Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels in the Theory of Evolving Systems Modeling

The method for solution to the Volterra integral equations of the first kind is proposed. Such equations appear in the theory of developing systems and they have kernels with discontinuities of the first kind. We construct the characteristic algebraic equation. Analytically and numerically we study the regular case when characteristic equation has no positive roots and the solution to the integral equation is unique. In the case of irregular characteristic equation has natural roots, and the solution contains arbitrary constants. We prove existence theorems and construct their asymptotics. The theoretical results are illustrated by numerical calculations.