Numerical solution of linear time delay systems using Chebyshev-tau spectral method

Mohammad Mousa Abadiyan | Sayed Masuleh

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=43096
Date :  2017/06/13
Publish in :    Applications and Applied Mathematics: An International Journal

Link :  https://www.pvamu.edu/mathematics/wp-content/uploads/sites/49/29_R911_AAM_Momeni_SM_052316_Posted_061217_pp_445_469.pdf
Keywords :solution, systems, Chebyshev-tau, spectral, method

Abstract :

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In this paper, a hybrid method based on method of steps and a Chebyshev-tau spectral method for solving linear time delay systems of differential equations is proposed. The method first converts the time delay system to a system of ordinary differential equations by the method of steps and then employs Chebyshev polynomials to construct an approximate solution for the system. In fact, the solution of the system is expanded in terms of orthogonal Chebyshev polynomials which reduces the solution of the system to the solution of a system of algebraic equations. Also, we transform the coefficient matrix of the algebraic system to a block quasi upper triangular matrix and the latter system can be solved more efficiently than the first one. Furthermore, using orthogonal Chebyshev polynomials enables us to apply fast Fourier transform for calculating matrix-vector multiplications which makes the proposed method to be more efficient. Consistency, stability and convergence analysis of the method are provided. Numerous numerical examples are given to demonstrate efficiency and accuracy of the method. Comparisons are made with available literature.