Spectral Scheme for Nonlinear Volterra Integro-Differential Equation

Abstract

Nonlinear problems widely exist in many aspects of the natural field. The nonlinear situation makes it difficult for most existing solvers to deal with. Therefore, constructing an efficient and accurate solver is a challenge. In this paper, a Legendre spectral method is developed for the nonlinear Volterra integro-differential equation. The error analysis is also provided to justify the spectral rate of convergence for the errors of approximate solution and approximate derivative decay exponentially in both the $L^2$ norm and the infinity norm. In the end, numerical results are displayed to confirm the effectiveness of the Legendre spectral analysis.