A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent

Weakly singular Volterra integral equations with variable exponent have integral kernels of the form (t-s)(-alpha(t)) with 0 <= alpha(t) <1. A Nystr & ouml;m method based on product integration and interpolation by piecewise polynomials is constructed and analysed for the numerical solution of this class of problems. To deal with the weak singularity of typical solutions of such problems, suitably graded meshes are used. A rigorous error analysis proves convergence of the computed solution; moreover, super convergence is obtained if the quadrature points are well chosen. These results also imply error bounds for the solution of a related iterated collocation method. Numerical experiments demonstrate the sharpness of our theoretical results for the Nystr & ouml;m method.