# Nikolai A. SidorovIrkutsk State University | ISU · Institute of Mathematics and Information Technologies

Nikolai A. Sidorov

PhD DSc

## About

108

Publications

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Introduction

Research interests: differential-operator and kinetic models
Methods and techniques: branching theory of nonlinear equations; bifurcation theory; Lyapunov–Schmidt branching equations; Vlasov-Maxwell equations; Conley index;
singular problems; regularization; approximate methods; functional equations

## Publications

Publications (108)

This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the error analysis of the formulation to validate the accuracy of...

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory...

The necessary and sufficient conditions of existence of the nonlinear operator equations' branches of solutions in the neighbourhood of branching points are derived. The approach is based on reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory bas...

In this paper, we study the stationary boundary value problem derived from the magnetic (non) insulated regime on a plane diode. Our main goal is to prove the existence of non-negative solutions for that nonlinear singular system of second-order ordinary differential equations. To attain such a goal, we reduce the boundary value problem to a singul...

This volume provides a comprehensive introduction to the modern theory of differential-operator and kinetic models including Vlasov-Maxwell, Fredholm, Lyapunov-Schmidt branching equations to name a few. This book will bridge the gap in the considerable body of existing academic literature on the analytical methods used in studies of complex behavio...

Исследованы дифференциальные уравнения с неклассическими начальными условиями в случае необратимости оператора в главной части уравнения. Приведены необходимые и достаточные условия существования неограниченных решений с полюсом $p$-го порядка в точках, в которых оператор, стоящий в главной части дифференциального уравнения, не имеет обратного. На...

The brief review of monograph [Sidorov N., Sidorov D., Sinitsyn A. Toward General Theory of Differential-Operator and Kinetic Models. Book Series: World Scientific Series on Nonlinear Science Series A vol. 97, eds. Prof. L. Chua, World Scientific, Singapore, London, 2020] is provided. This monograph devoted to the state-ofthe-art methods for the no...

Finding the optimal parameters and functions of iterative methods are among the
main problems of the Numerical Analysis. For this aim, a technique of the stochastic arithmetic (SA) is used to control of accuracy of Taylor-collocation method for solving first kind weakly regular integral equations (IEs). Thus, the CESTAC (Controle et Estimation Stoc...

The system of differential and operator equations is considered. This system is assumed to enjoy an equilibrium point. The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process. The sufficient conditions of the global classical s...

This volume provides a comprehensive introduction to the modern theory of differential-operator and kinetic models including Vlasov–Maxwell, Fredholm, Lyapunov–Schmidt branching equations to name a few. This book will bridge the gap in the
considerable body of existing academic literature on the analytical methods used in studies of complex behavio...

The branching theory of nonlinear parameter-dependent equations enabled various essential applications in natural sciences and engineering over the course of the last hundred years. V.I. Yudovich pioneered the application of symmetry methods in branching theory. A series of applications of the Lyapunov-Schmidt method, Conley index theory, and the c...

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...

This article considers the nonlinear dynamic model formulated as the system of differential and operator equations. This system is assumed to enjoy an equilibrium point. The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process....

This paper is an attempt to give the review of a part of our results in the area of singular partial differential equations. Using the results of the theory of complete generalized Jordan sets we consider the reduction of the PDE with the irreversible linear operator $B$ of finite index in the main differential expression to the regular problems. E...

The theory of complete generalized Jordan sets is employed to reduce the PDE with the irreversible linear operator $B$ of finite index to the regular problems. It is demonstrated how the question of the choice of boundary conditions is connected with the $B$-Jordan structure of coefficients of PDE. The various approaches shows the combination alter...

We consider a linear inhomogeneous wave equation and linear inhomo-geneous heat equation with initial and boundary conditions. It is assumed that the inhomogeneous terms describing the external force and heat source in the model are decomposed into Fourier series uniformly convergent together with the derivatives up to the second order. In this cas...

The dynamical model based on the differential equation with a nonlinear operator acting in Banach spaces and a nonlinear operator equation with respect to two elements from different Banach spaces is considered. It is assumed that the system has stationary state (rest points or equilibrium). The Cauchy problem with the initial condition with respec...

The dynamical model consisting of the differential equation with a non- linear operator acting in Banach spaces and a nonlinear operator equation with respect to two elements from different Banach spaces is considered. It is assumed that the system has stationary solutions (rest points). The Cauchy problem with the initial condition with respect to...

