An adaptive Huber method for nonlinear systems of Volterra integral equations with weakly singular kernels and solutions

Numerical methods for solving nonlinear systems of weakly singular Volterra integral equations (VIEs) possessing weakly singular solutions appear almost nonexistent in the literature, except for a few treatments of single first kind Abel equations. To reduce this gap, an extension is presented, of the adaptive Huber method designed for VIEs with singular kernels such as K(t,τ)=(t−τ)−1/2 and K(t,τ)=exp[−λ(t−τ)](t−τ)−1/2 (where λ≥0) and a variety of nonsingular kernels. The method was thus far restricted to bounded solutions having at least two derivatives. Under a number of assumptions specified, the extension applies to solutions Uμ(t) that can be written as sums of singular components cμt−1/2 (with unknown coefficients cμ), and nonsingular components U¯μ(t). In the solution process, factor t−1/2 is handled analytically, whereas cμ and U¯μ(t) are determined numerically. Computational experiments reveal that the extended method determines singular solutions equally well as the unextended method determined nonsingular solutions. The method is intended primarily for a class of VIEs encountered in electroanalytical chemistry, but it can also be of interest to other application areas.