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We consider integration Clein–Gordon and Dirac equations in Bianchi IX cosmology model. Using the noncommutative integration method we found the new exact solutions for Taub universe.

Noncommutative integration method for Bianchi IX model is based on the use of the special infinite-dimensional holomorphic representation of the rotation group, which is based on the nondegenerate orbit adjoint representation, and complex polarization of degenerate covector. The matrix elements of the representation of form a complete and orthogonal set and allow you to use the generalized Fourier transform. Casimir operator for rotation group under this transformation becomes constant. And the symmetry operators generated by the Killing vector fields in the linear differential operators of the first order from one dependent variable. Thus, the relativistic wave equation on the rotation group allow non-commutative reduction to ordinary differential equations. In contrast to the well-known method of separation of variables, noncommutative integration method takes into account the non-Abelian algebra of symmetry operators and provides solutions that carry information about the non-commutative symmetry of the task. Such solutions can be useful for measuring the vacuum quantum effects and the calculation of the Green’s functions by the splitting-point method.

The work for the Taub model compared the solutions obtained with the known, which are obtained by separation of variables. It is shown that the non-commutative solutions are expressed in terms of elementary functions, while the known solutions are defined by the Wigner function. And commutative reduced by the Klein–Gordon equation for Taub model coincides with the equation, reduced by separation of variables. A commutative reduced by the Dirac equation is equivalent to the reduced equation obtained by separation of variables.

Modification of the lattice Boltzmann method for computation of viscous incompressible fluid flows is proposed. The method is based on the splitting of differential operator in Navier–Stokes equation and on the idea of instantaneous Maxwellisation of distribution function. The method is based on explicit schemes and didn’t have any problems with parallelization of computations. The stability of the method is demonstrated using von Neumann method in a wide range of input parameter values. The efficiency of the method proposed is demonstrated on the solution of the problem of 2D lid-driven cavity flow.

Modification of the lattice Boltzmann method for computation of viscous Newtonian fluid flows is considered. Modified method is based on the splitting of differential operator in Navier–Stokes equation and on the idea of instantaneous Maxwellisation of distribution function. The problems for the system of lattice kinetic equations and for the system of linear diffusion equations are solved while one time step is realized. The efficiency of the method proposed in comparison with the ordinary lattice Boltzmann method is demonstrated on the solution of the problem of planar flow in cavern in wide range of Reynolds number and various grid resolution.

Modeling of gas-particle suspensions with large pressure and density gradients is of practical interest in the study of volcanic phenomena, explosions at different altitudes, as well as in technogenic problems related to the operation of space technology and the formation of space debris. This work presents numerical and analytical investigations of the expansion of gas suspensions with a high ratio (up to six orders of magnitude) of pressures and densities. For numerical modeling, a high-resolution hybrid large-particle method was employed. Under the conditions considered, the accuracy of the method was confirmed by comparison with asymptotically exact solutions. The study examined the wave and structural characteristics of concentrated gas suspension expansion depending on particle volume fraction, particle size, and initial pressure ratio. It was found that the polytropic index and sound speed in the gas suspension depend not only on temperature but also on pressure and particle concentration. With increasing pressure, both the polytropic index and sound speed rise, while with increasing particle volume fraction they decrease. In the case of an arbitrary discontinuity decay, an unusual effect is observed compared with “pure” gas dynamics: the relative velocity of the mixture in the uniform flow region decreases as the initial pressure increases. This is explained by the nonlinear dependence of the sound speed in a gas-dispersed mixture on pressure. With increasing particle size (Stokes number), the mixture flow splits into gaseous and dispersed components. At the initial moment, the contact discontinuity separating the mixture from the rarefied gas region splits into two contact boundaries: gaseous and dispersed. A practical conclusion is that when the particle size changes by two orders of magnitude, the gas-dynamic parameters of the mixture in the rarefaction wave region and up to the medium interface remain close to each other. During spatial expansion, the initial cylindrical shape of the dispersed medium successively transforms into a cross-section resembling a hexagon. At the next stage of expansion, the particles redistribute to form a bilateral conical structure. Eventually, a dispersed formation close to a spherical shape emerges.

