Vol 29, No 1 (2025)

This study investigates an inverse problem involving the determination of the kernel function in a multidimensional integrodifferential pseudo-parabolic equation of the third order. The study begins with an analysis of the direct problem, where we examine an initial-boundary value problem with homogeneous boundary conditions for a known kernel. Employing the Fourier method, we construct the solution as a series expansion in terms of eigenfunctions of the Laplace operator with Dirichlet boundary conditions. A crucial component of our analysis involves deriving a priori estimates for the series coefficients in terms of the kernel function norm, which play a fundamental role in our subsequent treatment of the inverse problem.
For the inverse problem, we introduce an overdetermination condition specifying the solution value at a fixed spatial point (pointwise measurement). This formulation leads to a Volterra-type integral equation of the second kind. By applying the Banach fixed-point principle within the framework of continuous functions equipped with an exponentially weighted norm, we establish the global existence and uniqueness of solutions to the inverse problem. Our results demonstrate the well-posedness of the problem under
consideration.

In this study, a three-dimensional gas-dynamic model of an ideal incompressible fluid is proposed, where the solution is sought in the form of a linear velocity field with inhomogeneous deformation. The problem is formulated in both Eulerian and Lagrangian variables. Exact solutions are obtained for a special linearity matrix, generalizing previously known solutions. The equations of world lines for these solutions are derived, the trajectories of fluid particle motion are constructed, and the evolution of the initial spherical particle volume is investigated. The equations of constant pressure surfaces are presented and their time dynamics is analyzed. Special attention is paid to the analysis of particle motion in an ideal incompressible fluid and to obtaining new, more general solutions.

The paper investigates the stability of circular vertical layered cylindrical shells completely filled with a quiescent compressible fluid subjected to hydrostatic and external static loads. The behavior of the elastic structure and the fluid medium is described within the framework of the classical shell theory and Euler equations. The linearized equations of motion of the shell and the corresponding geometrical and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation is transformed to a system of differential equations using the method of generalized differential quadrature. The solution of the formulated boundary value problem is reduced to the calculation of natural vibration frequency in terms of Godunov's orthogonal sweep method. For this purpose, a stepwise procedure is applied in combination with a subsequent refinement by the Muller method. The reliability of the obtained results is verified through a comparison with known numerical solutions. The dependence of the critical external pressure on the ply angle of simply supported, rigidly fixed and cantilevered two-layer and three-layer cylindrical shells is analyzed in detail. The influence of the combined static pressure on the optimal ply angle providing an increase of the stability boundary is evaluated.

We present a novel approach within linear elasticity theory for modeling the spatial distribution of enhanced crustal deformations during earthquake preparation. Our model utilizes the Lamé differential equation system, representing the seismic source as a concentrated force system acting at a point within an elastic half-space. The associated boundary value problem is solved analytically using Green’s functions. The framework computes anomalous pre-seismic deformations at each surface point and quantifies their occurrence frequency relative to background tidal deformation thresholds.
The method was validated using the Global Centroid-Moment-Tensor Catalog for the Kamchatka Peninsula seismic zone. Simulations of deformation patterns preceding earthquakes (1976–2020) reveal:

• Deformation anomalies predominantly align with the primary coastal fault system;
• Peak occurrence frequencies (0.6–0.8) correlate with densely populated regions;
• Distinct temporal variability, with high-activity phases (0.6–0.8) interspersed with low-activity intervals (0.1–0.2).

This approach provides a robust tool for investigating pre-seismic deformation patterns and identifying multidisciplinary precursor phenomena in active tectonic regions.

This study presents a novel theoretical model of steady-state ion transport through cation-exchange membrane systems. Unlike existing theoretical approaches, the proposed model relates modifications in the equilibrium constant not only to the electric potential gradient, but also to spatial charge distribution. Analysis of the Poisson equation confirms the significant dependence of ion dissociation kinetics on local space charge density within the membrane structure.
The developed mathematical model, incorporating this dependence, enables a more accurate description of diffusion-migration processes in cationexchange membranes. The obtained results provide a more precise description of ion behavior under steady-state transport conditions — a crucial factor for developing advanced membrane materials and technological processes. The proposed model can be applied in various technological fields employing ion-exchange membrane systems, including water treatment processes and energy converters.
A key advantage of the proposed model is its capability for comprehensive consideration of critical ion transport parameters: solution ionic strength, temperature conditions, and membrane structural-functional characteristics. This enables more accurate prediction of membrane system performance in actual technological processes.
In particular, application of this model in membrane water purification systems allows optimization of demineralization processes, thereby enhancing water treatment efficiency while reducing energy consumption in the technological cycle.
Thus, the developed model offers new opportunities for both fundamental research and practical optimization of mass transfer processes in ionexchange membrane systems.

Triply periodic surfaces (TPS) and their minimal analogs (TPMS) are currently widely used in various fields, including mechanics, biomechanics, aerodynamics, hydrodynamics, and radiophysics. In this context, the problem of establishing correlations between the topological and geometric properties of surfaces and their physical characteristics arises. To address this problem, it is necessary to introduce a measure of similarity between surfaces with different topological and geometric features. This work focuses on describing TPS and TPMS in terms of a specific metric space of descriptors. The problem is solved using the mathematical framework of image recognition theory. A descriptor is constructed based on a set of eigenvectors and eigenvalues of the Beltrami–Laplace operator and a joint Bayesian model. A metric based on a probabilistic measure of surface similarity is introduced in the descriptor space. The effectiveness of the method developed in this work has been tested on 51 surfaces of class P. The accuracy of predicting the surface type is 92.8 %. The developed machine learning model enables the determination of whether a given surface belongs to the class of P-surfaces.

This study presents an algorithm for analyzing spectral data through mathematical modeling, constructing prognostic models, and selecting optimal wavelength intervals for designing LED-based multisensor systems. The algorithm is implemented in Python and validated using experimental data from aqueous solutions of inorganic salts.
Key methodological aspects include:
– Application of multivariate calibration methods (PLS regression and multiple linear regression);
– Utilization of Shapley values to identify informative spectral wavelengths;
– Systematic enumeration to determine optimal wavelength intervals.
The developed model enables accurate prediction of two- and threecomponent systems in metal salt solutions using partial spectral data rather than full-spectrum analysis. Cross-validation demonstrates that:
– The model achieves comparable accuracy to full-spectrum approaches;
– The solution remains computationally efficient while maintaining predictive reliability.
The results confirm the model’s adequacy for quantitative spectral analysis, particularly in resource-constrained environments where partial spectral data acquisition is advantageous.