I. V. Boykov

Penza State University · higher and applied mathematics

The paper is devoted to the approximate calculation of Riemann definite integrals, singular and hypersingular integrals over closed and open non-rectifiable curves and fractals. The conditions of existence for the Riemann definite integrals over non-rectifiable curves and fractals are provided. We give a definition of a singular integral over non-r...

Background. Developing exact and stable algorithms for solution of inverse problems of mathematical physics is at the leading edge of modern numerical mathematics thanks to rapidly increasing number of applications of such problems in physical and technical sciences as well as some properties of such problems that significantly complicate their sol...

Background. The study is devoted to the analysis of stability in the sense Lyapunov Cohen-Grossberg neural networks with time-dependent delays. To do this, we study the stability of the steady-state solutions of systems of linear differential equations with coefficients depending on time and with delays, time dependent. The cases of continuous and...

The paper is devoted to approximate methods for solution of direct and inverse problems for parabolic equations. An approximate method for the solution of the initial problem for multidimensional nonlinear parabolic equation is proposed. The method is based on the reduction of the initial problem to a nonlinear multidimensional intergral Fredholm e...

Background. Singular integral equations in degenerate cases describe many processes in natural science and technology. The theory of these equations has been studied quite well, but as far as the authors know, there are currently no analytical methods for solving them. In this regard, there is a need to construct approximate methods for solving sin...

Background. The study is devoted to the analysis of stability in the sense Lyapunov steady state solutions for systems of linear parabolic equations with coefficients depending on time, and with delays depending on time. The cases of continuous and impulsive perturbations are considered. Materials and methods. A method for studying the stability of...

In this paper we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficie...

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator...

Background. When solving many physical and technical problems, a situation arises when only operators (functionals) from the objects under study (signals, images, etc.) are available for observations (measurements). It is required to restore the object from the known operator (functional) from the object. In many cases, the correlation (autocorrela...

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular inte...

We consider the problems of information-measuring equipment, modeled by ordinary differential equations, when some physical variable cannot be measured, but its value can be determined by the functional (or operator) of another physical variable available for measurement. Direct application of models with ordinary differential equations for recover...

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonli...

The article is devoted to the issue of construction of an optimal with respect to order passive algorithms for evaluating Cauchy and Hilbert singular and hypersingular integrals with oscillating kernels. We propose a method for estimating lower bound errors of quadrature formulas for singular and hypersingular integral evaluation. Quadrature formul...

The work is devoted to the approximate methods for solution direct and inverse problems of gravity exploration on bodies with a fractal structure. It is known that in order to construct mathematical models adequate to the geological reality, it is necessary to take into account the orderliness inherent in geological environments. One of the manifes...

The paper consists of three parts. The first one is devoted to approximate methods for evaluating Riemann integrals, singular and hypersingular integrals on closed non-rectifiable curves and fractals in the complex plane. An integral on non-rectifiable curves or fractals is defined as a double integral over a region that bounded by a non-rectifiabl...

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M), Ω¯ur,γ(Ω,M), Ω=[−1,1]l, l=1,2,…,M=Const, and γ is a real positive number. The functions that belong to classes Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M) have bounded derivatives up to t...

We describe the continuous operator method for solution nonlinear operator equations and discuss its application for investigating direct and inverse scattering problems. The continuous operator method is based on the Lyapunov theory stability of solutions of ordinary differential equations systems. It is applicable to operator equations in Banach...

Background. Ambartsumian’s equation and its generalizations are one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. An analytical solution to this equation is currently unknown, and the development of approximate methods is urgent. To solve the Ambartsumian’s equation, sever...

Background. In recent decades the theory for solution of inverse and ill-posed problems has become one of the most important and fast-growing branch of modern mathematics. A relevancy of this theory is due to not only significant growth in the number of applications of inverse and ill-posed problems in different fields of physical and technical sci...

We study the stabilization of linear nonstationary dynamical systems by introducing a linear nonstationary feedback with delay. We obtain sufficient stabilization conditions for the systems and propose a method for constructing stabilizing matrices.

The work is devoted to a review of analytical and numerical methods for solving linear hypersingular integral equations. Hypersingular integral equations of the first and second kind on closed and open integration intervals are considered. Particular attention is paid to equations with second-order singularities, since these equations are most in d...

