{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:01:57Z","timestamp":1760144517340,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,27]],"date-time":"2024-04-27T00:00:00Z","timestamp":1714176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-RP23095"],"award-info":[{"award-number":["IMSIU-RP23095"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo\u2013Hadamard differential equations. Since the Caputo\u2013Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L2 and L\u221e. In addition, we provide numerical results to support our theoretical analysis.<\/jats:p>","DOI":"10.3390\/fractalfract8050262","type":"journal-article","created":{"date-parts":[[2024,4,29]],"date-time":"2024-04-29T08:49:24Z","timestamp":1714380564000},"page":"262","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Non-Polynomial Collocation Spectral Scheme for Systems of Nonlinear Caputo\u2013Hadamard Differential Equations"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3376-7238","authenticated-orcid":false,"given":"Mahmoud A.","family":"Zaky","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"},{"name":"Department of Applied Mathematics, Physics Research Institute, National Research Centre, Dokki, Cairo 12622, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6168-2282","authenticated-orcid":false,"given":"Ibrahem G.","family":"Ameen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo 11884, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7016-5324","authenticated-orcid":false,"given":"Mohammed","family":"Babatin","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9832-1424","authenticated-orcid":false,"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey"},{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 102, Lebanon"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8478-4575","authenticated-orcid":false,"given":"Magda","family":"Hammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Information Science, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilfer, R. 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