The linear system of partial differential equations is considered. It is assumed that there is the irreversible linear operator in the main part of the system, which enjoy the skeletal decomposition. The differential operators is such system are assumed to have a sufficiently smooth coefficients. In the concrete situations the domains of such diffe...

The linear system of partial differential equations is considered. It is assumed that there is an irreversible linear operator in the main part of the system. The operator is assumed to enjoy the skeletal decomposition. The differential operators is such system are assumed to have a sufficiently smooth coefficients. In the concrete situations the d...

Abstract. Existence theorems about bifurcation points of solutions for nonlinear
operator equation in Banach spaces are proved. The sufficient conditions
of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system
are obtained. The analytical method of Lyapunov-Schmidt-Trenogon is
employed.

Existence theorems about bifurcation points of solutions for non-
linear operator equation in Banach spaces are proved. The su�cient conditions
of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell sys-
tem are obtained. The analytical method of Lyapunov-Schmidt-Trenogon is
employed.

The linear PDE ${\mathbf B} {\mathbf L} (\frac{\partial}{\partial x}) u ={\mathbf L}_1(\frac{\partial}{\partial x})u +f(x)$ with nonclassic conditions on boundary $\partial \Omega$ is considered. Here ${\mathbf B}$ is linear noninvertible bounded operator acting from linear space $E$ into $E,$ $x=(t,x_1,\dots, x_m) \in \Omega, $ $\Omega \subset {\m...

The regularization method of linear integral Volterra equations of the first kind is considered. The method is based on the perturbation theory. In order to derive the estimates of approximate solutions and regularizing operator norms we use the Banach-Steinhaus theorem, the concept of stabilising operator, as well as abstract scheme for constructi...

We suggest method based on the skeleton decomposition of linear operators in
order to reduce ill-posed degenerate differential equations to the non-classic
initial-value problem enjoying unique solution

One of the most common problems of scientific applications is computation of
the derivative of a function specified by possibly noisy or imprecise
experimental data. Application of conventional techniques for numerically
calculating derivatives will amplify the noise making the result useless. We
address this typical ill-posed problem by applicatio...

We obtain sufficient conditions for the existence and uniqueness of continuous solutions of Volterra operator equations of the first kind with piecewise determined kernels. For the case in which the solution is not unique, we prove existence theorems for the parametric families of solutions and present their asymptotics in the form of logarithmic p...

We consider generalized solutions of polynomial integral Volterra equations of the first kind that arise in a control problem for nonlinear dynamical processes of the "input-output" type. We prove the existence theorem and propose a method for constructing generalized solutions. We establish that the number of solutions equals the number of roots o...

The review of existence theorems of bifurcation points of solutions for
nonlinear operator equation in Banach spaces is presented.
The sufficient conditions of bifurcation of solutions of boundary-value
problem for Vlasov-Maxwell system are considered. The analytical method of
Lyapunov-Schmidt-Trenogin is employed.

The sufficient conditions for existence and uniqueness of continuous
solutions of the Volterra operator equations of the first kind with piecewise
continuous kernel are derived. The asymptotic approximation of the parametric
family of solutions are constructed in case of non-unique solution. The
algorithm for the solution's improvement is proposed...

We consider a nonlinear operator equation with a Fredholm linear operator in the principal part. The nonlinear part of the equation depends on the functionals defined on an open set in a normed vector space. We propose a method of successive asymptotic approximations to branching solutions. The method is used for studying the nonlinear boundary val...

We consider the nonlinear operator equation B(λ)x + R(x, λ) = 0 with parameter λ, which is an element of a linear normed space Λ. The linear operator B(λ) has no bounded inverse for λ = 0. The range of the operator B(0) can be nonclosed. The nonlinear operator R(x, λ) is continuous in a neighborhood of zero and R(0, 0) = 0. We obtain sufficient con...

Рассматривается нелинейное операторное уравнение B(λ)x+R(x,λ)=0 с параметром λ, являющимся элементом линейного нормированного пространства Λ. Линейный оператор B(λ) не имеет ограниченного обратного при λ=0. Область значений оператора B(0) может быть незамкнутой. Нелинейный оператор R(x,λ) непрерывен в окрестности нуля, R(0,0)=0. Получены достаточны...

