The Operational matrices with respect to generalized Laguerre polynomials and their applications in solving linear di erential equations with variable coe cients

Abstract

In this paper, a new and e cient approach based on operational matrices with respect to the gener-<br />alized Laguerre polynomials for numerical approximation of the linear ordinary di erential equations<br />(ODEs) with variable coe cients is introduced. Explicit formulae which express the generalized La-<br />guerre expansion coe cients for the moments of the derivatives of any di erentiable function in terms<br />of the original expansion coe cients of the function itself are given in the matrix form. The main<br />importance of this scheme is that using this approach reduces solving the linear di erential equations<br />to solve a system of linear algebraic equations, thus greatly simplify the problem. In addition, several<br />numerical experiments are given to demonstrate the validity and applicability of the method.