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We generalize the notion of the minimal realization from the linear theory to the nonlinear case. We justify a relationship of this notion with the notion of first integral of a system. We suggest algorithms for the computation of first integrals and minimal realizations. [ABSTRACT FROM AUTHOR]

We consider the problem on the convergence rate of biorthogonal expansions of functions in systems of root functions of a wide class of ordinary second-order differential operators defined on a finite interval. These expansions are compared with expansions of the same functions in Fourier trigonometric series in an integral or uniform metric on any interior compact subset of the main interval. We find the dependence of the equiconvergence rate of resulting expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions. [ABSTRACT FROM AUTHOR]

We prove that the estimates earlier obtained by the authors for the deviations of quasi-integrals of a nonstationary three-dimensional linear differential system with continuously differentiable skew-symmetric coefficient matrix from the corresponding integrals in the stationary case, which are attainable on different nontrivial solutions of the system, are not simultaneously attainable on any single nontrivial solution and at the same time. In the course of proof, we establish new estimates, which are more accurate than those obtained earlier in Mem. Differential Equations Math. Phys., 2007, vol. 41, pp. 157–162. The suggested method for constructing such estimates is also of interest. [ABSTRACT FROM AUTHOR]

In the space L2(ℝ), we consider the self-adjoint extension $$ \mathcal{L} $$ of the Sturm-Liouville operator ly = − y″ + q( x) y whose potential q is uniformly locally integrable on ℝ, i.e., satisfies the condition . We study the problem on the equiconvergence rate of the spectral expansion associated with $$ \mathcal{L} $$ of a function f ∈ L1(ℝ) with the Fourier integral on the entire real line. We obtain uniform estimates of the equiconvergence rate under some additional conditions on f or q. [ABSTRACT FROM AUTHOR]

Using the Sturm-Liouville operator with a complex potential as an example, we analyze the spectral instability effect for operators that are far from being self-adjoint. We show that the addition of an arbitrarily small compactly supported function with an arbitrarily small support to the potential can substantially change the asymptotics of the spectrum. This fact justifies, in a sense, the necessity of well-known sufficient conditions for the potential under which the spectrum of the operator is localized around some ray. For an operator with a logarithmic growth, we construct a perturbation that preserves the asymptotics of the spectrum but has infinitely many poles inside the main sector. [ABSTRACT FROM AUTHOR]

In the space L2(ℝ2), we consider the operator . We study the spectrum of H and, for V ∈ C(ℝ2), prove the trace formula where c0 = π−1 $$ \smallint _{\mathbb{R}^2 } $$ V( x) dx and the µ are the eigenvalues of H. [ABSTRACT FROM AUTHOR]

We express singular integrals of functions involving solutions of transmission problems via linear combinations of these functions with fractional-rational coefficients. [ABSTRACT FROM AUTHOR]

The article examines the inverse problems for a quadratic pencil of Sturm-Liouville operators on the entire line in the case of periodic and finite-gap periodic coeffecients. The previous studies emphasized the inverse problems for the Sturm-Liouville operator with finite-gap periodic and nonperiodic potentials on the half-line and on the entire line. In this scope, it focuses on the half-line area where it obtain the system Dubrovin-Trubowitz differential equations.

Suggests a locally one-dimensional difference scheme for the solution of a two-dimensional parabolic problem with an integral condition. Features of the economical algorithm for the scheme's implementation; Solvability of parabolic problems with a nonlocal integral condition; Unique solution for explicit difference schemes for the two-dimensional diffusion problem.

In the present paper, the author show that the Gegenbauer polynomials with an appropriate weight are orthogonal in the sense of the inner product and nonorthogonal in the inner product corresponding to the norm. The author introduced the norm 3.2, one of the theorem on differential equations, and consider positive integral operators of the form 2.2, one of the theorem of differential equations. The author constructs an orthonormal basis in the Sobolev-Slobodetskii spaces from Gegenbauer polynomials.

