Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method

Abstract

Herein, novel basis orthogonal polynomials have developed. These developed polynomials have been used to find the approximation solutions for some types of linear and non-linear ordinary differential equations by direct numerical method. This numerical method depends on the Chebyshev polynomials' derivatives. We shall present these solutions in the form of a finite sum of the Chebyshev polynomials' derivatives and unknown coefficients involving these polynomials. By substituting into the differential equation, the given differential equation will be converted into a system of algebraic equations. The obtained algebraic system can be solved easily to get the values of the spectral expansion constant. In addition, an algorithm for the approximated process has been designed to be easily used in the coding process. Consequently, some ordinary differential equations have been solved via the introduced Chebyshev polynomials' derivatives. Finally, the approximated solutions have been compared with exact and other methods solutions to illustrate the efficiency and accuracy of the used method.