Convergence Analysis of Legendre-Collocation Methods for Nonlinear Volterra Type Integro Equations

A Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors in $L^2$-norm and $L^\infty$-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.