{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T18:13:50Z","timestamp":1776449630576,"version":"3.51.2"},"reference-count":39,"publisher":"Wiley","license":[{"start":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T00:00:00Z","timestamp":1680480000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005307","name":"Direction G\u00e9n\u00e9rale de la Recherche Scientifique et du D\u00e9veloppement Technologique","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100005307","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2023,4,3]]},"abstract":"<jats:p>In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra\u2212Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev\u2212Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach\u2019s efficiency, applicability, and performance.<\/jats:p>","DOI":"10.1155\/2023\/6401067","type":"journal-article","created":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T16:09:25Z","timestamp":1680538165000},"page":"1-13","source":"Crossref","is-referenced-by-count":3,"title":["Theoretical and Numerical Study for Volterra\u2212Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind"],"prefix":"10.1155","volume":"2023","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9907-774X","authenticated-orcid":true,"given":"Zineb","family":"Laouar","sequence":"first","affiliation":[{"name":"Laboratoire des Math\u00e9matiques Appliqu\u00e9es et Didactique, Ecole Normale Sup\u00e9rieure El Katiba Assia Djebar, Constantine, Algeria"},{"name":"Centre Universitaire Abdelhafid Boussouf, Mila, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6051-8237","authenticated-orcid":true,"given":"Nouria","family":"Arar","sequence":"additional","affiliation":[{"name":"Laboratoire des Math\u00e9matiques et Sciences de La D\u00e9cision (LAMASD), Universit\u00e9 Fr\u00e8res Mentouri, Constantine 25017, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7142-7026","authenticated-orcid":true,"given":"Abdellatif","family":"Ben Makhlouf","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, P.O. 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