We consider generalized solutions of polynomial integral Volterra equations of the first kind that arise in a control problem
for nonlinear dynamical processes of the “input-output” type. We prove the existence theorem and propose a method for constructing
generalized solutions. We establish that the number of solutions equals the number of roots o...

We construct parametric families of small branching solutions to nonlinear differential equations of the nth order near branching points. We use methods of the analytical theory of branching solutions of nonlinear equations and
the theory of differential equations with a regular singular point. We illustrate the general existence theorems with an e...

The main solutions in sense of Kantorovich of nonlinear Volterra
operator-integral equations are constructed. Convergence of the successive
approximations is established through studies of majorant integral and majorant
algebraic equations. Estimates are derived for the solutions and for the
intervals on the right margin of which the solution has b...

The branches of solutions of a nonlinear integral equation of Volterra type in a Banach space are constructed by the successive
approximation method. We consider the case in which a solution may have an algebraic branching point. We reduce the equation
to a system regular in a neighborhood of the branching point. Continuous and generalized solution...

The branches of a solution of the nonlinear integral equation
$$
u\left( x \right) = \int\limits_a^b {K\left( {x,s} \right)q\left( {s,u\left( s \right),\lambda } \right)ds}
$$
, where q(s, u, λ) = u(s) + Σ
i=2∞
q
io
(s)u
i
+ Σ
i=0∞ Σ
k=1∞
q
ik
(s)u
i
λ
k
and λ is a parameter, are constructed by successive approximations. Under consideration is t...

We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(α(t)),t) = 0 with a functional perturbation α(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional
perturbation of the argument. We show that the solutions of such equations can have not onl...

In this paper we derived the explicit structure of generalized solutions of the Volterra integral equations of the rst kind. The solution contains singular and regular components. These components can be constructed separately. On the rst stage we construct the singular component of the solution by solving the special linear algebraic system. On th...

Classes of nonlinear integral Volterra equations occurring in identifying dynamic systems are studied. A solution to a nonlinear
system of integral Volterra equations of the first kind is constructed in the class of generalized functions with a point
support in the form of a sum of singular and regular parts. In obtaining a singular part of the sol...

The continuous solutions for BVP of third order nonlinear differential equations appears in [1] mathematical model of the melt spinning process. The existence theorem is proved for such BVP. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The continuous solutions for BVP of third order nonlinear differential equations appears in [1] mathematical model of the melt spinning process. The existence theorem is proved for such BVP. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Deterministic approaches to the Volterra models learning is considered. First approach is based on kernels identification procedure and the second one employs product indegration method. Generalized solutions of linear and nonlinear integral Volterra equations are also addressed covering the discontinuous solutions cases. (© 2008 WILEY-VCH Verlag G...

We give iterative methods for calculating branching solutions of nonlinear equations using explicit and implicit parametrization in a neighbourhood of a branch point. We consider the regularization problem for these schemes in the presence of errors of calculation. We construct modifications of these methods developing results in (R. Zh. Mat., 1991...

Necessary conditions for inheriting the interlacing property of a non-linear equation by the branching system are obtained. The case when the pair of linear operators interlacing the equation consists of projections or parametric families of linear operators is considered. New conditions are presented which allow one to reduce the number of the equ...

Volterra integral equations of the first kind are studied in terms of generalized functions. The solutions consist of singular and regular compo-nents which can be constructed separately. The singular component is con-structed as solution of the special linear algebraic system. The regular com-ponent is constructed as solution of the special Volter...

A method of construction of generalized solutions with a point carrier in the singular part is proposed for nonlinear Volterra integral-functional equations ∫ 0 t K(t,s)(x(s)+ax(αs)+g(s l x(s),s))ds=f(t) with sufficiently smooth kernel and function f; α and a are constants, and 0<|α|<1. The solution is constructed as a sum of singular and regular c...

The paper discusses continuous and generalized solutions of partial differential equations having operator coefficients which operate in Banach spaces. The operator at the higher derivative with respect to time is Fredholm. We apply Lyapunov-Schmidt’s ideas and the generalized Jordan sets techniques to reduce the partial operator differential equat...

Paper addresses generalized solution of the Volterra inte-gral equations of the first kind. The explicit structure of the solution is derived. Applications of the proposed ap-proach include blind identification of nonlinear dynamic systems from time domain input-output data in heat and power engineering. The resluts will be also of interest to biom...

