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

In this paper, a new and efficient approach based on operational matrices with respect to the generalized Laguerre polynomials for numerical approximation of the linear ordinary differential equations (ODEs) with variable coefficients is introduced. Explicit formulae which express the generalized Laguerre expansion coefficients for the moments of the derivatives of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. The main importance of this scheme is that using this approach reduces solving the linear differential equations to solve a system of linear algebraic equations, thus greatly simplify the problem. In addition, several numerical experiments are given to demonstrate the validity and applicability of the method.