Most papers are machine-translated to English; some are originally in English.
Showing 50 of 634 papers
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…
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…
A spatial contact problem regarding the pressure of a strip punch on an elastic foundation with an elastic modulus $E=E_m z^m$ ($0
It is proved that in linear boundary value problems, the property of a parameter being an eigenvalue or a regular value is preserved under small changes in the delay. Bibliography: 6 items.
Two-dimensional autonomous systems of the form \begin{equation} \dot{x}=P(x,y),\quad\dot{y}=Q(x,y)\tag{1} \label{1} \end{equation} are considered, where the right-hand sides have isolated singular points $O_j$ of the pole type. The concept of a "quasi-residue" $J_j$ of a singular point $O_j$ is intr…
For a certain class of autonomous differential equations of the $n$-th order, a three-parameter family of solutions is constructed, the derivative of which vanishes at infinity in the form of a Dirichlet series that converges uniformly and absolutely on the real semiaxis, starting from a certain neg…
In this work, necessary and sufficient conditions are found for the existence of closed algebraic curves of the second order among the trajectories of the equation \begin{equation}yy'=Q_4(x,y)\tag{1}, \label{1} \end{equation} where $Q_4(x,y)$ is a polynomial of the fourth degree. At the same time, i…
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…
The article is a continuation of the author's work [1] and is devoted to the derivation of two types of asymptotic expansions for Macdonald integrals $n$ of order $M_n(x,y)$. Two expansions of the first type are asymptotic as $|y|\sqrt{1+x^2}\to\infty$. The first expansion is an expansion in terms o…
The problems of choosing controls $u_1(t)$ and $u_2(t)$ that bring linear systems $$\frac{dx}{dt}=Ax+bu_1,\quad\frac{dx}{dt}=Ax+bu_2$$ to the equilibrium state $x=0$ in a given time $a\le t\le T$ are considered, subject to the minimization of the following control intensity estimates: $$J(u_1^0)=\ma…
A number of problems in mathematical physics lead to the consideration of divergent integrals. Cauchy and Hadamard developed an algorithm that allows assigning a well-defined finite part to certain divergent integrals. This article proposes a method for introducing the finite part of singular functi…
The paper specifies the integrability criteria for three nonlinear second-order differential equations. Bibliography: 8 items.
The paper investigates a singular Cauchy problem for the generalized wave equation $$ z_{xx}=z_{ss}+\frac{a}{s}z_s+b^2z,\quad z(x,0)=\tau(x),\quad \tau_s(x,0)=0.\tag{1}$$ Using the integral representation of its solution, the author constructs basis series expansions of two types for $z(x,\lambda s;…
A number of theorems are proved on the topological structure of the phase portrait of the system of ordinary differential equations \begin{equation} \begin{aligned}\dot{x}&=y\\dot{y}&=-g(x,y)\end{aligned}\biggr}\tag{1}, \end{equation} which the author calls relaxation if: (I) the function $g(x,y)$ i…
A system of differential equations \begin{equation}\frac{dx}{dz}=\sum_{j=0}^pa_j(z)y^{p-j},\quad\frac{dy}{dz}=\sum_{j=0}^kb_j(z)x^{k-j}\tag{1}, \end{equation} is considered, where $a_j$, $b_j$ are holomorphic functions and $k\ge p\ge2$. Necessary and sufficient conditions are provided for the movabl…
This paper considers the problem of free transverse vibrations of a finite visco-elastic rod where one end is fixed and the other is free. Mathematically, the problem is formulated as seeking a solution to the equation: \begin{equation} \frac{\partial^2u}{\partial t^2}+a\frac{\partial^4u}{\partial x…
We consider the equation \begin{equation} \frac{\partial u}{\partial t} = \int_0^1 k\biggl(x,y,u(y,t),\frac{\partial u(y,t)}{\partial t}t\biggr)\,dy \tag{1} \label{1} \end{equation} where $$k(x,y,u_1,u_2t)=\sum_{l+k+j\ge1}^\infty k_{l,k,j}(x,y)u_1^l u_2^k t^j,$$ subject to the initial condition \beg…
The article considers a system of fourth-order differential equations describing, in particular, the motion of coupled Froude pendulums of equal masses suspended on a shaft rotating at a speed $\Omega$. The paper provides upper and lower bounds for the values of the speed $\Omega$ in the case of the…
Sufficient conditions are established for the dissipativity of systems \begin{gather}\dot{x}=y,\quad\dot{y}=-yf(x,y,t)-g(x)+e(t),\tag{1}\\dot{x}=y-f(x,y,t),\quad\dot{y}=-g(x)\tag{2}\end{gather} in the case where the function $f$ satisfies substantially different constraints for $x>x_0>0$ and for $x<…
A topological classification of the singular points of the system $$dx^i=\sum_{\mu=1}^{n-1}\sum_{j=1}^na^i_{j\mu}x^j\,dt^{\mu}\quad(i=1,\dots,n),$$ is provided, assuming that the conditions of complete integrability are satisfied, and the $n\times n$-matrices $(a^i_{j\mu})$, $\mu=1,\dots,n-1$ have n…
We consider the Dirichlet and Neumann boundary value problems for the equations \begin{align}L_u&\equiv\Delta u=v(x,y)\tag{1}\label{1},\L_u&\equiv\Delta u-\lambda u=v(x,y)\tag{2}\label{2}\end{align} in the square $R[0 \le x, y \le \pi]$. Approximate solutions are sought in the form \begin{equation} …
The problem of the minimax time $T$ until encounter with respect to a subset of selected coordinates is considered for two linear controllable objects described by identical equations: \begin{gather}\dot{y}=Ay+Bu\quad\dot{z}=Az+Bv,\notag\y_{i_k}(\tau+T^0)=z_{i_k}(\tau+T^0),\quad T^0=\min_u\max_vT_{u…
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 …