We apply the generalized Jordan sets techniques to reduce partial differential-operator equations with the Fredholm operator in the main expression to regular problems. In addition this techniques has been exploited to prove a theorem of existence and uniqueness of a singular initial problem, as well as to construct the left and right regularizator...

Constructing nonlinear parameter-dependent mathematical models is essential
in modeling in many scientific research fields. The investigation of branching
(bifurcating) solutions of such equations is one of the most important aspects
in the analysis of such models. The foundations of the theory of bifurcations
for the functional equations were laid...

The theorem on the existence of bifurcation points of the stationary solutions for the Vlasov-Maxwell system with bifurcation direction is proved.

For the Vlasov-Maxwell system, sufficient conditions are obtained for the existence of bifurcation points λ0 ∈ℝ+ corresponding to distribution functions of the form
$f_i (r, v) = \lambda \overset{\lower0.5em\hbox{$f_i (r, v) = \lambda \overset{\lower0.5em\hbox{
.
It is assumed that the values of the scalar and vector potentials of the electromag...

General theorems for potentiality of branching equation (BEq) are established. With the aid of M. Morse theory in its strengthening Conley’s variant applied to potential BEq the general theorems about bifurcation points and surfaces are established. Thus we extend the ideas of maximal using of finite-dimensionality of BEq. Note here that the idea o...

The study of the Vlasov-Maxwell system is reduced to nonlinear elliptic equations in the stationary case and hyperbolic equations in the nonstationary case. On this basis, theorems on the existence of solutions and sufficient conditions of Lyapunov stability are obtained. The cases are considered when electromagnetic fields and distribution functio...

Using group-theoretic methods (MR 80d: 58072, 83m: 58082), the authors construct the general form of the branching equation, symmetric with respect to fundamental representations of the rotation group, and on the basis of this form they propose an iterative method for calculating families of small branching solutions in a neighborhood of a bifurcat...

The study of a class of nonstationary solutions for the Vlasov-Maxwell integro-differential system of equations is reduced to a nonlinear hyperbolic equation. It is shown that the electromagnetic fields and $n$ components of the distribution function are expressed by means of the solution of this equation. In the case of a two-component model the L...

The problem of potentiality of the branching equation is considered. Theorems of existence of bifurcation points are stated and a connection between group symmetry and potentiality of a branching equation is exposed.

A set of stationary solutions of the Vlasov-Maxwell equations is constructed for the single-particle distribution function of a hot plasma. The electromagnetic field is found explicitly by solving the Liouville equation

This book is devoted to approximative methods for computing bifurcating solutions of nonlinear equations in Banach spaces. The problem of computing solutions near a bifurcation point is unstable both in analytical and computational senses. This difficulty is overcome by means of regularization methods. The contents of the book are reflected by the...

We consider a method for constructing the solutions of a linear Fredholm operator equation that is regularized by means of a special perturbation of the equation by a linear operator.

In [1, 2], V. A. Trenogin and the author proposed a method for constructing regularizing equations (R.E.) for ill-posed problems in the theory of branching [3, 4]. Here we consider the problem of choosing initial approximations, that are optimal in a certain sense, to simple solutions of R.E. in a neighborhood of a branching point.

M. K. Gavurin proposed (RZh Mat. 4B383, 1962) an iteration method based on the theory of perturbations for finding the eigenvalues and eigenvectors of linear operators. The method was developed later by F. Kunert (RZh Mat. 5B815, 1968). In the present note, using results of V. A. Trenogin (RZh Mat. 1B706, 1970), this method is extended to problems...

The existence theorems of bifurcation points of solutions for non-linear operator equation in Banach spaces are proved. Because of these theorems, the sufficient conditions of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system are obtained.

We consider the reduction of the degenerate dierence-dieren tial equations with Fredholm operator in the main expression to the regular problems. It is shown how the question of the choice of boundary conditions is connected with the Jordan struc- ture of operator coecien ts of the equations. The problem of the choice of boundary conditions is solv...

We consider methods of reduction of dieren tial operator equations with the Fredholm operator in the main expression to regular problems. Relation between the initial conditions choice problem and the Jordan structure of operator coecien ts of equations is shown. The theorem of existence and uniqueness of the initial prob