Recent Papers

The theory of Macdonald integrals. I. Recurrent relations. Uniformly convergent series

Solution of problems in the theory of heat and mass transfer

The paper presents the application of the contour integral method to solving mixed problems for a system of differential equations of heat and mass transfer under molecular and molar transport of energy and matter in an arbitrary three-dimensional domain. Furthermore, the contour integral method is …

Some properties of functions which satisfy elliptic equations of higher order

A regular solution of the equation \begin{equation} L_nL_{n-1}\dotsb L_1u=0, \label{1} \end{equation} is considered, where $$L_i=A_i\frac{\partial^2}{\partial x^2}+2B_i\frac{\partial^2}{\partial x\partial y}+C_i\frac{\partial^2}{\partial y^2}+D_i\frac{\partial}{\partial x}+E_i\frac{\partial}{\partia…

The classification of trajectories of a dynamical system with cylindrical phase space

A system of differential equations containing angular coordinates is considered. The phase space of such a system is cylindrical. Based on the behavior of trajectories on covering spaces, a classification of the trajectories of the system under consideration is introduced. Conditions for the absence…

The choice of the right-hand sides in the system of differential equations of the gradient method

Methods are proposed for constructing the right-hand sides of a system of differential equations for the gradient method, ensuring acceleration of the convergence process. In this context, the system of differential equations describing the motion of a phase space point toward an extremal point is c…

The existence and uniqueness theorem for the Cauchy problem of a hyperbolic equation with self-regulating delay

For a hyperbolic equation of the form $$U_{tx}=f(t,x,U(t,x),U(t-\tau,x),U_t(t,x),U_t(t-\tau,x),U_x(t,x)U_x(t-\tau,x))$$ with an initial function $\varphi(t,x)$ defined for $(t,x)\in[t_0-\tau_0,t_0]\times\Omega$ and with a delay $\tau=\tau(t,x,U,U_t,U_x)$ that depends not only on the independent vari…

Solution of a completely hyperbolic equation with initial data given on a line of degeneracy

A solution is sought for an equation $$L_1L_2\dotsb L_n=f\quad(y>0),$$ which is strictly hyperbolic in a certain bounded domain $D$, where $$L_iU\equiv y^{a_i}U_{xx}-U_{yy}+a_i(x,y)U_x+b_i(x,y)U_y+c_i(x,y)U\quad(i=1,2,\dots,n),$$ satisfying the initial data $$\quad\lim_{y\to0}\frac{\partial^iU(x,y)}…

Некоторые теоремы о предельных циклах

The equation $\ddot{x}=f(x)-R(x,\dot{x})$ with a discontinuous right-hand side, which can be interpreted as a generalized pendulum equation, is considered. This work is devoted to the issues of the existence and relative positioning of global limit cycles. Estimates of the distance between two globa…

The topological equivalence of systems of total differential equations in neighborhoods of closed trajectories

A system of equations \begin{equation} dx=\sum_{i=1}^np_i(x)\,dt^i, \tag{1} \label{1} \end{equation} is considered, where $x$ and $p_i(x)$ are $(n+1)$-dimensional vectors for which the conditions of complete integrability are satisfied. It is assumed that the system \eqref{1} possesses a closed traj…

The connection between the stability of characteristic exponents and almost reducibility of systems with almost periodic coefficients

The article proves the following theorems: 1) If $A(t)$ is recurrent, then the system $\dot{x}=A(t)x$ can be reduced to the triangular form $\dot{u}=P(t)u$ with a recurrent matrix $P(t)$ by a Perron transformation $x=U(t)u$ with a recurrent matrix $U(t)$. 2) If $A(t)$ is almost periodic, then for th…

Limit of the solution to the wave equation for an inhomogeneous medium as $t\to\infty$

The non-stationary problem \begin{equation} u_{xx}=k^2(x)u_{tt}, \tag{1} \label{1} \end{equation} is investigated, where $k(x)=k_0$ for $x<0$ and $k(x)=k_1$ for $x>x_0$, under the initial condition $u_0(x,t) = \mu(t-k_0x)$ for $t<0$, where $\mu(z)=0$ for $z<0$. It is shown that under the condition $…

On the problem of the analytic construction of regulators

The paper considers the well-known problem of the analytical design of a controller for a linear controlled system. However, in contrast to existing developments, this study examines a more complex nonlinear case rather than a quadratic optimality criterion. For the aforementioned criterion, the pro…

On the question of stability of mixed problems for a quasi-linear equation

A remark on limit cycles of an equation

The basic existence theorems of space elasticity theory

A problem on a pairing of equations of parabolic and hyperbolic types when time derivatives occur in the boundary conditions

We consider the problem of finding a solution to the following system of equations: $$\frac{\partial u^{(1)}}{\partial t}=\sum_{l=0}^2C_{0l}^{(1)}(x)\frac{\partial^lu^{(1)}}{\partial x^l}+f^{(1)}(x,t),\quad x\in(a_1,b_1),\quad t\in(0,T),$$ $$\frac{\partial^2 u^{(2)}}{\partial t^2}=\sum_{\substack{{k…

