Eurasian Mathematical Journal

Boundary value problem for hyperbolic integro-differential equations of mixed type

A.T. Assanova — 2025-06-30

The boundary value problem for a system of hyperbolic integro-differential equations of
mixed type with degenerate kernels is considered on a rectangular domain. This problem is reduced
to a family of boundary value problems for a system of integro-differential equations of mixed type
and integral relations. The system of integro-differential equations of mixed type is transferred to a
system of Fredholm integro-differential equations. For solving the family of boundary value problems
for integro-differential equations Dzhumabaev’s parametrization method is applied. A new concept
of a general solution to a system of integro-differential equations with parameter is developed. The
domain is divided into N subdomains by a temporary variable, the values of a solution at the interior
lines of the subdomains are considered as additional functional parameters, and a system of
integro-differential equations is reduced to a family of special Cauchy problems on the subdomains for
Fredholm integro-differential equation with functional parameters. Using the solutions to these problems,
a new general solutions to a system of Fredholm integro-differential equations with parameter
is introduced and its properties are established. Based on a general solution, boundary conditions,
and the continuity conditions of a solution at the interior lines of the partition, a system of linear
functional equations with respect to parameters is composed. Its coefficients and right-hand sides
are found by solving the family of special Cauchy problems for Fredholm integro-differential equations
on the subdomains. It is shown that the solvability of the family of boundary value problems
for Fredholm integro-differential equations is equivalent to the solvability of the composed system.
Methods for solving boundary value problems are proposed, which are based on the construction and
solving of these systems. Conditions for the existence and uniqueness of a solution to the boundary
value problem for a system of hyperbolic integro-differential equations of mixed type with degenerate
kernels are obtained.

Notes on the generalized Gauss reduction algorithm

Y. Baissalov — 2025-06-30

The hypothetical possibility of building a quantum computer in the near future has forced a revision of the foundations of modern cryptography. The fact is that many difficult algorithmic problems, such as the discrete logarithm, factoring a (large) natural number into prime factors, etc., on the complexity of which many cryptographic protocols are based these days, have turned out to be relatively easy to solve using quantum algorithms.
Intensive research is currently underway to find problems that are difficult even for a quantum computer and have potential applications for cryptographic protocols. Our article contains notes related to the so-called generalized Gauss algorithm, which calculates the reduced basis of a twodimensional lattice [8], [2]. Note that researchers are increasingly putting forward difficult algorithmic problems from lattice theory as candidates for the foundation of post-quantum cryptography. The majority of algorithmic problems related to lattice reduction become NP-hard as the lattice dimension increases [3], [1]. Fundamental problems such as the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), and Bounded Distance Decoding (BDD) are conjectured to remain hard even for quantum algorithms [4], [6]. Although the generalized Gauss reduction algorithm applies to two-dimensional lattices, where exact analysis is feasible (dimensions 3 and 4 are studied in [7], [5]), understanding such low-dimensional reductions provides important insights into the structure and complexity of lattice-based cryptographic constructions.

Nikol’skii-Besov spaces with a dominant mixed derivative and with a mixed metric: interpolation properties, embedding theorems, trace and extension theorems

K.A. Bekmaganbetov — 2025-06-30

In this work, we define Nikol’skii-Besov spaces with a dominant mixed derivative and with a mixed metric. The interpolation properties of these spaces with respect to the anisotropic interpolation method are studied, sharp embedding theorems of different metrics are proved, and sharp trace and extension theorems are proved.

Local and 2-local 1/2-derivations of solvable Leibniz algebras

U. Mamadaliyev — 2025-06-30

We show that any local 1/2-derivation on solvable Leibniz algebras with model or abelian nilradicals, whose dimensions of complementary spaces are maximal, is a 1/2-derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimensional complementary spaces are 1/2-derivations. Moreover, a similar problem concerning 2-local 1/2-derivations of such algebras is investigated.

Factorization method for solving systems of second-order linear ordinary differential equations

I.N. Parasidis — 2025-06-30

We consider in a Banach space the following two abstract systems of first-order and second-order linear ordinary differential equations with general boundary conditions, respectively,

X'(t) - A_0(t) X(t) = F(t), \Phi(X) = \sum_{j=1}^{n} M_j \,\Psi_j(X)

and

X''(t) - S(t)X'(t) - Q(t)X(t) = F(t), \Phi(X) = \sum_{j=1}^{n} M_j \Psi_j(X),

where X(t)=col(x_1 (t),…,x_m (t)) denotes a vector of unknown functions, F(t) is a given vector and A_0 (t), S(t), Q(t) are given matrices, Φ, Ψ_1, . . . , Ψ_n, Θ_1, . . . , Θ_r are vectors of linear bounded functionals, and M_1, . . . , M_n, C, N_1, . . . , N_r are constant matrices. We first provide solvability conditions and a solution formula for the first-order system. Then we construct in closed form the solution of a special system of 2m first-order linear ordinary differential equations with constant coefficients when the solution of the associated system of m first-order linear ordinary differential equations is known. Finally, we construct in closed form the solution of the second-order system in the case in which it can be factorized into first-order systems.

An inverse problem for 1d fractional integro-differential wave equation with fractional time derivative

A.A. Rahmonov — 2025-06-30

This paper is devoted to obtaining a unique solution to an inverse problem for a onedimensional time-fractional integro-differential equation. First, we consider the direct problem, and the unique existence of the weak solution is established, after that, the smoothness conditions for the solution are obtained. Secondly, we study the inverse problem of determining the unknown coefficient and kernel, and the well-posedness of this inverse problem is proved. The local existence and global uniqueness results are based on the Fourier method, fractional calculus, properties of the Mittag-Leffler function, and Banach fixed point theorem in a suitable Sobolev space.

International conference "Actual Problems of Analysis, Differential Equations and Algebra" (EMJ-2025), dedicated to the 15th anniversary of the Eurasian Mathematical Journal

V.I. Burenkov — 2025-06-30

From January 7 to January 11, 2025 at the Non-profit joint-stock company “L.N. Gumilyov Eurasian National University” (ENU) the International Conference “Actual Problems of Analysis, Differential Equations and Algebra” (EMJ-2025) was held. The conference was dedicated to the 15th anniversary of the Eurasian Mathematical Journal (EMJ).
The purposes of the conference were to discuss the current state of development of mathematical scientific directions, expand the number of potential authors of the Eurasian Mathematical Journal and further strengthen the scientific cooperation between the Faculty of Mechanics and Mathematics of the ENU and scientists from other cities of Kazakhstan and abroad.