An adaptive Huber method for non-linear systems of weakly singular second kind Volterra integral equations

Abstract

Numerical methods for systems of weakly singular Volterra integral equations are rarely considered in the literature, especially if the equations involve non-linear dependencies between unknowns and their integrals. In the present work an adaptive Huber method for such systems is proposed, by extending the method previously formulated for single weakly singular second kind Volterra equations. The method is tested on example systems of 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. The magnitude of the errors, and practical accuracy orders, observed for IE systems, are comparable to those for single IEs. In cases when the solution vector is not differentiable at t=0, the estimation of errors at t=0 is found somewhat less reliable for IE systems, than it was for single IEs. The stability of the IE systems solved appears to be sufficient, in practice, for the numerical stability of the method.