Aleksandr Tynda

Penza State University · Higher and Applied Mathematics

The Volterra integral-functional series is the classic approach for nonlinear black box dynamical system modeling. It is widely employed in many domains including radiophysics, aerodynamics, electronic and electrical engineering and many others. Identifying the time-varying functional parameters, also known as Volterra kernels, poses a difficulty d...

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...

The objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the rest of the information is known. The resulting in...

The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gausstype quadrature formula is used to approximate integrals during the discretization of the proposed projection method. The estimate of accuracy of approximate solution is obtained. Sto...

The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gauss-type quadrature formula is used to approximate integrals during the discretisation of the proposed projection method. The estimate of accuracy of approximate solution is obtained. St...

Volterra integral equations find their application in many areas, including mathematical physics, control theory, mechanics, electrical engineering, and in various industries. In particular, dynamic Volterra models with discontinuous kernels are effectively used in power engineering to determine the operating modes of energy storage devices, as wel...

Volterra integral equations find their application in many areas, including mathematical physics, control theory, mechanics, electrical engineering, and in various industries. In particular, dynamic Volterra models with discontinuous kernels are effectively used in power engineering to determine the operating modes of energy storage devices, as wel...

The aim of this study is to present a novel method to find the optimal solution of the reverse osmosis (RO) system. We apply the Sinc integration rule with single exponential (SE) and double exponential (DE) decays to find the approximate solution of the RO. Moreover, we introduce the stochastic arithmetic (SA), the CESTAC method (Controle et Estim...

The aim of this study is to present a novel method to find the optimal solution of the reverse osmosis (RO) system. We apply the Sinc integration rule with single exponential (SE) and double exponential (DE) decays to find the approximate solution of the RO. Also, we introduce the stochastic arithmetic (SA), the CESTAC method (Controle et Estimatio...

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of...

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical me...

The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified Newton-Kantorovich iterative process for the integral operators linearization. On each step of the iterative process the linea...

Further growth in renewable energy and planned electrification and decentralization of transport and heating loads in future power systems will result in a more complex unit commitment problem (UCP). This paper proposes an adaptive approach to load leveling problem using novel dynamic models based on the Volterra integral equations of the first kin...

В данной работе на основе интегральных динамических моделей предлагается универсальный подход к проблеме рационального использования аккумулирования энергии для разных характеристик аккумуляторных батарей, таких как: КПД, емкость, уровень заряда, ограничения на скорость зарядки и разрядки, максимальное количество циклов разрядки. Предлагаемый подхо...

In this paper we investigate the systems of nonlinear Volterra integral equations of the first kind with kernels having jump discontinuities along the set of smooth curves. The necessary theory concerning the existence and uniqueness of solutions of such systems is described. An iterative numerical method is proposed, based on the linearization of...

In this paper we suggest several methods for numerical treatment of integral dynamical systems described by nonlinear integral equations of the special form. This first part of the paper is devoted to the construction of iterative numerical algorithm for the systems of nonlinear Volterra-type equations related to the Vintage Capital Models (VCM)....

Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the optimal approximation problem, the orders of the Babenko and Kolmogorov n-widths of compact sets fr...

We propose the numerical methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equation of the first kind. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays) which starts at the origin. In order to linearize these equations we use the modified Newton-Kan...

Integral equations are in the core of many mathematical models in physics, economics and ecology. Volterra integral equations of the first kind with jump discontinuous kernels play important role in such models and they are considered in this article. Regularization method and sufficient conditions for existence and uniqueness of the solution of su...

The numerical method for solution of the weakly regular scalar Volterra integral equation of the 1st kind is proposed. The kernels of such equations have jump discontinuities on the continuous curves which starts at the origin. The mid-rectangular quadrature rule is employed. The accuracy of proposed numerical method is $\mathcal{O}(1/N).$

In this paper we develop a numerical method for solving volume integral equation of the effective dielectric permittivity based on the "dead layer" model. This model links the effective dielectric permittivity to the electric field distribution inside the granule and the "dead layer", the ferroelectric dielectric permittivity, the dielectric permit...

Abstract. Objective: the main aim of this paper is the construction of the optimal with respect to accuracy order methods for weakly singular Volterra integral equations of different types. Methods: since the question of construction of the accuracyoptimal numerical methods is closely related with the optimal approximation problem, the authors appl...

The article estimates the diameters of Kolmogorov and Babenko class functions which have the solutions of Volterra integral functions with singular kernels. A distinctive feature of these classes is an unlimited growth of function derivative modules when approaching a definitial domain boundary. For these function classes the authors have built loc...

This paper is devoted to numerical treatment of rheological models in the context of nonlinear heritable creep theory. An approximate method for nonlinear weakly singular Volterra integral equations with Rzhanitsyn's kernel used in rheological models of viscoelastic continuum is suggested. In conclusion we adduce some numerical results demonstratin...

This paper is devoted to the construction of iterative numerical algorithm for the systems of nonlinear Volterra-type equations of the special form related to the Vintage Capital Models (VCMs). One of the unknown functions of these equations is in the lower limit of integration. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper we investigate the possibility of a construction of direct numerical methods for systems of nonlinear Volterra–type integral equations. One of the unknown functions of such systems is in the limit of integration. Two numerical schemes based on the piecewise constant and piecewise linear approximation of the exact solutions are suggest...

The paper is dedicated to the numerical solution of 2D weakly singular Volterra integral equations with two different kernels. In order to weaken a singularity influence on the numerical computations we transform these equations in both cases into equivalent equations. The piecewise polynomial approximation of the exact solution is then applied. Fo...

We construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the “block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods...

11 @t vl l . In the case of the class Q r; (;M), by % (t; 0) we denote the distance from the point t to the intersection 0 of the boundary of the domain with coordinate planes given by the formula % (t; 0 )=m inijtij; in the case of the class Q r;(;M), by % (t; 0) we denote the distance from the point t to the origin, that is, %(t; 0) = p t 2 + +t...

ones, taking into account given properties of the input functions. This paper consists of the review of the recent results of the authors and report of the present research on this topic. We suggest the optimal on accuracy spline-collocation methods with special meshes for the numerical solution of mul- tidimensional weakly singular VIE. For this p...