Uniqueness theorems for a system of multidimensional Volterra integral equations

Expansion formulas for a boundary value problem with a complex parameter

A priori estimate of solutions of boundary value problems for second order ordinary nonlinear differential equations

Problems $$N[y]=y''f(t,y,y')=0,\\alpha_0 y(a)+\alpha_1 y'(a)=A,\quad\beta_0 y(b)+\beta_1 y'(b)=0$$ are considered under the assumption that $f(t,y,y')$ satisfies the Carathéodory condition, the Lipschitz condition with respect to $y$, and there exists a continuous $\partial f(t,y,y')/\partial y'$. A…

An investigation of a system of equations describing the motion of a spherical pendulum in the case of the presence of a resistance

The system of differential equations \begin{gather}\theta=x,\notag\\dot{x}=-\alpha x-\frac{g}{l}\sin\theta+y^2\sin\theta\cos\theta+L,\tag{1}\\dot{y}=-\alpha y+2xy\operatorname{ctg}\notag\theta\end{gather} is investigated using Lyapunov functions. It is shown that for $0<\alpha<2\sqrt{\frac{q}{l}}$, …

The solution of a boundary value problem by the method of$R$-functions

The work is devoted to a topical issue related to the practical calculation of electrostatic fields. One of the authors of this article previously introduced functions that allow for the exact satisfaction of boundary conditions for domains of practically arbitrary shape. These functions, while bein…

Conditions for the existence of recurrent trajectories in dynamic systems with a cylindrical phase space

The behavior of solutions to differential equations containing angular coordinates is investigated. Dynamical systems defined by such equations are often referred to as phase systems. In the first part of the work, a third-order equation is studied. Using a specially constructed Lyapunov function, a…

Solvability of an integro-differential equation

Analysis of a certain class of systems of differential-difference equations by the method of separation of motions

An approximate method for analyzing systems of equations describing controllers with digital computers is presented. The method consists of the artificial introduction of a “small” parameter for a subset of derivatives or differences, which allows for reducing the order of the equations under consid…

A periodic boundary value problem for the differential equation $y^{(n)}+f(t,y,y',\dots,y^{(n-1)})=0$

The paper considers the conditions for the existence and uniqueness of the solution to the specified problem, while simultaneously providing estimates for both the solution and its derivatives. The main results are Theorems $1$, $2$, $3$, and $4$. The periodic boundary value problem for third- and f…

Oscillations of a pendulum with relay control

The differential equation $$\ddot{x}+a\dot{x}+f(x)=-u_0\operatorname{sign}(\dot{x}-\varphi(x)),$$ is considered, where $a>0$; $u_0>0$; $f(x)$ and $\varphi(x)$ are periodic and everywhere continuously differentiable functions that vanish at $x=0$ and $x=+\pi$. This equation describes, in particular, …

On the theory of singular integral equations

Solvability of certain boundary value problems for elliptic systems with a degeneration of the order on the boundary

On averaging in systems of integro-differential equations

A system of nonlinear integro-differential equations of the form \begin{equation} \frac{dx}{dt}=\varepsilon f(t,x,\int_0^t\varphi(t,s,x(s))\,ds),\tag{1} \label{1} \end{equation} is considered, where $\varepsilon>0$ is a small parameter. The system \eqref{1} is associated with a system of averaged eq…

The theory of characteristic vectors and its application to the study of the asymptotic behavior of solutions of differential systems. II

The study of characteristic vectors, initiated in the author's previous article, is continued. The concept of a superior vector is introduced, which is used to investigate the distribution of characteristic vectors of linear differential systems under small perturbations. This leads to stability cri…

Global solution of the Tricomi problem for a class of nonlinear differential equations of mixed type

On a boundary value problem for an equation of mixed type with two lines of parabolic degeneration

Inverse problems of potential theory

Expansion of an arbitrary function in an integral with respect to the squares of Legendre functions with complex index

On a class of solutions of the sixth Painlevé equation

Solutions of long wave type for quasilinear elliptic equations in an unbounded strip

An elliptic system of partial differential equations with a singular point at the origin

Controllability in a Hilbert space

Complex eigenvalues of a non-selfadjoint operator

Incorrect problems with a closed non-invertible operator

The first boundary value problem for quasilinear parabolic systems with retarded argument

Differential equations and inequalities with discontinuous right hand sides. II

Generalized Emden–Fowler equation

Asymptotic stability of the zero solution of a non-autonomous system of two differential equations. I

A problem of N. P. Erugin on integrability by quadratures of a system of ordinary differential equations

Integration in finite form of certain systems of ordinary second order differential equations

The construction of a family of integral lines situated in a prescribed region

Families of extremals

Asymptotic behavior of integrals of quasiperiodic functions

A letter to the editors: Corrections