Solutions of linear dynamic systems by generalized orthogonal polynomials

TLDR: In this paper, generalized orthogonal polynomials that represent all types of orthogonality functions are introduced and the operational matrix for integration of a generalized polynomial is derived and then applied to solve the equations of linear dynamic systems.

Abstract: Generalized orthogonal polynomials that represent all types of orthogonal polynomial are introduced in this paper. Using the idea of orthogonal polynomial functions that can be expressed by power series, and vice versa, the operational matrix for integration of a generalized orthogonal polynomial is first derived and then applied to solve the equations of linear dynamic systems. The characteristics of each kind of orthogonal polynomial in relation to solving linear dynamic systems is demonstrated. The computational strategy for finding the expansion coefficients of the state variables is very simple, straightforward and easy. The operational matrix is simpler than those of conventional orthogonal polynomials. Hence the expansion coefficients are more easily calculated from the proposed recursive formula when compared with those obtained from conventional orthogonal polynomial approximations.