Institute of Mathematics and Mathematical Modeling

ABSTRACT This paper addresses a multi‐characteristic problem for a higher‐order hyperbolic partial differential equation with a piecewise‐constant generalized argument. By introducing appropriate auxiliary unknowns, the original formulation is transformed into a parameter‐dependent problems for a system of first‐order differential equations with a piecewise‐constant generalized argument and accompanying integral relations. The analysis and resolution of this auxiliary set of problems are carried out by means of the parametrization Dzhumabaev method. A novel procedure for constructing solutions on subdomains is proposed. Sufficient conditions guaranteeing the existence and uniqueness of solutions to the resulting parameter‐dependent systems with piecewise‐constant generalized argument are established, and algorithms for their computation are developed; their convergence is rigorously justified. On this basis, conditions ensuring the unique solvability of the initial multi‐characteristic problem for higher‐order partial differential equations with piecewise‐constant generalized argument are derived.

ABSTRACT The article addresses a two‐point boundary value problem for an integro‐differential equation with a delay argument. The problem is studied by means of the Dzhumabaev parametrization method. Within this framework, additional parameters and new functions are introduced, reducing the original boundary value problem to an equivalent problem for a system of integro‐differential equations with delay involving these additional parameters. On the basis of the parametrization method, sufficient conditions for solvability are established in terms of the coefficient matrices, the initial vector functions, the delay, and the boundary conditions. Furthermore, an algorithm is proposed for constructing the solution of the two‐point boundary value problem for the integro‐differential equation with delay.

ABSTRACT This paper is devoted to studying inverse source problems for the linear Kelvin–Voigt equations with memory, which govern the flow of incompressible viscoelastic fluids. The memory term is represented by an integral with convolution and reflects the elastic properties of the fluid. The inverse problems consist of recovering the spatial distribution of external forces by measurements for the velocity and gradient of pressure at the final moment . These inverse problems are investigated in the following cases: with and without the memory term in the momentum equation, and under both sticking and sliding boundary conditions. In all cases, under suitable conditions on the data of the problem, the existence, uniqueness as well as the stability of strong solutions were established.

This study presents the results of numerical and experimental investigations into the impact of uneven fiber distribution in fiber-reinforced composites. A voxel-based model was developed using a Gaussian distribution with deviations of 5%, 10%, and 15% from an ideal hexagonal structure. Numerical simulations assessed the effect of microstructural statistical deviations on the elastic properties of composites. Computer vision methods were applied to analyze micrographs of real samples, enabling a comparison between modeled and actual material structures. The results demonstrated a significant influence of fiber misalignment on local stress concentrations and mechanical properties, validated by quantitative image analysis. This approach enhances the accuracy of predicting composite behavior under varying microstructural conditions.

Строятся и исследуются транспортные решения бикватернионного волнового уравнения, которое является бикватернионным обобщением уравнений Дирака и Максвелла. Данные уравнения описывают электромагнитные поля излучателей электромагнитных волн и электро-гравимагнитных волн, движущихся в фиксированном направлении с постоянной скоростью, которая меньше скорости распространения волн в электромагнитной среде (световой скорости). Построены фундаментальные и обобщенные транспортные решения, которые описывают поля движущихся обьектов при досветовых скоростях. С использованием преобразования Фурье обобщенных функций построена бикватернионная функция (бифункция) Грина в подвижной системе координат, которая описывает поле, порождаемое подвижным точечным излучателем на оси $z$. Определены плотность энергии и вектор Пойнтинга этого поля. Исследовано влияние скорости движения на полевые характеристики.

We prove the following submajorization inequalities for τ-measurable operators in a semifinite von Neumann algebra (M, τ).Let x₀, …, xₙ₋₁ be τ-measurable operators,f : [0, ∞) → [0, ∞) be a nonnegative function, and let α₀, …, αₙ₋₁ be positive real numbers such that (∑ⱼ₌₀ⁿ⁻¹ αⱼ xⱼ = 0), (∑ⱼ₌₀ⁿ⁻¹ αⱼ = 1), and αℓ ≥ ½ for some ℓ ∈ {0, …, n−1}. Then: (i) If f(0) = 0 and g(t) = f(√t) is convex, then ∑j,k ∈ S_ℓ [ f( √( αⱼ αk / (4 αℓ (1 − αℓ)) ) · |xⱼ + xk| ) +f( √( αⱼ αk (2 αℓ − 1) / (4 αℓ (1 − αℓ)) ) · |xⱼ − xk| ) ] ≼ ∑ⱼ₌₀ⁿ⁻¹ αⱼ f(|xⱼ|), where Sℓ = { 0, …, n−1 } ∖ { ℓ }. (ii) If h(t) = f(√t) is concave, then the reverse inequality holds: ∑ⱼ₌₀ⁿ⁻¹ αⱼ f(|xⱼ|) ≼ ∑j,k ∈ S_ℓ [ f( √( αⱼ αk / (4 αℓ (1 − αℓ)) ) · |xⱼ + xk| ) +f( √( αⱼ αk (2 αℓ − 1) / (4 αℓ (1 − αℓ)) ) · |xⱼ − xk| ) ], where Sℓ = { 0, …, n−1 } ∖ { ℓ }.