{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T22:44:59Z","timestamp":1779403499184,"version":"3.53.1"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T00:00:00Z","timestamp":1760659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62303118"],"award-info":[{"award-number":["62303118"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["2023AH050429"],"award-info":[{"award-number":["2023AH050429"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["2024AH051311"],"award-info":[{"award-number":["2024AH051311"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012469","name":"Doctoral Foundation of Fuyang Normal University","doi-asserted-by":"crossref","award":["2021KYQD0034"],"award-info":[{"award-number":["2021KYQD0034"]}],"id":[{"id":"10.13039\/501100012469","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012469","name":"Doctoral Foundation of Fuyang Normal University","doi-asserted-by":"crossref","award":["2025KYQD0095"],"award-info":[{"award-number":["2025KYQD0095"]}],"id":[{"id":"10.13039\/501100012469","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Bagley\u2013Torvik equation (BTE) is an important model in mathematical physics and mechanics, but obtaining its analytical solution remains challenging. For its numerical treatment, the presence of composite functions in the generalized BTE poses additional difficulties, and efficient approaches for handling nonlinear terms are still lacking in the literature. This study proposes an improved numerical method for the fractional BTE with integral boundary conditions. By employing an integration technique, the original problem is transformed into a weakly singular Fredholm\u2013Hammerstein (F\u2013H) integral equation of the second kind. To address the nonlinear terms, an enhanced piecewise Taylor expansion scheme is developed to construct the discrete form, while the uniqueness of the solution is proven using the contraction mapping theorem in Banach spaces. The convergence and error analyses are rigorously carried out, and numerical experiments confirm the accuracy and efficiency of the proposed approach.<\/jats:p>","DOI":"10.3390\/sym17101755","type":"journal-article","created":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T07:33:50Z","timestamp":1760686430000},"page":"1755","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Numerical Solutions for Fractional Bagley\u2013Torvik Equation with Integral Boundary Conditions"],"prefix":"10.3390","volume":"17","author":[{"given":"Xueling","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jing","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junlin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yufeng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236000, China"},{"name":"Interdisciplinary Research Institute, School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114562","DOI":"10.1016\/j.chaos.2024.114562","article-title":"A bio inspired learning scheme for the fractional order kidney function model with neural networks","volume":"180","author":"Sabir","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.matcom.2024.03.008","article-title":"Harmonic resonance and bifurcation of fractional Rayleigh oscillator with distributed time delay","volume":"221","author":"Zhang","year":"2024","journal-title":"Math. 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