{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T10:05:24Z","timestamp":1766138724409,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,8]],"date-time":"2024-06-08T00:00:00Z","timestamp":1717804800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Qassim University","award":["QU-APC-2024-9\/1"],"award-info":[{"award-number":["QU-APC-2024-9\/1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the present paper, we consider an effective computational method to analyze a coupled dynamical system with Caputo\u2013Fabrizio fractional derivative. The method is based on expanding the approximate solution into a symmetry Haar wavelet basis. The Haar wavelet coefficients are obtained by using the collocation points to solve an algebraic system of equations in mathematical physics. The error analysis of this method is characterized by a good convergence rate. Finally, some numerical examples are presented to prove the accuracy and effectiveness of this method.<\/jats:p>","DOI":"10.3390\/sym16060713","type":"journal-article","created":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T10:45:56Z","timestamp":1718016356000},"page":"713","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Approach Based on the Haar Wavelet Collocation Method for Solving a Coupled System with the Caputo\u2013Fabrizio Fractional Derivative"],"prefix":"10.3390","volume":"16","author":[{"given":"Bachir","family":"Dehda","sequence":"first","affiliation":[{"name":"Laboratory of Operator Theory and PDE: Foundations and Applications, University of El Oued, El Oued 39000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6629-9963","authenticated-orcid":false,"given":"Fares","family":"Yazid","sequence":"additional","affiliation":[{"name":"Laboratory of Pure and Applied Mathematics, Amar Telidji University, Laghouat 03000, Algeria"}]},{"given":"Fatima Siham","family":"Djeradi","sequence":"additional","affiliation":[{"name":"Laboratory of Pure and Applied Mathematics, Amar Telidji University, Laghouat 03000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7895-4168","authenticated-orcid":false,"given":"Khaled","family":"Zennir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Sciences, University of 20 A\u00f4ut 1955, Skikda 21000, Algeria"}]},{"given":"Keltoum","family":"Bouhali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1466-8821","authenticated-orcid":false,"given":"Taha","family":"Radwan","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia"},{"name":"Department of Mathematics and Statistics, Faculty of Management Technology and Information Systems, Port Said University, Port Said 42511, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2761","DOI":"10.1007\/s12190-023-01859-7","article-title":"Numerical solution of a class of Caputo\u2013Fabrizio derivative problem using Haar wavelet collocation method","volume":"69","author":"Dehda","year":"2023","journal-title":"J. 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