":false,"given":"Abdelaziz","family":"Mennouni","sequence":"additional","affiliation":[{"name":"Department of Mathematics, LTM, University of Batna 2, Mostefa Ben Boula\u00efd, Fesdis, Batna 05078, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1115\/1.3167615","article-title":"On the appearance of the fractional derivative in the behavior of real materials","volume":"51","author":"Bagley","year":"1984","journal-title":"ASME Trans. J. Appl. Mech."},{"key":"ref_2","first-page":"50","article-title":"LMI stability test for initialized fractional order control systems","volume":"18","year":"2019","journal-title":"Appl. Comput. 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Algor."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s40324-018-0148-2","article-title":"Application of M\u00fcntz\u2013Legendre polynomials for solving the Bagley\u2013Torvik equation in a large interval","volume":"75","author":"Rahimkhani","year":"2018","journal-title":"SeMA J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1186\/s13662-020-03110-0","article-title":"Numerical solution of the Bagley-Torvik equation using shifted Chebyshev operational matrix","volume":"2020","author":"Ji","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1023\/A:1021973025166","article-title":"Numerical Solution of the Bagley-Torvik Equation","volume":"42","author":"Diethelm","year":"2002","journal-title":"BIT Numer. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Setia, A., Liu, Y., and Vatsala, A.S. (2014, January 7\u20139). 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Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers."},{"key":"ref_13","first-page":"63","article-title":"A new efficient strategy for solving the system of Cauchy integral equations via two projection methods","volume":"14","author":"Mennouni","year":"2022","journal-title":"Transylv. J. Math. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1186\/s13662-016-0989-x","article-title":"A fractional-order Legendre collocation method for solving the Bagley-Torvik equations","volume":"2016","author":"Mohammadi","year":"2016","journal-title":"Adv. Differ. 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