Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation

This paper is concerned with obtaining approximate solution and approximate derivatives up to order k of the solution for neutral kth-order Volterra integro-differential equation with a regular kernel. The solution of the equation, for analytic data, is smooth on the entire interval of integration. The Legendre collocation discretization is proposed for this equation. In the present paper, we restate the initial conditions as equivalent integral equations instead of integrating two sides of the equation and provide a rigorous error analysis which justifies that not only the errors of approximate solution but also the errors of approximate derivatives up to order k   of the solution decay exponentially in L2L2 norm and L∞L∞ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.