Vol 64, No 10 (2024)

The Cauchy problem for the regular transfer equation is considered. For this task, using the Richardson technique, a difference scheme of an increased order of accuracy is constructed on three nested grids, converging in a uniform norm with a third order of convergence rate.

One of the possible formulations of the inverse problems of gravimetry and magnetometry is considered, which consists in finding at a given depth hypothetical point sources of corresponding potential fields for these fields measured on the Earth’s surface. The uniqueness of the solution of such inverse problems is established. For the numerical solution of their discretized variants, a new algorithm is used based on improving the condition number of the problem matrix using the minimum pseudo-inverse matrix method (MPM algorithm). The algorithm is tested on model problems of gravity and magnetic exploration with their separate solution. A variant of the MPM algorithm for the joint solution of these inverse problems is also proposed and tested. In conclusion, the algorithm is used for separate and joint processing of some well-known gravity and magnetic exploration data for the Kursk magnetic anomaly.

A stepwise optimal control problem is considered, described by a set of Volterra type difference and integrodifferential equations and a Boltz type functional. Previously, similar problems were investigated for the case of differential as well as ordinary difference equations. Assuming the openness of the control areas, using a modified version of the increment method, the first and second variations of the quality functional are calculated. With the help of these variations, an analogue of the Euler equation and a number of constructively verifiable necessary conditions for second-order optimality are proved.

An algorithm is proposed that is sufficiently convenient for numerical implementation to construct a differentiable control function that guarantees the transfer of a wide class of nonlinear stationary systems of ordinary differential equations from the initial state to the origin of coordinates over an unfixed period of time. A constructively sufficient Kalman type condition has been found to guarantee the specified transfer. The effectiveness of the algorithm is obvious in solving a specific practical problem and its numerical modeling.

The initial boundary value problem for the Sobolev type equation of the theory of ion-acoustic waves in plasma is considered. This problem comes to an equivalent abstract integral equation. The local solvability of this equation is proved by the contraction mapping principle. Next, a “bootstrap” method is used to increase the smoothness of the solution. Finally, the test function method with a certain sufficient condition is used to obtain a finite time blow-up result and an upper bound on the blow-up time is found.

A nonlinear partial differential evolutionary equation is considered, which is obtained as a natural generalization of the well-known Cahn–Hilliard–Oono equation from a physical point of view. The terms responsible for accounting for convection and dissipation have been added to the generalized version. A new version of the equation is considered together with homogeneous Neumann boundary conditions. For such a boundary value problem, local bifurcations of codimension 1 and 2 are studied. In both cases, questions about the existence, stability, and asymptotic representation of spatially inhomogeneous equilibrium states, as well as invariant manifolds formed by such solutions to the boundary value problem, are analyzed. To substantiate the results, the methods of the modern theory of infinite-dimensional dynamical systems, including the method of integral manifolds, the apparatus of the theory of Poincare normal forms, are used. The differences between the results of the analysis of bifurcations in the Neumann boundary value problem are indicated with conclusions in the analysis of the periodic boundary value problem studied by the authors of the article in previous publications.

A comparative analysis of the accuracy of the CABARET (second-order) scheme with the WENO5 and A-WENO (fifth-order in space and fourth-order in time) schemes is carried out when calculating various Riemann problems for a non-convex system of conservation laws of a two-phase filtration model with an active impurity. It is shown that when calculating these problems, the CABARET scheme has significantly higher accuracy compared to the WENO schemes, especially in those areas of precise solution where centered rarefaction waves are adjacent to shock waves.