Numerical Solutions of Monic Chebyshev Polynomials on Large Scale Differentiation

In this paper, a new formula of the spectral differentiation matrices is presented. Therefore, the numerical solutions for higher-order differential equations are presented by expanding the unknown solution in terms of monic Chebyshev polynomials. The resulting systems of linear equations are solved directly for the values of the solution at the extreme points of the Chebyshev polynomial of order N. The round-off errors during the calculations of differentiation matrices elements are studied. A number of numerical examples are provided in order to show the advantages of the suggested differentiation matrices through comparisons with other works.