Solving Linear Fractional Differential Equations with Time Delay by Steps Chebyshev-Tau Scheme

Sayed Hodjatollah Momeni- Masuleh | Mohammad Mousa-Abadian

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URL :   http://research.shahed.ac.ir/wsr/webpages/Report/PaperView.aspx?PaperID=148275
Date :  2021/01/30
Publish in :    Iranian Journal of Science and Technology, Transactions A: Science
DOI :  https://doi.org/10.1007/s40995-020-01058-0
Link :  http://dx.doi.org/10.1007/s40995-020-01058-0
Keywords :Linear fractional differential system Time-delay systems, Method of steps, Tau method, Shifted Chebyshev polynomials

Abstract :

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In this article, we obtain a new formula for obtaining an exact expression for the fractional derivatives of the shifted Chebyshev polynomials in the desired interval α,β based on shifted Chebyshev polynomials themselves. We propose a novel scheme to solve linear fractional differential equations with the order ν and a single time delay. The proposed scheme consists of two stages. The first stage converts the linear fractional differential equations with a time delay to linear fractional differential equations using the method of steps. It employs the shifted Chebyshev polynomials to construct an approximate solution for the system of equations on the second stage. The proposed scheme’s convergence analysis has been performed, suggesting that the proposed scheme has a spectral rate of convergence for sufficiently smooth solutions. Also, the stability analysis is provided. Numerical benchmark problems are examined to show the proficiency and accuracy of the proposed scheme.