This article presents derivation of the asymptotic stability of a differential system with linear coppel-conti approximation.

Discusses numerical schemes of discrete vortex type for a class of singular integral equations with closed contours. Modified discrete vortex method; Higher order of approximation of the integrals; Closed Lyapunov contour on the complex plane.

Describes two forms of stability change for integral manifolds. Features of the use of attracting-repulsive integral manifolds analogus to duck trajectories for the modeling and control of critical combustion modes; Analysis of the existence of continuous attracting-repulsive integral surfaces for some class of regular systems and systems with singular perturbations.

The article focuses on the numerical solution of Lippmann-Schwinger Equation. The article presents the quadrature method based on sanikidze quadrature formulas, the sanikidze representation for the densities of singular integrals, and the method of discrete singularities.

The author considers a non-local boundary value problem for an elliptic-hyperbolic equation. In the hyperbolic part of the domain, our problem coincides with the Gellerstedt problem. A non-local boundary condition of the Samarskii-Bitsadze type is posed in the elliptic part. The author proves the existence and uniqueness of a regular solution of the problem. For a special domain, the authors write out the solution of the problem in quadratures.

The article presents a discourse on weakly degenerate hyperbolic equations. In the present paper, the authors prove the unique solvability of problem and construct a closed-form expression for the solution. To state the main results and their proofs the authors need a number of auxiliary results about Bessel operator functions introduced in previous papers of the journal. In what follows, the authors define by induction for arbitrary negative, give a precise description of their domains, and clarify their properties.

The article presents a discourse on optimal control of an elliptic Equation with a non-smooth non-linearity. By the Sobolev theorem, there are continuous embeddings that are given in the article.

The article presents the solution of boundary value problems for the fractional diffusion equation by the green function method. The authors use the green function method to obtain a general representation of solutions and construct green functions of the first, second, and mixed boundary value problems.

The article focuses on the bifurcation of an invariant torus from an equilibrium under commensurability conditions. The authors study the problem on the bifurcation of an s-dimensional invariant torus of system.

The article presents information on the interior Ellipsoidal bounds for the Attainability Sets of linear delay Control Systems. By a straightforward application of the formulas describing the bounds for integrals of ellipsoids, it obtains exhaustive interior bounds for the attainability set of the delay problem in a finite-dimensional space. By transferring the technique of estimating integrals over ellipsoids to function spaces and sets of ellipsoidal type, it proves exhaustive interior bounds for the attainability set of the delay system in a function space.

The article calculates the invariant algebraic curves of polynomial dynamical systems. The article uses various differential equations and integral calculus to solve the problem.

The article presents information on the numerical solution of hypersingular and singular Integral Equations on the Circle. The article analyzes the properties, including stability, of discrete operators approximating hypersingular and singular integrals operators in the entire scale of spaces of fractional quotients and present theorems on the stability of a hypersingular discrete operator in the uniform metric. It proves the convergence of quadrature formulas in the upper part of the scale of spaces of fractional quotients. The article also proves the existence and uniqueness of the solutions of these equations under natural conditions, the convergence of approximate solutions to the exact solution in spaces of fractional quotients, and estimates of the convergence rate in spaces of fractional quotients and in the uniform metric with the use of a discrete analog of the Sobolev embedding theorem.

Presents abstracts of reports presented in the spring semester 2003 seminar on qualitative theory of differential equations at Moscow State University. "Boundary homogenization in domains with singular density," by G.A. Chechkin; "On effectiveness of perturbations in the class of linear Hamiltonian systems," by I.N. Sergeev; "Trajectories of invariant sets of holomorphic mappings," by I.L. Zinchenko.

We suggest a new algorithm for constructing first integrals of a class of third-order differential equations. The algorithm is based on the use of necessary and sufficient conditions for the existence of such integrals. [ABSTRACT FROM AUTHOR]

The article presents a class of asymptotically equivalent spatially homogeneous discrete Boltzmann Equations. It is possible to reduce the number of parameters in system while preserving its main asymptotic properties.