ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.25.202301.565-577

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 65R20

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

A. N. Tynda

Penza State University (Penza, Russian Federation)

Abstract. 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.

Key Words: integro-differential equations, nonlinear Volterra integral equations, delay arguments, the Newton-Kantorovich method, linearization, direct discretization

For citation: A. N. Tynda. Methods of numerical analysis for some integral dynamical systems with delay arguments. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:1(2023), 565–577. DOI:

Submitted: 01.12.2022; Revised: 10.02.2023; Accepted: 24.02.2023

Information about the author:

Aleksandr N. Tynda, Associate Professor, Department of Higher and Applied Mathematics, Penza State University (40 Krasnaya St., Penza 440026, Russia), Ph. D. (Phys.-Math.), ORCID:,

The author have read and approved the final manuscript.

Conflict of interest: The author declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.