Analytical and computational methods for a class of nonlinear singular integral equations

Applied Numerical Mathematics, 114 (2017), 2-17
https://doi.org/10.1016/j.apnum.2016.06.001

We consider a class of nonlinear singular Hammerstein Volterra integral equations. In general, these equations will have kernels containing both an end point and an Abel-type singularity, with exact solutions being typically nonsmooth. Under certain conditions, a uniformly convergent iterative solution is obtained on a small interval near the origin. In this work, two product integration methods are proposed and analyzed where the integral over a small initial interval is calculated analytically, allowing the optimal convergence rates to be achieved. This is illustrated by some numerical examples.