A new Legendre polynomial-based approach for non-autonomous linear ODEs

Abstract

We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind d dt ũ(t) = f (t)ũ(t), ũ(−1) = 1, with f (t) an analytic function. The method is based on a new expression for the solution ũ(t) given in terms of a convolution-like operation, the ⋆-product. This expression is represented in a finite Legendre polynomial basis translating the initial problem into a matrix problem. An efficient procedure is proposed to approximate the Legendre coefficients of ũ(t) and its truncation error is analyzed. We show the effectiveness of the proposed procedure through some numerical experiments. The method can be easily generalized to solve systems of linear ODEs.