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

In this paper, a new and efficient approach is applied for numerical approximation of the linear differential equations with variable coefficients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansion coefficients for the moments of 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 simplifying the problem. In addition, two experiments are given to demonstrate the validity and applicability of the method.