{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T02:01:44Z","timestamp":1767924104015,"version":"3.49.0"},"reference-count":58,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,30]],"date-time":"2021-12-30T00:00:00Z","timestamp":1640822400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>We propose a fractional-order shifted Jacobi\u2013Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann\u2013Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.<\/jats:p>","DOI":"10.3390\/fractalfract6010019","type":"journal-article","created":{"date-parts":[[2021,12,30]],"date-time":"2021-12-30T21:41:21Z","timestamp":1640900481000},"page":"19","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel"],"prefix":"10.3390","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9043-9644","authenticated-orcid":false,"given":"Mohamed A.","family":"Abdelkawy","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 11625, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4044-3335","authenticated-orcid":false,"given":"Ahmed Z. M.","family":"Amin","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, Technology, Universiti Kebangsaan Malaysia, UKM, Bangi 43600, Selangor, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"additional","affiliation":[{"name":"LAETA\/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4237-7140","authenticated-orcid":false,"given":"Ishak","family":"Hashim","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, Technology, Universiti Kebangsaan Malaysia, UKM, Bangi 43600, Selangor, Malaysia"}]},{"given":"Mohammed M.","family":"Babatin","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,30]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"A Jacobi spectral collocation scheme for solving Abel\u2019s integral equations","volume":"1","author":"Abdelkawy","year":"2015","journal-title":"Prog. 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