A spectral collocation method for multidimensional nonlinear weakly singular Volterra integral equation

This paper is concerned with the convergence properties of Chebyshev spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation of the second kind with a weakly singular kernel. We consider the case that the underlying solution is sufficiently smooth. The Chebyshev collocation discretization is proposed for this equation. In the present paper, we provide a rigorous error analysis which justifies that the errors of approximate solution decay exponentially in weighted L2 norm and L∞ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.