 # Vol 26, No 2 (2020)

## Статьи

### ON BOUNDARY VALUE PROBLEM FOR GENERALIZED ALLER EQUATION

Gekkieva S.K., Karmokov M.M., Kerefov M.A.

#### Abstract

The mathematical models of fluid filtration processes in porous media with a fractal structure and memory are based on differential equations of fractional order in both time and space variables. The dependence of the soil water content can significantly affect the moisture transport in capillary-porous media. The paper investigates the generalized Aller equation widely used in mathematical modeling of the processes related to water table dynamics in view of fractal structure. As a mathematical model of the Aller equation with
Riemann — Liouville fractional derivatives, a loaded fractional order equation is proposed, and a solution to the Goursat problem has been written out for this model in explicit form.

Vestnik of Samara University. Natural Science Series. 2020;26(2):7-14 7-14  ### ON SMOOTHNESS OF SOLUTION OF ONE NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION

Kirichek V.A.

#### Abstract

In this paper we consider a nonlocal problem with integral boundary condition for hyperbolic equation. The conditions of the problem contain derivatives of the first order with respect to both x and t,, which can be interpreted as an elastic fixation of the right end rod in the presence of a certain damper, and since the conditions also contain integral of the desired solution, this condition is nonlocal. It is known that problems with nonlocal integral conditions are non-self-adjoint and, therefore, the study of solvability encounters difficulties that are not characteristic of self-adjoint problems. Additional difficulties arise also due to the fact that one of the conditions is dynamic. The attention of the article is focused on studying the
smoothness of the solution of the nonlocal problem. The concept of a generalized solution is introduced, and the existence of second-order derivatives and their belonging to the space L2 are proved. The proof is based
on apriori estimates obtained in this work.

Vestnik of Samara University. Natural Science Series. 2020;26(2):15-22 15-22  ### FRACTAL MAGMAS AND PUBLIC-KEY CRYPTOGRAPHY

Tsvetov V.P.

#### Abstract

In this paper, we deal with magmas – the simplest algebras with a single binary operation. The main result of our research is algorithms for generating chain of finite magmas based on the self-similarity principle of its Cayley tables. In this way the cardinality of a magma’s domain is twice as large as the previous one for each magma in the chain, and its Cayley table has a block-like structure. As an example, we consider a cyclic semigroup of binary operations generated by a finite magma’s operation with a low-cardinality domain, and a modify the Diffie-Hellman-Merkle key exchange protocol for this case.

Vestnik of Samara University. Natural Science Series. 2020;26(2):23-49 23-49  ### THE EFFECT OF METAL ALLOY MICROSTRUCTURE ON THE THICKNESS DISTRIBUTION IN THE CIRCULAR PLATE UNDER THE SUPERPLASTIC BLOW-FORMING

Beliakova T.A., Goncharov I.A.

#### Abstract

The gas-blow forming of sheet specimens under the superplastic conditions is the widespread and intensive developing technological method for metal-forming of modern alloys (mainly on the aluminium and titanium base). The investigation of evolution of the microstructure parameters in the deformation process is necessary to obtain the defect-free constructions with required mechanical properties. In the
present paper the blow-forming process of the thin circular plate to the hemispherical form is modelled. The stress-strain relations used include the parameters of the grain size and of the hardening depending on the
accumulated plastic strain. It is shown that taking into account the evolution of the microstructure parameters in the superplastic deformation can significantly improve the estimation of the optimal characteristics of the manufacturing process and eventually can provide the most thickness-uniform engineering product.

Vestnik of Samara University. Natural Science Series. 2020;26(2):50-62 50-62  ### DYNAMIC PROBLEM FOR A THIN-WALLED BAR WITH A MONOSYMMETRIC PROFILE

Elekina E.N., Vronskaja E.A.

#### Abstract

The paper presents an analytical solution to the dynamic problem for a thin-walled elastic rod, the
cross-section of which has one axis of symmetry. The solution is constructed for an arbitrary dynamic load and two types of boundary conditions: hinged support in constrained torsion and free warping of the end sections of the rod; rigid fastening with constrained torsion and absence of warping. The peculiarity of the mathematical model lies in the fact that the differential equations of motion contain a complete system of
inertial terms. Spectral expansions obtained as a result of using the method of integral transformations are represented as an effective method for solving linear non-stationary problems in mechanics. The structural
algorithm of the method of finite multicomponent integral transformations proposed by Yu.E. Senitsky is used.

Vestnik of Samara University. Natural Science Series. 2020;26(2):63-69 63-69  