A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations

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This paper provides with a generalization of the work by Chelyshkov (Electron. Trans. Numer. Anal. 25(7): 17–26, 2006), who has introduced sequences of orthogonal polynomials over [0,1] which can be expressed in terms of Jacobi polynomials. We develop a new approach of product integration algorithm based on these orthogonal polynomials including the numerical quadratures for solving the nonlinear weakly singular Volterra integral equations. The convergence analysis of the proposed scheme is derived and numerical results are given showing a marked improvement in comparison with recent numerical methods.