Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials

Abd-Elhameed, Waleed Mohamed; Youssri, Youssri Hassan; Atta, Ahmed Gamal

doi:10.22124/jmm.2024.25844.2295

Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials

Document Type : Research Article

Authors

¹ Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia & Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

² Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt & Faculty of Engineering, Egypt University of Informatics, Knowledge City, New Administrative Capital, Egypt

³ Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt

10.22124/jmm.2024.25844.2295

Abstract

Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy.

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