A. T. Alymbaev, A. Bapa Kyzy , F. K. Sharshembieva

PERIODIC SOLUTIONS OF A SECOND-ORDER NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION

Introduction

Periodic solutions of a second-order nonlinear volterra integro-differential equation. Constructing 2π-periodic solutions for a second-order nonlinear Volterra integro-differential equation using Green's function and the projection-iteration method. Includes error estimation.

Abstract

The article considers the problem of constructing a $2 \pi$-periodic solution of a quasilinear second-order integro-differential equation. Using the Green's function of bounded solutions on the number line, the integro-differential equation is reduced to an integral equation. A $2 \pi$-periodic solution to the integral equation is found using the projection-iteration method. A $2 \pi$-periodic solution is sought as the limit of successive $2 \pi$-periodic functions representable as a Fourier series. An estimate of the error of the difference between the exact and approximate solutions is obtained. Received: February 18, 2024Revised: April 29, 2024Accepted: May 9, 2024

Review

This article investigates the important problem of constructing $2\pi$-periodic solutions for a class of quasilinear second-order Volterra integro-differential equations. The study of periodic solutions is fundamental in understanding the long-term behavior of dynamical systems across various scientific and engineering disciplines. The authors propose a constructive method to address this complex analytical challenge, which holds significant theoretical interest and practical implications for modeling systems exhibiting oscillatory behavior. The methodology employed is systematic and multi-faceted. The authors initiate their approach by skillfully reducing the original integro-differential equation to an equivalent integral equation, leveraging the Green's function specifically designed for bounded solutions on the number line. This transformation simplifies the problem considerably. Subsequently, the projection-iteration method is utilized to determine a $2\pi$-periodic solution for the derived integral equation. A key aspect of this iterative process is the representation of the successive $2\pi$-periodic functions, which converge to the solution, as Fourier series. A crucial strength of the presented work lies in the derivation of an error estimate, providing a quantitative measure of the difference between the exact and the approximate solutions, thereby validating the accuracy and reliability of the proposed method. Overall, this paper offers a rigorous and well-articulated approach to a challenging problem in nonlinear analysis. The integration of Green's function techniques with the projection-iteration method, augmented by Fourier series representation, provides a robust framework for tackling integro-differential equations. The explicit construction of periodic solutions, coupled with the essential provision of error bounds, significantly enhances the practical utility and theoretical credibility of the findings. This work is a valuable contribution to the field and will be of interest to researchers engaged in the analytical and numerical treatment of differential and integral equations, particularly those focused on periodic phenomena in complex systems.

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