Methods of numerical analysis for some integral dynamical systems with delay arguments

The aim of this work is to construct direct and iterative numerical methods for solving functional equations with hereditary components. Such equations are a convenient tool for modeling dynamical systems. In particular, they are used in population models structured by age with a finite life span. Models based on integro-differential and integral equations with various kinds of delay arguments are considered. For nonlinear equations, the operators are linearized according to the modified Newton-Kantorovich scheme. Direct quadrature and simple iteration methods are used to discretize linear equations. These methods are constructed in the paper: an iterative method for solving a nonlinear integro-differential equation on the semiaxis $(-\infty, 0]$, a direct method for solving the signal recovery problem, and iterative methods for solving a nonlinear Volterra integral equation with a constant delay. Special quadrature formulas based on orthogonal Lagger polynomials are used to approximate improper integrals on the semiaxis. The results of numerical experiments confirm the convergence of suggested methods. The proposed approaches can also be applied to other classes of nonlinear equations with delays.