A pseudospectral scheme and its convergence analysis for high-order integro-differential equations

The main purpose of this work is to develop an integral pseudospectral scheme for solving integro-differential equations. We provide new pseudospectral integration matrices (PIMs) for the Legendre-Gauss and the flipped Legendre-Gauss-Radau points, respectively, and present an efficient and stable approach to computing the PIMs via the recursive calculation of Legendre integration matrices. Furthermore, we provide a rigorous convergence analysis for the proposed pseudospectral scheme in both L∞ and L2 spaces via a linear integral equation, and the spectral rate of convergence is demonstrated by numerical results.