Construction of implicit multistep methods for solving integral algebraic equations

This paper discusses techniques for construction of implicit stable multistep methods for solving systems of linear Volterra integral equations with a singular matrix multiplying the leading part, which means that systems under consideration comprise Volterra equations of the first kind as well as Volterra equations of the second kind. Methods for solving first kind Volterra equations so far have been justified only for some special cases, for example, for linear equations with a kernel that does not vanish on the diagonal for all points of the segment. We present a theoretical analysis of solvability of the systems under study, single out classes of two- and three-step numerical methods of order two and three, respectively, and provide examples to illustrate our theoretical assumptions. The experimental results indicate that the stability of the methods can be controlled by some weight parameter that should be chosen from a prescribed interval to provide the necessary stability of the algorithms.