Vol 29, No 2 (2025)

 The Khalouta integral transform is a powerful method for solving various types of equations, including integro-differential equations and integral equations. It can also be applied to initial and boundary value problems associated with ordinary differential equations and partial differential equations with constant coefficients. The main objective of this paper is to derive solutions to systems of linear Caputo fractional Volterra integro-differential equations using the Khalouta integral transform.
To solve such systems using this technique, it is essential to establish and define several key properties of the Khalouta integral transform, which are crucial for deriving the transformation of the Caputo fractional derivative appearing in the systems. Several numerical examples are presented and solved by using the Khalouta integral transform method to demonstrate the applicability of the proposed approach. The results obtained from these numerical examples confirm that the proposed method is highly efficient and provides exact solutions for systems of linear fractional Volterra integro-differential equations in a straightforward manner.

Currently, local boundary value problems for hyperbolic-type differential equations have been studied in considerable details. However, mathematical modeling of various real-world processes leads to nonlocal boundary value problems for nonlinear hyperbolic differential equations, which remain insufficiently investigated. This paper is devoted to a general integral boundary value problem in a characteristic rectangle for hyperbolic equations. Under natural conditions on the input data, we construct the Green’s function and establish uniqueness criteria for the solution. The proofs of the main results demonstrate the essential nature of the imposed conditions: their violation makes it impossible to construct the Green’s function and leads to the loss of required solvability properties. For a special case, by using Banach’s contraction mapping principle, we obtain sufficient conditions for the existence and uniqueness of the boundary value problem solution. A specific example is provided to illustrate the obtained results.

In the present paper, cubic approximations of energy forms for the potentials of force and couple stresses in hemitropic micropolar elastic solids are proposed and discussed. H/E/A-representations for these potentials were introduced in earlier studies. However, the A-form allows us to obtain a cubic approximation for the stress potential in the form of a polynomial combination of invariants, comprising integer powers of individual and joint base rational algebraic invariants and pseudoinvariants. Some of these pseudoinvariants have “pseudo-tensor pre-images” that are sensitive to mirror reflections and inversions of three-dimensional space.
Within the framework of this study, a complete irreducible set of individual and joint linear, quadratic, and cubic integer rational algebraic invariants is obtained for a set consisting of the symmetric and antisymmetric parts of the asymmetric strain tensor and the wryness tensor. A cubic energy form for a hemitropic micropolar solid is determined, and a complete set of 37 constitutive moduli is specified. Additionally, constitutive equations for force and couple stresses in arbitrary curvilinear coordinates are derived, including quadratic corrections.

One of the key challenges in linear regression analysis is ensuring robust parameter estimation under stochastic data heterogeneity. In such cases, classical least squares estimates lose their stability. This problem becomes particularly acute with error distributions having heavier tails than normal distribution. Among various approaches to enhance regression robustness, replacing quadratic loss functions with convex-concave ones has been considered, though direct application leads to multimodal objective functions, significantly complicating the optimization problem.

This study aims to analyze properties of variationally-weighted quadratic and absolute approximations for non-convex loss functions. We propose an approach based on replacing the original non-convex regression problem with iterative application of weighted least squares and least absolute deviations methods, effectively implementing variationally-weighted approximations for non-convex loss functions. Each iteration of the weighted least absolute deviations method employed descent algorithms along nodal lines.

Through Monte Carlo simulations with various loss functions, we demonstrate that the weighted least absolute deviations method outperforms least squares in computational efficiency while maintaining comparable estimation accuracy. When multiple regression assumptions are violated simultaneously, either the weighted least absolute deviations method or the generalized least absolute deviations method (implemented as a generalized descent algorithm) proves preferable for achieving acceptable accuracy. We provide computational complexity estimates and execution time analyses depending on sample size and number of regression parameters.

This study is devoted to the development of Parker instability in the short-wavelength range of large-scale magnetic field oscillations (wavenumber $m>20$) within the upper layers of the solar convective zone. The strongly nonlinear ascent of the magnetic arc's apex forms a needle-like structure that penetrates the solar atmosphere at hypersonic velocities. During the magnetic field's rise, photospheric and chromospheric layers experience an abrupt vertical impact, generating a train of circular diverging shock waves propagating along the solar surface. This phenomenon, reliably detected by modern observational instruments, is called as a ''sunquake''. The onset of diverging shock wave generation serves as a precursor to flare activity within the active region. The paper provides numerical estimates for the spatial and temporal resolution requirements of observational instrumentation needed to study hypersonic magnetic flux emergence in the solar chromosphere.

The paper deals with a procedure for constructing solutions to the problem of stabilizing periodic perturbations of equilibrium states in the onedimensional Broadwell model. The solution procedure employs the Fourier method to solve the system of equations for the Fourier coefficients of the variables. In the Fourier transform space, the system reduces to a projection onto a single variable, enabling expression of the remaining Fourier coefficients $u_{k,l}$, $v_{k,l}$, $w_{k,l}$ through $z_{k,l}$ by using state equations.
The linearization of the $z$-projection plays a crucial role in studying the stabilization rate, representing in this case an integro-differential operator described in terms of the Paley–Wiener theorem. The discrepancy between the right and left sides of the one-dimensional system creates obstacles in the Fourier method when constructing annihilators of secular terms for the corresponding projection. These obstacles prevent obtaining solutions for arbitrary initial data describing periodic perturbations of the equilibrium position. It is established that the arising obstacles are identical for different projections.

The article presents a stochastic model for forecasting dynamics of gross regional product (GRP), developed using statistical data from Samara Region for the period 1998–2023. The model enables assessment of investment impact on regional economic development. To describe GRP dynamics, we propose a stochastic differential balance equation that relates GRP indicators to regional production resource (RPR) volumes. Within the study, we have: (1) estimated RPR volumes, (2) constructed theoretical trajectories of GRP and RPR dynamics, and (3) derived mathematical expectation curves for their growth. Numerical analysis demonstrates the model’s high consistency with empirical data.