A numerical method for solving singular integral algebraic equations with weakly singular kernels

Statements of many applied problems often include differential equations and Volterra integral equations of the first and second kind. By joining such equations together, we obtain a system of integral differential equations with a singular matrix multiplying the leading part. Such systems are commonly referred to as singular integral differential equations. If they do not contain an integral part, then they are called differential-algebraic equations. If there is no term with a derivative, then they are usually called integral algebraic equations. Such mathematical problem statements arise in simulation of processes occurring in electrical and hydraulic circuits, various dynamic systems, in particular, multibody systems. Therefore, qualitative study and numerical solution of such problems are quite relevant, and the results of research remain in demand in practice. In this paper, on the basis of the theory of matrix pencils, as well as using research schemes developed for differential algebraic and integral algebraic equations, the conditions for the existence and uniqueness of the solution of singular integral-differential equations with a weakly singular kernels are analyzed and a numerical method for their solution is proposed. The method was coded in MATLAB and tested on model examples.