Bulletin KRASEC. Physical and Mathematical Sciences

In 1978, the journal Differential Equations published an article by A. M. Nakhushev, which provided a technique for correctly formulating a boundary value problem for a class of second-order parabolic-hyperbolic equations in an arbitrary bounded domain  $\Omega$ with a smooth or piecewise smooth boundary $\Sigma$. The boundary value problem investigated in the above-mentioned work is currently called the first boundary value problem for a second-order mixed parabolic-hyperbolic equation. Within the framework of this work, the first boundary value problem for a third-order model parabolic-hyperbolic equation in a mixed domain is formulated and investigated in the sense in which it was formulated and investigated by A. M. Nakhushev for second-order equations. In one part of the mixed domain, the equation under consideration coincides with a degenerate hyperbolic equation of the first kind of the second order, and in the other part it is an inhomogeneous third-order equation with multiple characteristics of parabolic type. For various values of the parameter $\lambda$ included in the equation under consideration, theorems of existence and uniqueness of a regular solution of the problem under study are proved. To prove the uniqueness theorem, the method of energy integrals is used in conjunction with the method of A.M. Nakhushev. To prove the existence theorem, the method of integral equations is used. In terms of the Mittag-Leffler function, the solution to the problem is found and written out in explicit form.

The paper considers a linear ordinary differential equation with a fractional derivative that contains an involution operator in the subordinate term. The equation under consideration is a model equation and belongs to the class of differential equations that need to be investigated due to the study of boundary value problems for fractional differential equations containing a composition of left- and right-hand fractional differentiation operators. The latter arise when modeling various physical and geophysical processes and, in particular, are of great importance when describing dissipative oscillatory systems. For the equation under consideration, the initial value problem in a unit interval is investigated. The main result of the paper is a theorem of existence and uniqueness of a solution to the problem under consideration. Sufficient conditions that ensure unique solvability of the problem under consideration are formulated in terms of constraints on the coefficient and the right-hand side of the equation under consideration. A fundamental solution is constructed, its various representations are obtained, and its main properties are studied. An explicit representation of the solution to the problem under consideration is found in terms of the fundamental solution. 

The paper is devoted to a class of stochastic two-mode hereditary models of the cosmic dynamo. The models include two magnetic field generators — large-scale and turbulent ($\alpha$-effect). The influence of the magnetic field on the motion of the medium is presented through the suppression of the $\alpha$-effect by a functional of the field components, which introduces memory (hereditary) into the model. The model describes the dynamics of only large-scale components, but takes into account the possible impact of small-scale modes using a stochastic term. This term models the influence of possible spontaneous synchronization of small-scale modes. The paper also presents a numerical scheme for solving the integro-differential equations of the model. The numerical scheme consists of two parts, for the differential part the Adams «predictor-corrector» method of the fourth order is used, and for the integral part the Simpson method.The main result of the work is a generalized model of a dynamo system, with an additive addition of a random correction to the $\alpha$-generator. Taking into account such a correction significantly diversifies the dynamic modes in the model.

Radon is an inert radioactive gas, and studies of its variations in relation to seismicity are considered promising for the development of earthquake prognosis methods. A network of observation points has been deployed on the Kamchatka peninsula, where radon volumetric activity (RVA) is monitored using accumulation chambers with gas-discharge counters. Analysis of RVA data within the framework of radon monitoring is one of the methods of searching for precursors of seismic events. This is due to the fact that changes in the stress-strain state of the geo-environment, through which the gas flows, affect the RVA. The change in radon transport intensity due to changes in the stress-strain state of the geosphere is described by a fractional differentiation operator of constant real order $\alpha$, which is related to the permeability of the geosphere. It is known that the RVA in the storage tank with sensors is also affected by the air exchange rate ${\lambda }_0$, the effect of which should be taken into account in the study of the radon transport process. The aim of the research is to study the accumulation of radon in the chamber, which consists in the identification of the values of the parameters ${\lambda }_0$ and $\alpha$ by solving the corresponding inverse problem. As a result of the research it was shown that for the hereditary $\alpha$-model of radon transport by the Levenberg-Mackwardt method with the involvement of experimental data of RVA it is possible to determine the optimal values of its parameters ${\lambda }_0$ and $\alpha$ . The obtained model curves agree well with the RVA data obtained within the framework of the well-known classical model of radon transport in an accumulation chamber.