A Nyström interpolant for some weakly singular nonlinear Volterra integral equations (Q455891)

In order to obtain an accurate approximation of the solution of a second kind weakly singular nonlinear Volterra-Hammerstein integral equation, defined by a completely continuous operator, a Nyström type interpolant of the solution based on Gauss-Radau nodes and Lagrange interpolation fundamental polynomials is proposed. The convergence of the method is proved showing the convergence of the sequence of the Nyström interpolants to the unique solution of the integral equation with a rate of convergence that depends on the quadrature rule used.NEWLINENEWLINE In the attempt to improve the order of convergence of the numerical method for the case of integral equations with nonlinearity of algebraic kind, the smoothing transformation technique is used. In this context, for the transformed integral equation, the numerical method is reformulated and its convergence is proved obtaining a better order of convergence. The nonlinear integral equation of Abel-type is included in this approach. Four test numerical examples confirm the theoretical results and illustrate the performances of the proposed method.