{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T19:20:45Z","timestamp":1769023245170,"version":"3.49.0"},"reference-count":39,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2022,12,31]],"date-time":"2022-12-31T00:00:00Z","timestamp":1672444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,3,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This study presents a spectral method for solving the two-dimensional variable-order fractional optimal control problems (2D-VOFOCPs). In this work, a dynamic system with variable-order fractional derivatives appears. The Caputo derivative, which is one of the most widely used and essential types of fractional derivatives, has been used to construct operational matrices. The shifted Gegenbauer polynomials are used as orthogonal bases. For this purpose, at first, the control and state functions are approximated by the shifted Gegenbauer polynomials with unknown coefficients. Then, by substituting the approximated functions into initial and boundary conditions, the dynamical system and the objective function, an algebraic equation system is achieved. The solution of the obtained system of the algebraic equation is equivalent to the solution of 2D-VOFOCP. Furthermore, the convergence of the method is studied. Eventually, two numerical examples are presented to illustrate the applicability and accuracy of the proposed method.<\/jats:p>","DOI":"10.1093\/imamci\/dnac031","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T08:10:36Z","timestamp":1672647036000},"page":"1-19","source":"Crossref","is-referenced-by-count":11,"title":["A numerical approach for solving a class of two-dimensional variable-order fractional optimal control problems using Gegenbauer operational matrix"],"prefix":"10.1093","volume":"40","author":[{"given":"Farzaneh","family":"Soufivand","sequence":"first","affiliation":[{"name":"Department of Mathematics, Payame Noor University , Tehran , Iran"}]},{"given":"Fahimeh","family":"Soltanian","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Payame Noor University , Tehran , Iran"}]},{"given":"Kamal","family":"Mamehrashi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Payame Noor University , Tehran , Iran"},{"name":"Mathematics Unit, School of Science and Engineering, University of Kurdistan Hewler (UKH) , Erbil, Kurdistan Region, Iraq"}]}],"member":"286","published-online":{"date-parts":[[2022,12,31]]},"reference":[{"key":"2023031512451958300_","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s11071-004-3764-6","article-title":"A general formulation and solution scheme for fractional optimal control problem","volume":"38","author":"Agrawal","year":"2004","journal-title":"Nonlinear Dynam."},{"key":"2023031512451958300_","doi-asserted-by":"crossref","first-page":"011010","DOI":"10.1115\/1.2814055","article-title":"A quadratic numerical scheme for fractional optimal control problems","volume":"130","author":"Agrawal","year":"2008","journal-title":"Journal of Dynamic Systems, Measurement, and Control"},{"key":"2023031512451958300_","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1177\/1077546307077467","article-title":"A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems","volume":"13","author":"Agrawal","year":"2007","journal-title":"J. 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