Adaptive numerical approach based upon Chebyshev operational vector for nonlinear Volterra integral equations and its convergence analysis

In the current paper, we present a direct numerical scheme to approximate a second-kind nonlinear Volterra integral equations (NVIEs). The scheme is based upon shifted Chebyshev polynomials and its operational matrices which eventually leads to the sparsity of the coefficients matrix of obtained system. The main idea of the proposed approach is based on a useful property of Chebyshev polynomials that yields to construct a new operational vector. This vector eliminates any requirement of using projection methods and also enhances the accuracy vs. other methods applied projection methods. The constructive technique and the convergence analysis of this approach under the Lw2-norm are also described. Numerical experiments and comparisons confirm the applicability and the validity of the presented scheme.