Chebyshev polynomial derivative-based spectral tau approach for solving high-order differential equations

In this paper, Chebyshev polynomial derivative-based spectral schemes are constricted for solving linear and non-linear ordinary differential equations. Linearization relation and some essential integrated formulae concerning the basis functions are provided to deal with the spectral tau method. Unlike the regular weight function, another modified weight is introduced. Also, different patterns and results have been obtained regarding the relation between the Jacobi polynomials, ultraspherical polynomials, Chebyshev polynomials, and their derivatives. Moreover, the algebraic systems of the spectral expansion for solving the Riccati, Lane–Emden equations, and water contamination model are discussed. Error bounds are introduced, studied, and proven. Finally, several real applications are numerically solved using 2ndDCh polynomial-based spectral tau method. The obtained results are compared with different methods to confirm the accuracy and efficiency of the schemes.