Vol 211 (2022)
Статьи
Boundary-value problem for an integro-differential equation of mixed type
Abstract
For a two-point boundary-value problem for a system of integro-differential equations of mixed type, we obtain conditions for unique solvability in terms of the solvability of the Cauchy problem and a hybrid system.
3-13
On the solvability of some boundary-value problems for the fractional analog of the nonlocal
Abstract
In this paper, we examine methods for solving the Dirichlet boundary-value problem and the periodic boundary-value problem for one class of nonlocal second-order partial differential equations with involutive argument mappings. The concept of a nonlocal analog of the Laplace equation is introduced. A method for constructing eigenfunctions and eigenvalues of the spectral problem based on separation of variables is proposed. The completeness of the system of eigenfunctions is examined. The concept of a fractional analog of the nonlocal Laplace equation is introduced. For this equation, boundary-value problems with the Dirichlet and periodic conditions are considered. The well-posedness of these problems is verified and the existence and uniqueness of the solution of boundary-value problems are proved.
14-28
Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. II. Potential force fields
Abstract
In many problems of dynamics, systems arise whose position spaces are four-dimensional manifolds. Naturally, the phase spaces of such systems are the tangent bundles of the corresponding manifolds. Dynamical systems considered have variable dissipation, and the complete list of first integrals consists of transcendental functions expressed in terms of finite combinations of elementary functions. In this paper, we prove the integrability of more general classes of homogeneous dynamical systems with variable dissipation on tangent bundles of four-dimensional manifolds. The first part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. I. Equations of geodesic lines// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory. — 2022. — V. 210. — P. 77–95.
29-40
Systems with dissipation with a finite number of degrees of freedom: analysis and integrability. I. Primordial problem from dynamics of a multidimensional rigid body in a nonconservative field of forces
Abstract
This paper is the first part of a survey on the integrability of systems with a large number n of degrees of freedom. The review consists of three parts. In this first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. In the second and third parts, which will be published in the next issues, we consider more general dynamical systems on the tangent bundles to the n-dimensional sphere and other smooth manifolds of a sufficiently wide class. Theorems on sufficient conditions for the integrability of the considered dynamical systems in the class of transcendental functions are proved.
41-74
Structure of the diagnostic space in problems of differential diagnostics
Abstract
In this paper, we discuss the generalized concepts of a malfunction and a neighborhood of a reference malfunction. We introduce the concept of a generalized diagnostic space and examine its mathematical structure, which formalizes the continuity of processes in the diagnostic space. We show that in the diagnostic space, reference malfunctions and the corresponding differential equations are nondegenerate. The generalized problem of differential (topological) diagnostics is considered.
75-82
On one integro-differential equation with fractional Hilfer operator and nonlinear maximums
Abstract
In this paper, we discuss the unique solvability of the initial-value problem for a nonlinear fractional integro-differential equation of the Hilfer type with a degenerate kernel and nonlinear maximums. USing a simple integral transformation based on the Dirichlet formula, we reduce the initial-value problem to a nonlinear, fractional integral equation of the Volterra type with nonlinear maximums. The theorem of existence and uniqueness of a solution of the initial-value problem considered is proved. The stability of solutions with respect to the parameter and the initial data is also proved. Illustrative examples are given.
83-95
On one loaded mixed-type integro-differential equation with fractional Gerasimov–Caputo operators
Abstract
In this paper, we examine the unique solvability of a boundary-value problem for a loaded mixed-type integro-differential equation with fractional Gerasimov–Caputo operators, spectral parameters, and small coefficients of mixed derivatives. The solution of the problem is obtained in the form of a Fourier series. The unique solvability of the problem for regular values of the spectral parameters is proved. The continuous dependence of the solution of the boundary-value problem on small parameters and on given functions is studied for regular values of the spectral parameters.
96-113
Boussinesq integro-differential equation with integral conditions and a small coefficient of mixed derivatives
Abstract
In this paper, we prove the unique solvability of a nonlocal boundary-value problem for a high-order, three-dimensional, linear Boussinesq integro-differential equation with a degenerate kernel and general integral conditions and construct a solution in the form of a Fourier series. The absolute and uniform convergence of the resulting series and the possibility of term-by-term differentiation of the solution with respect to all variables are established. A criterion for the unique solvability of the boundary-value problem in the case of regular values of the parameter is obtained. For irregular values of the parameter, an infinite set of solutions is constructed in the form of a Fourier series.
114-130