Background. Hypersingular integrals are now finding more and more fields of application – aerodynamics, elasticity theory, electrodynamics and geophysics. Moreover, their calculation in an analytical form is possible only in very special cases. Therefore, approximate methods for calculating hypersingular integrals are an urgent problem in computati...

Background. Ambartsumian equation and its generalizations are one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. Ambartsumian equation plays an important role in the study of light scattering in media of infinite optical thickness. Nowadays the analytical solution of this e...

Background. Despite Gibbs effect (phenomenon) was discovered almost 170 years ago, the amount of works devoted to its research and the construction of methods of its suppression has not weakened until recently. This is due to the fact that the Gibbs effect has a negative impact on the study of many wave processes in hydrodynamics, electrodynamics,...

In the paper we investigate approximate methods for solving linear and nonlinear hypersingular integral equations defined on the number axis. We study equations with the second-order singularities because such equations are widely used in problems of natural science and technology. Three computational schemes are proposed for solving linear hypersi...

Background. A problem of determination of unknown boundary condition often appears in different fields of physics and technical sciences in cases when direct measuring of field characteristics at some part of the boundary is difficult or even impossible. Examples of such problems can be found in applications of geophysics, nuclear physics, inverse...

Background. The theory of solving inverse problems of mathematical physics is one of the most actively developing branches of modern mathematics. The interest of researchers in such problems is primarily due to the large number of their applications that have appeared in recent years in connection with the rapid development of physics and technolog...

Non-stationary continuous and discrete dynamical systems are modeled, respectively, by Volterra integral equations of the first kind and their discrete analogues. Algorithms for the exact restoration of the impulse response (in analytical form) for continuous systems and the transient characteristic for discrete systems are constructed. We study an...

We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today,hypersingular integral equations of this type are widely used in physics and technology. The...

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its...

Background. This work is devoted to the study of sets of functions in which the condition of unique solvability of degenerate polysingular integral equations is satisfied, and to the construction of approximate methods for solving polysingular integral equations in degenerate cases. Nowadays, the study of many sections of singular integral equation...

Background. Parabolic differential equations of mathematical physics play very important role in mathematical modeling of the wide range of phenomena in physical and technical sciences. In particular, parabolic equations are widely used for modeling diffusion processes, processes of fluid dynamics as well as biological and ecological phenomena. The...

Background. The work is devoted to the study of sets of functions in which the condition for the unique solvability of degenerate singular integral equations is satisfied. At present, the study of many sections of singular integral equations can be considered completed. An exception is singular integral equations that vanish on manifolds with a mea...

The article is devoted to the modeling of some problems arising in the analysis and synthesis of automatic control systems by methods of singular integral equations. The problems of determining the impulse response functions of dynamic systems in the presence of noise, problems of optimal extrapolation and filtering of the signal, which lead to the...

The paper presents a brief review of the authors' results on methods for reconstructing input signals of dynamic systems. Continuous and discrete systems with distributed and lumped parameters are considered. Mathematical models of these systems are differential and difference equations, as well as integral equations. The main focus of the work is...

The problem of recovering a value of the constant coefficient in heat equation for one- and two-dimensional cases is considered in the paper. This inverse coefficient problem has broad range of applications in physics and engineering, in particular, for modelling heat exchange processes and for studying properties of materials and designing of engi...

There are the Conference papers

We propose a method for the recovery of input signals of eddy-current displacement transducers under thermal-shock action. The recovery of the input signals was performed with correction of the output data of a primary measuring transducer. The procedure of correction is based on the methods of identification of nonlinear dynamic systems described...

We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential equations. Well known that many types of differential equations can be solved in quadratures. So, we can receive analyt...

The paper is devoted to the analysis of stability in the sense of Lyapunov steady-state solutions of systems of nonlinear differential equations with coefficients and with time delays. The cases of continuous and impulsive perturbations are considered.

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich ar...

The paper describes an unconventional method of solving the amplitude-phase problem. The main properties of the Hilbert transform in the discrete and continual cases for one-dimensional and two-dimensional mappings are considered. These transformations are widely used to solve amplitude-phase problem. A numerical method for solving of two-dimension...

