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. Manuscript profile

The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we needan efficient and accurate computational method for the solution of fractional differential equations.This paper presents a numerical method for solving a class of linear and nonlinear multi-order fractional differential equations with constant coefficients subject to initial conditions based on the fractional order Chebyshev functions that this function is defined as follows:\begin{equation*}\overline{T}_{i+1}^{\alpha}(x)=(4x^{\alpha}-2)\overline{T}_{i}^{\alpha}(x)\overline{T}_{i-1}^{\alpha}(x),\,i=0,1,2,\ldots,\end{equation*}where $\overline{T}_{i+1}^{\alpha}(x)$ can be defined by introducing the change of variable $x^{\alpha},\,\alpha>0$, on the shifted Chebyshevpolynomials of the first kind. This new method is an adaptation of collocationmethod in terms of truncated fractional order Chebyshev Series. To do this method, a new operational matrix of fractional order differential in the Hilfer sense for the fractional order Chebyshev functions is derived. By using this method we reduces such problems to those ofsolving a system of algebraic equations thus greatly simplifying the problem. At the end of this paper, several numerical experiments are given to demonstrate the efficiency and accuracy of the proposed method. Manuscript profile