Multistep methods for numerical solution of integral algebraic equations of index-2

Many processes in various eco-biological and physical systems are described by interconnected Volterra integral equations of the first and the second kinds. Such equations may be written as a system with an identically singular principal part, in other words, in the form of integral algebraic equation. The article studies the class of linear integral algebraic equations, for which the sufficient conditions for existence of a unique continuous solution are formulated in terms of matrix polynomials. The fundamental difference between the problems under consideration and the Volterra integral equations of the first and the second kinds is noted. There are few works on the qualitative theory of integral algebraic equations, and numerical methods for their solution are underdeveloped. Many methods for numerical solution of Volterra integral equations are not applicable or lead to divergent processes. We have proposed one- and two-step methods based on modifications of Adams-Bashforth and Adams-Moulton formulas for the selected class of problems. The results of numerical experiments confirm the efficiency of the algorithms.