Vol 70, No 2 (2024): Functional spaces. Differential operators. Problems of mathematics education

We consider strongly elliptic differential-difference equations with mixed boundary conditions in a bounded domain. There are homogeneous Dirichlet conditions on a part of the boundary, and boundary conditions of the third kind on the other part of the boundary. We establish the connection between these problems and nonlocal mixed problems for strongly elliptic differential equations. We prove the uniqueness and the smoothness of their solutions.

In this paper, we present a method for constructing vector fields whose phase portraits have finite sets of prescribed special trajectories (limit cycles, simple and complex singular points, separatrices) and prescribed topological structures in limited domains of the phase plane. The problem of constructing such vector fields is a generalization of a number of well-known inverse problems of the qualitative theory of ordinary differential equations. The proposed method for solving it expands the possibilities of mathematical modeling of dynamic systems with prescribed properties in various fields of science and technology.

We consider a second-order quasilinear elliptic equation with an integrable right-hand side in the space  Rⁿ. Restrictions on the structure of the equation are formulated in terms of a generalized N -function. In the nonreflexive Muzilak-Orlicz-Sobolev spaces, the existence of a renormalized solution in the space  Rⁿ is proved.

Tsunami after the March 11, 2011, as well as the other recent events, have shown that destructive tsunami waves generated by earthquakes continue to pose a significant risk to coastal populations adjacent to subduction zones, where most of tsunami sources are located. In some places along these coasts, the tsunami run-up heights can reach 30 m or more, causing destruction and casualties. However, the wave heights maxima are distributed very nonuniformly along the coast with sharp local peaks in amplitude. Since for near-shore events the tsunami wave arrival time at the nearest coastal point after an earthquake is on the order of 20 minutes, a quick (within 1-2 minutes) correct assessment of the distribution of maximum wave heights along the coast will allow warning services take evacuation actions exactly where needed. Modern modelling tools allowing quickly calculate wave parameters with sufficient accuracy if the wave characteristics at the initial time instance are known. However, this requires calculations in spatial steps of several meters, which is time-consuming even when using supercomputers. In addition, in the case of a strong earthquake, power outages are possible, which does not guarantee that numerical modelling can be started immediately after the seismic event. The use of large, hundreds of meters resolution calculation grid does not allow estimate correctly the tsunami wave heights near the shore. Fine grids entail the growth of the duration of computing time. The resolution of this contradiction dictates the necessity to choose the optimal correlation between grid spacing (results precision) and calculation time. In this paper the dependence of the calculated tsunami wave parameters depending on the grid spacing is studied. Obtained results will be used for optimal selection of application zones of meshes with different spacing. Computational experiments were carried out on a personal computer (PC) using hardware acceleration - a specialized FPGA-based microchip (FPGA being Field Programmable Gates Array), used with the computer as a coprocessor. As a result, a sufficiently high performance of calculations is achieved. Calculation of wave parameters near the shore on the computational grid of 3000×2500 nodes takes less than 1 min. In addition, the proposed solution does not depend on possible power supply failures.