Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences

In the course of a series of studies spanning the fifty-year period from 1889 to 1939, all double hypergeometric series of the second order were systematically investigated. A significant contribution to the study of hypergeometric functions of two variables was made by Horn, who proposed their classification into two types: complete and confluent. Horn’s final list comprised fourteen complete (non-confluent) functions of two variables and twenty distinct confluent functions, which represent limiting cases of the complete ones. In 1985, Srivastava and Karlsson completed the classification of all possible second-order complete hypergeometric functions of three variables, while a similar systematic classification for their confluent counterparts remains incomplete. Thus, the theory of confluent hypergeometric functions of three variables has not yet been fully developed, and the study of functions of four variables represents an area for future research.
This paper investigates certain confluent hypergeometric functions of three and four variables, establishing their new properties and applying them to the solution of the Dirichlet problem for the three-dimensional Helmholtz equation with three singular coefficients. 
Fundamental solutions of the aforementioned Helmholtz equation are expressed in terms of a confluent hypergeometric function of four variables, while an explicit solution to the Dirichlet problem in the first octant is constructed using a function of three variables, which is derived as a trace of the four-variable confluent function. A theorem on the computation of limiting values of multivariate functions is proved, and transformation formulas for these functions are established. These results are employed to determine the singularity order of fundamental solutions and to validate the correctness of the solution to the Dirichlet problem.
The uniqueness of the solution to the Dirichlet problem is proved using the maximum principle for elliptic equations.

Numerical modeling of impact interaction between a non-deformable conical body and a porous layer has been performed. The porous layer is reprsented as an assembly of discrete elements, whose motion and deformation are described using a mesh-free discrete element method (DEM). This approach interprets elements as particles with defined elastic properties, enabling effective simulation of processes involving large displacements and material discontinuity, unlike conventional mesh-based methods. The fundamental principles of DEM, which has gained widespread adoption due to advances in computational technologies, are presented. The numerical model and calculation methodology are described in detail. Simulation results are presented for normal high-velocity interaction between a deformable porous medium (composed of particles) and an elastic rod with a conical contact surface. Coulomb friction at the interface between the porous medium and conical surface is accounted for. The contact forces exerted by the discrete medium on the elastic conical body are evaluated. The numerical results are compared with experimental data obtained from reverse ballistic experiments where a container with porous material is projected against a stationary rod at various initial velocities.

The present study is devoted to employing the theory of algebraic invariants for deriving an approximation of the potential of force and couple stresses of the fourth degree for a nonlinear hemitropic micropolar elastic solid. The complete set of irreducible invariants for a system of two asymmetric second-rank tensors in the form of invariant traces is studied using the theory of integer rational algebraic invariants (semi-invariants).
As a result, a set of 86 invariant traces is obtained. This set comprises 8 individual invariants, 17 doublets, 44 triplets, and 17 quadruplets. From these 86 elements, 39 invariants were selected according to the rule of increasing algebraic degrees: 2 linear invariants, 6 quadratic, 12 cubic, and 19 quartic. The 39 fourth-degree invariants are divided into four groups based on the following rules: products of linear invariants with each other, products of quadratic invariants with each other, products of linear and quadratic invariants, pairwise products of linear and cubic invariants, and proper fourth-degree invariants.
The potential of force and couple stresses of a hemitropic micropolar elastic solid is constructed, containing quadratic, cubic, and quartic algebraic terms. Thus, the micropolar potential contains a total of 124 constitutive modules. Formulas for calculating all 39 invariants in mixed tensor components are provided. As a result, 87 quartic corrections to the cubic potential of force and couple stresses of a nonlinear hemitropic micropolar elastic solid are obtained.

This paper presents a new exact solution describing the inhomogeneous distribution of velocity and pressure fields in the problem of isothermal steady shear flow of a viscous incompressible fluid. The obtained exact solutions remain valid when the kinematic viscosity is replaced by the turbulent viscosity in the Navier–Stokes equations.
It is shown that in the class of functions that are linear in some coordinates, a joint inhomogeneous solution for the velocity field can have only a specific structure—with constant spatial accelerations. In this case, either only two specific accelerations vanish, or all four spatial accelerations equal zero (homogeneous velocity field, Ekman solution). No other joint solutions exist in the specified class.
The case of two nonzero spatial accelerations is analyzed in detail, and the complete exact solution is provided. To understand the main properties of this solution, the corresponding boundary value problem is investigated and comprehensive illustrative material is presented.

The paper present a mathematical model of self-consistent relaxation in a perturbed region, based on the nonlinear Vlasov–Poisson system, which describes the interaction of a stationary absorbing charged conductor (of spherical or cylindrical geometry) with a free-molecular multicomponent low-temperature plasma. The high dimensionality of kinetic equations posed significant challenges for numerical implementation. To overcome these, we developed a system of curvilinear coordinates with nonholonomic constraints that reduces the phase volume of the problem; the derivation of the kinetic equation form in this coordinate system is provided. The employed numerical simulation method is described in detail.
The obtained results not only validate the adequacy of the proposed model and the correctness of numerical algorithms implementation, but also demonstrate substantial practical relevance. The kinetic nature of the model enables detailed investigation of plasma state and self-consistent electric field in the near-surface region. Specifically, for the case of a spherical body in three-component plasma, we demonstrate significant nonequilibrium in particle distribution within the perturbed zone and reveal characteristic features of spatial distribution and dynamics for particles with different charge signs.

This study investigates an initial-boundary value problem for a bounded domain in thermal interaction with an external medium, incorporating memory effects through the Caputo time-fractional derivative. Heat transfer through the lateral surface is modeled as a negative heat source in the governing differential equation. An a priori estimate for the solution is established. The solution is derived by using an operational method based on the Laplace transform in time.

An extension of a previously developed method for predicting creep and long-term strength is presented. The method is based on known information concerning the behavior of a pre-tested specimen (a leader specimen or prototype) under conditions of viscous material failure and is eventually generalized to the case of exposure to an aggressive environment—hydrogen charging of metallic specimens and structural components with varying degrees of introduced hydrogen. The advantages of the developed method over more complex known models are noted. Results of the calculation of creep and long-term strength for hydrogen-charged specimens made of VT6 alloy at a temperature of 600 ∘C are presented. The research results demonstrate that the developed method not only allows us to predict the creep and longterm strength curves but even enables to plan the optimal experiments to obtain a series of creep curves under constant stress.