Most papers are machine-translated to English; some are originally in English.
Showing 50 of 634 papers
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 $…
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…
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…
A generalized Gylden problem is considered, i.e., the problem of the motion of a point of variable mass in a nonstationary central force field whose reduced force law (the ratio of force to mass) is expressed by an arbitrary function $f_{\text{pr}}(t,r)$ of time $t$ and the distance $r$ of the point…
The paper considers a necessary and sufficient condition for the validity of a theorem analogous to S. A. Chaplygin's theorem on differential inequalities for an elliptic difference equation. Effective estimates of the domain of applicability of the mentioned theorem to difference and differential e…
For the system \begin{equation} x(t)=\int_0^tK(t,s,x(s))\,ds+f(t)\tag{1}, \label{1} \end{equation}, where the vector function $K(t,s,x)$ has the form $$K(t,s,x)=K_1(t,s,x)-K_2(t,s,x),$$ and $K_j(t,s,x)$ ($j=1,2$) are continuous and non-decreasing with respect to $x$ in the domain $0<s\le t\le b$, $|…
For the third Painlevé equation, the nature of possible singular points of its solutions is investigated, the question regarding the number and residues of movable poles of the solutions is resolved, and necessary and sufficient conditions for the existence of rational solutions are specified. Bibli…
A quasilinear autonomous system with many degrees of freedom and delays is considered. Under the assumption that the generating system possesses a multi-frequency periodic regime, the solution of the system is constructed using the asymptotic method of N. M. Krylov and N. N. Bogolyubov. The possibil…
In the peer-reviewed work, the following one-dimensional mixed problem is investigated: \begin{gather}\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}-\alpha\frac{\partial^3u}{\partial t\partial x^2}=\lambda F[t,x,U,U_x,U_t,U_{xx},U_{tx}],\tag{1}\label{1}\U(t,0)=U(t,\pi)=0,\quad U(0…
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…
The Galerkin method is used to solve the mixed problem $$\begin{aligned} \frac{\partial^2u}{\partial t^2}&=\frac{\partial}{\partial x}\biggl(p(x)\frac{\partial u}{\partial x}\biggr)+f(t,x,u,u_t,u_x), \ u(0,x)&=\varphi_0(x),\quad u_t(0,x)=\varphi_1(x), \ u(t,0)&=u(t,\pi)=0, \end{aligned} \tag{1}$$ wh…
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 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…
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…
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…
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…
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, …
The paper investigates a system of three equations of strongly elliptic type, supplemented by a boundary condition for one variable and a periodicity condition for the second. This system corresponds to the problem of determining the displacements of a momentless lattice cylindrical shell from an in…
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 Lauricella method is applied to reduce the boundary value problem of the equilibrium of a part of a cylindrical shell, bounded by a simply connected smooth contour, to integral equations. The components of the displacement vector and the rotation angles of the shell normal are specified on the c…
The paper presents a comparison of the errors of Störmer's method in the case of a single differential equation and Adams' method in the case of the corresponding system of first-order differential equations. The arguments are conducted within the framework of linearized error theory. The considerat…
