Collocation methods for third-kind VIEs

The spline collocation method in piecewise polynomial spaces is applied to a class of third-kind Volterra integral equations (VIEs). Under certain conditions the operator associated with the equivalent secondkind VIE is compact, and this guarantees that the resulting algebraical system is uniquely solvable for all sufficiently small mesh diameters. However, the solvability of this system is not ensured, both on uniform and classical graded meshes, when the operator is noncompact (which typically is the case). Hence, we introduce a modified graded mesh to overcome the solvability problem. For such meshes we establish results on the optimal order of global convergence of the collocation solutions for third-kind VIEs. Numerical tests confirm the validity of these results.