An adaptive Huber method for weakly singular second kind Volterra integral equations with non-linear dependencies between unknowns and their integrals

Abstract

Numerical methods for weakly singular Volterra integral equations with non-linear dependencies between unknowns and their integrals, are almost non-existent in the literature. In the present work an adaptive Huber method for such equations is proposed, by extending the method previously formulated for the first kind Abel equations. The method is tested on example integral equations involving integrals with kernels K(t, τ) = (t − τ)^−1/2, K(t, τ) = exp[−λ(t − τ)](t − τ)^−1/2 (where λ > 0), and K(t, τ) = 1. By controlling estimated local discretisation errors, the integral equation can be solved adaptively on a discrete grid of nodes in the independent variable domain, in a step-by-step fashion. The practical accuracy order is close to 2. The accuracy can be varied by varying the prescribed local error tolerance parameter tol, although the actual errors tend to be larger than tol. Approximations to off-nodal solution values can also be computed, with a comparable accuracy. The method appears numerically stable when partial derivatives, of the non-linear function representing the equation, with respect to the unknown and its integral(s), are of the same sign. The stability of the method in the opposite case may be debated.