Vol 69, No 3 (2023)


On global weak solutions of the Vlasov-Poisson equations with external magnetic field

Belyaeva Y.O., Skubachevskii A.L.


We consider the first mixed problem for the system of Vlasov-Poisson equations with a given external magnetic field in a bounded domain. This problem describes the kinetics of high-temperature plasma in controlled thermonuclear fusion plants and is considered with respect to unknown functions: electric field potential, distribution functions of positively charged ions and electrons. Additionally, we assumed that the distribution functions of charged particles satisfy the condition of mirror reflection from the boundary of the domain under consideration. We prove the existence of global weak solutions of such a problem.

Contemporary Mathematics. Fundamental Directions. 2023;69(3):383-398
pages 383-398 views

Smoothness of generalized solutions of a boundary-value problem for a second-order differential-difference equation with mixed boundary conditions

Ivanov N.O.


We consider a boundary-value problem with mixed boundary conditions for a second-order differential-difference equation on a finite interval (0 ,d). We prove existence of a generalized solution of the problem and study the conditions on the right-hand side of the differential-difference equation ensuring the smoothness of the generalized solution over the entire interval.

Contemporary Mathematics. Fundamental Directions. 2023;69(3):399-417
pages 399-417 views

Mathematical model of matter transfer in a helical magnetic field using boundary conditions at infinity

Lazareva G.G., Oksogoeva I.P., Sudnikov A.V.


The paper presents a mathematical model of plasma transfer in an open magnetic trap using the condition of zero plasma concentration at infinity. New experimental data obtained at the SMOLA trap at the Budker Institute of Nuclear Physics SB RAS were used. Plasma confinement in the plant is carried out by transmitting a pulse from a magnetic field with helical symmetry to a rotating plasma. The mathematical model is based on a stationary plasma transfer equation in an axially symmetric formulation. The stationary equation of the transfer of matter contains second spatial derivatives. The optimal template for the approximation of the mixed derivative based on the test problem is selected. The numerical implementation of the model by the establishment method and the method of successive over-relaxation is compared.

Contemporary Mathematics. Fundamental Directions. 2023;69(3):418-429
pages 418-429 views

Analytical solution of the space-time fractional reaction-diffusion equation with variable coefficients

Mahmoud E.I.


In this paper, we solve the problem of an inhomogeneous one-dimensional fractional differential reaction-diffusion equation with variable coefficients (1.1)-(1.2) by the method of separation of variables (the Fourier method). The Caputo derivative and the Riemann-Liouville derivative are considered in the time and space directions, respectively. We prove that the obtained solution of the boundary-value problem satisfies the given boundary conditions. We discuss the convergence of the series defining the proposed solution.

Contemporary Mathematics. Fundamental Directions. 2023;69(3):430-444
pages 430-444 views

Nonlinear differential-difference equations of elliptic and parabolic type and their applications to nonlocal problems

Solonukha O.V.


In this survey, we study boundary-value problems for nonlinear differential-difference equations of elliptic and parabolic types, as well as related nonlinear equations with nonlocal boundary conditions. The main feature of the equations under consideration is that the difference operator is located in the principal part of the nonlinear operator containing higher-order derivatives.

Contemporary Mathematics. Fundamental Directions. 2023;69(3):445-563
pages 445-563 views

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