Numerical Study of Index-v First kind Volterra Integral Equations as Ill-posed Problem

To quantify the level of difficulty in solving Volterra integral equations of the first kind (VIEs), the concept of an index is introduced. Volterra integral equations of the first kind are inherently ill-posed, a property they share with all first-kind equations. When the index exceeds 1, the ill-posedness of the solution is exacerbated. It is important to stress that a numerical scheme convergent for first-kind VIEs of a given index may not be applicable to those with a higher index. In this paper, we consider some common piecewise polynomial collocation methods and global spectral methods to solve index-???? first kind VIEs. We show that the proposed piecewise collocation methods (one-step, multi-step, implicit multi-step collocation method) in some references do not work properly to solve index-???? > 2, first kind VIEs while the spectral method can work properly.