We propose a method for solving linear and nonlinear hypersingular integral equations. For nonlinear equations the advantage of the method is in rather weak requirements for the nonlinear operator behavior in the vicinity of the solution. The singularity of the kernel not only guarantees strong diagonal dominance of the discretized equations, but a...

Sufficient conditions for the stability of steady-state solutions of systems of nonautonomous linear and nonlinear differential equations with time-dependent delay are obtained in terms of coefficients. These sufficient conditions are written as inequalities relating quantities that can be calculated directly from the right-hand side of the system...

We study the identification methods for the nonlinear dynamical systems described by Volterra series. One of the main problems in the dynamical system simulation is the problem of the choice of the parameters allowing the realization of a desired behavior of the system. If the structure of the model is identified in advance, then the solution to th...

Methods for reconstructing input signals of dynamical systems with distributed parameters based on nonparametric and parametric identification methods are considered. The proposed methods enable reconstruction of the input signals to within a computational error.

The paper presents two methods for calculating singular and hypersingular integrals with rapidly oscillating kernels. One method is based on the conversion of the mentioned integrals to ordinary differential equations and a numerical solution of the latter. The second method is to construct quadrature formulas of interpolation type. To obtain lower...

Книга посвящена аналитическим и численным методам идентификации динамических систем с сосредоточенными и распределенными параметрами. Рассматриваются динамические системы, описываемые системами обыкновенных дифференциальных уравнений, уравнений в частных производных, интегральными уравнениями Вольтерра и Фредгольма. Рассмотрены линейные и нелинейны...

In this paper we study stability and asymptotic stability in the Lyapunov sense of mathematical models in immunology. Tested models are based on different approaches. We propose a common method for obtaining criteria for stability and asymptotic stability of all considered models. This method employs the assertion of stability of solutions of syste...

In this paper we study stability and asymptotic stability in the Lyapunov sense of mathematical models in immunology. Tested models are based on different approaches. We propose a common method for obtaining criteria for stability and asymptotic stability of all considered models. This method employs the assertion of stability of solutions of syste...

Background. At the present time the theory and technology of antennas are one of most rapidly developing fields of radio engineering. Modern progress in the theory and technology of antennas is based on latest achievements in physics and mathematics. Due to the need for antenna miniaturization for mobile devices, the methods of fractal geometry are...

This paper is devoted to overview of the authors works for numerical solution of singular integral equations (SIE), polysingular integral equations and multi-dimensional singular integral equations of the second kind. The authors investigated onsidered iterative - projective methods and parallel methods for solution of singular integral equations,...

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Omega = [-1, 1] (l) , l = 1, 2,..., and having bounded partial derivatives up to the order r in Omega and the derivatives of jth order (r < j a parts per thousand currency sig...

Приближенные методы решения гиперсингулярных интегральных уравнений являются активно развивающимся направлением вы- числительной математики. Это связано с многочисленными приложениями гиперсингулярных интегральных уравнений в аэродинамике, электродинами- ке, физике и с тем обстоятельством, что аналитические решения гиперсингу- лярных интегральных у...

We propose a method for calculation of one-dimensional and two-dimensional hypersingular integrals, which allows to obtain a solution in the analytic form in a series of cases. Based on this method, we construct an approximate method for calculation of one-dimensional and twodimensional hypersingular integrals having singularities on rather complic...

Background. Recently the question of ensuring information security is particularly acute. Voice identification of personality hasn't become current so far because of a number of unresolved problems. One of the major problems is reliability of authentication. Now the probability of an error of recognition of speaker’s voice is rather high. There is...

Background. Main problems in development of algorithms and software for implementing voice authentication are the following: user's voice variations (voice can vary depending on health conditions, age, mood etc.); presence of a noise component. Solving these problems will allow to use the voice authentication technology to ensure the best protectio...

Considered collocation method and method of mechanical quadrature for solution of hypersingular integral equations od the first kind/ Considered all types of weight functions.

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...

This paper describes numerical schemes based on spline-collocation method and their justifications for approximate solutions of linear and nonlinear hypersingular integral equations with singularities of the second kind. Collocations with continuous splines and piecewise constant functions are examined for solving linear hypersingular integral equa...

A method for determining the parameters of linear systems described by differential equations with variable coefficients. The method is based on solving systems of integral equations with kernels that are impulse response functions of the system.

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...