The paper simulates a mixture of gases in a multi-stage micro-pump and evaluates its effectiveness at separating the components of the mixture. A device in the form of a long channel with a series of transverse plates is considered. A temperature difference between the sides of the plates induces a radiometric gas flow within the device, and the differences in masses of the gases lead to differences in flow velocities and to the separation of the mixture. Modeling is based on the numerical solution of the Boltzmann kinetic equation, for which a splitting scheme is used, i. e., the advection equation and the relaxation problem are solved separately in alternation. The calculation of the collision integral is performed using the conservative projection method. This method ensures the strict fulfillment of the laws of conservation of mass, momentum, and energy, as well as the important asymptotic property of the equality of the integral of the Maxwell function to zero. Explicit first-order and second-order TVD-schemes are used to solve the advection equation. The calculations were performed for a neon-argon mixture using a model of solid spheres with real molecular diameters and masses. Software has been developed to allow calculations on personal computers and cluster systems. The use of parallelization leads to faster computation and constant time per iteration for devices of different sizes, enabling the modeling of large particle systems. It was found that the value of mixture separation, i. e. the ratio of densities at the ends of the device linearly depends on the number of cascades in the device, which makes it possible to estimate separation for multicascade systems, computer modeling of which is impossible. Flows and distributions of gas inside the device during its operation were analyzed. It was demonstrated that devices of this kind with a sufficiently large number of plates are suitable for the separation of gas mixtures, given that they have no moving parts and are quite simple in manufacture and less subject to wear.

The difference scheme for numerical solution of a time-dependant system of two Schrödinger equations with the operator of a spin-orbit interaction for a two-component spinor wave function is offered on the basis of a split method for a time-dependant Schrödinger equations. The computer simulation of the external neutrons’ wave functions evolution with different values of the full moment projection upon internuclear axis and probabilities of their transfer are executed for head-on collisions of ¹⁸O and ⁵⁸Ni nuclei.

This paper addresses the problem of detecting small objects in visible-spectrum aerial imagery. High background variability and weak feature saliency make small object detection a non-trivial task. Under such conditions, classical computer vision algorithms based on hand-crafted descriptors exhibit low efficiency, prompting a shift towards deep neural network architectures, which demonstrate superior generalization capability and robustness to false positives. We selected a one-stage approach based on a neural network predictive model as the primary object detection method. Also, several neural network architectures belonging to this class were analyzed, outlining their advantages and disadvantages. As the baseline detector, we adopted YOLO version 11 and incorporated a multi-scale feature aggregation module (which combines information from neural network layers operating at different scales) and a dimension-aware selective integration module (which automatically determines the dimension (channel, height, or width) along which to process features and fuses them selectively). These modifications aim to both enhance computational efficiency, enabling deployment of neural network models onboard aerial vehicles for real-time image analysis, and improve small object detection accuracy. Given the complexity of image annotation, we used an open synthetic image database containing approximately 4 000 images for training and testing (with a 90/10 split, respectively). We extend the training set using various random augmentation techniques. To evaluate the performance of the resulting predictive models, we employed mean Average Precision across all classes, using both a fixed 50% intersectionover- union threshold and a varying threshold from 50% to 95%. Overfitting was monitored by analyzing loss curves during training process. The proposed modifications to the YOLO architecture reduced image processing time by a factor of two while maintaining detection accuracy.

A spatially distributed mathematical model is proposed for the evolution of five myocardial tissue states — healthy myocardium, inflammatory infiltrate, necrosis, replacement fibrosis, and interstitial fibrosis — in the form of a multicomponent parabolic reaction – diffusion system. At each spatial point the tissue state is described by a vector of Dirichlet concentration parameters, which allows one to encode both the expected tissue composition and the uncertainty of classification associated with transition zones of damage. For a truncated system that coincides with the original one inside an invariant region, local Lipschitz continuity and quasi-positivity of the extended reaction operator are established. Global existence of a weak solution, non-negativity, an explicit a priori upper bound, and a quantitative exponential lower bound are proved. These results imply invariance of the admissible region and hence existence of a global weak solution to the original model. Uniqueness is established in a strengthened class of solutions characterized by additional gradient regularity. For an auxiliary reversible transition matrix, a formal entropy balance is derived for smooth positive solutions; for the clinically motivated irreversible transition matrix, a modal spectral stability condition is formulated for the linearized problem. On the numerical side, a positivity condition is derived for the reaction substep of a splitting scheme relative to an adaptive upper trajectory bound, and numerical experiments are conducted for a reproducible calibrated computational variant. In the one-dimensional scenario, by day $60$ the replacement-fibrosis fraction at the lesion center increases from $0.091$ to $0.855$, while the total concentration parameter $\alpha_0$ rises from $11.0$ to $15.9$; in the two-dimensional scenario, on day $14$ the mean value of $\overline p_4^{}$ is $0.431$ in the lesion core versus $0.097$ in the healthy region. These results confirm the mathematical well-posedness of the proposed model and its applicability for the quantitative description of post-infarction scar formation and heterogeneous transition zones.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.