The Operational matrices with respect to generalized Laguerre polynomials and their applications in solving linear dierential equations with variable coecients

In this paper, a new and ecient approach based on operational matrices with respect to the gener-alized Laguerre polynomials for numerical approximation of the linear ordinary dierential equations(ODEs) with variable coecients is introduced. Explicit formulae which express the generalized La-guerre expansion coecients for the moments of the derivatives of any dierentiable function in termsof the original expansion coecients of the function itself are given in the matrix form. The mainimportance of this scheme is that using this approach reduces solving the linear dierential equationsto solve a system of linear algebraic equations, thus greatly simplify the problem. In addition, severalnumerical experiments are given to demonstrate the validity and applicability of the method.