The Cauchy problem on the real axis for the Gurtin–Pipkin equation with the Rabotnov kernel is considered. For some special case, it is proved that there is no propagation front in this problem. DOI 10.1134/S1061920824040137

A problem of the distributed control of oscillations of an effective (averaged) medium corresponding to a two-phase medium of slightly viscous liquids is considered. The averaged model is described by a boundary value problem for an integrodifferential equation. It is shown that for this model, it is impossible to bring the vibrations to a state of rest in finite time by applying force on the entire region.

The problem of simulation of gravitational internal waves from a point mass source moving in a stratified fluid is concerned. The fluid is non-uniformly stratified, that is, the buoyancy frequency is non-constant in the vertical direction. The computer program that allows to calculate the fluid velocity field, the vertical displacement, the pressure, the vertical displacement of the surface is developed. An explicit finite deferences scheme is used for calculations. An example of simulation of rectilinear motion of the source under and above a layer of jump in the buoyancy frequency is shown.

This paper presents the algorithms and results of calculations of the dynamics of a surface layer of a fluid under the action of currents that have emerged from a depth. Several approaches to model the velocity field in a horizontal flow round a fixed underwater obstacle are investigated. Formulas for calculating the velocity field on the free surface of an ideal homogeneous fluid are proposed. A computer program is developed that makes it possible to simulate the interaction of a stratified fluid flow with an underwater obstacle. The possibility of using asymptotic formulas for the far-field approximation to calculate the velocity field in a uniformly stratified fluid is studied.

We consider an initial-boundary problem describing the motion of a two-phase medium with a periodic structure. The first phase of the medium is an isotropic elastic material and the second phase is an incompressible viscoelastic Kelvin-Voigt fluid. This problem consists of second and fourth order partial differential equations, conditions of continuity of displacements and stresses at the phase boundaries, and homogeneous initial and boundary conditions. Using the Laplace transform method, we derive a homogenized problem, which is an initial boundary value problem for the system of fourth order partial integro-differential equations with constant coefficients. The coefficients and convolution kernels of the homogenized equations are found by using solutions of auxiliary periodic problems on the unit cube. In the case of a layered medium, the solutions of the periodic problems are written explicitly, and this allows us to find analytic expressions for the homogenized coefficients and convolution kernels. In particular, we establish that the type and properties of the homogenized convolution kernels depend on the volume fraction of the fluid layers inside the periodicity cell.

We consider the problem of the boundary control of one-dimensional oscillations of an effective (averaged) medium corresponding to a two-phase medium consisting of periodically alternating layers of elastic and viscoelastic materials with long-term memory or various viscoelastic materials with Kelvin–Voigt friction and long-term memory. The averaged model is described by a boundary value problem for an integrodifferential equation. It is shown that for this model, it is impossible to bring oscillations to a state of rest in finite time (in contrast to the equation of string oscillations) by a force acting at one end of the band. A hypothesis is formulated on the possibility of bringing the specified object to a state of rest with the help of force effects distributed along the entire length of the object.

The article deals with the issues of qualitative behavior and methods for calculating the spectra of natural vibrations of a layered composite consisting of elastic and viscoelastic phases with dissipation. As a viscoelastic phase, it is proposed to consider a viscoelastic material with exponential aftereffect kernels, with aftereffect kernels in the form of Rabotnov functions, a material with Kelvin-Voigt friction and a material with fractional Kelvin-Voigt friction. A general scheme is given for studying the qualitative properties of the spectra of one-dimensional oscillations and a method for calculating one-dimensional oscillations, which consists in reducing the spectral problem to solving algebraic equations. The question of the convergence of natural frequencies of oscillations of composite samples to the natural frequencies of averaged boundary value problems, which are non-self-adjoint, is discussed.

In this work we consider a mathematical model of the water treatment process and determine the effective characteristics of this model. At the microscopic length scale we describe our model in terms of a lattice random walk in a high-contrast periodic medium with absorption. Applying then the upscaling procedure we obtain the macroscopic model for total mass evolution. We discuss both the dynamic and the stationary regimes, and show how the efficiency of the purification process depends on the characteristics of the macroscopic model.

We consider an integro-differential fractional order equation describing one-dimensional eigenoscillations of a homogeneous viscoelastic medium. We show that the spectrum of this equation consists of N sequences of eigenvalues, where N = 4 or N = 5. Elements of these sequences are squared roots of equations of degree N. We study their limit behavior. We prove that there are only two sequences consisting of points with negative real part.

The paper considers the principles of constructing a mathematical model of water treatment based on the use of a biologically active layer, the bacteria of which absorb harmful impurities contained in water. A system of equations is presented on the basis of which a model of water purification is constructed in the simplest element, which is a rod covered with a biofilm. The system of equations is a system that includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on a part of the surface of this domain, connected to each other through a boundary condition and a potential in an equation of hyperbolic type. Next, an asymptotic analysis of this system is carried out, which allows us to reduce the model of an individual element to the solution of a simple ordinary differential equation. On this basis, a model of the entire water treatment device is proposed.

The paper considers the problem of damping vibrations of a membrane and a plate with the help of forces distributed over their entire area. The proposed method allows us to consider restrictions not only on the absolute value of the control, but also on the absolute value of the derivatives of the functions that specify the control. Sufficient conditions are given for the initial conditions under which the problem of bringing the system to rest in a finite time is solvable, and the time of bringing to rest is estimated.

Данная работа посвящена построению математической модели влияния вышедших из глубины течений на поле поверхностных ветровых волн. По аналогии с работой [1] выводится интегро-дифференциальное уравнение на величину отклонения свободной поверхности воды от положения равновесия. Далее осуществляется асимптотический анализ данного интегро-дифференциального уравнения на основе предположения о малости величины скорости возмущения поля течения по сравнению со скоростью движущегося потока, который взаимодействуя с неровностями дна вызывает возмущение потока. Асимптотический анализ с применением преобразования Фурье приводит к явной формуле для деформации спектра ветровой волны вышедшим на поверхность возмущенным течением. Спектр поверхностного волнения деформируется в широком диапазоне от миллиметровых волн до метровых, зона деформации принимает стабильную форму, в зоне деформации можно увидеть признаки неровностей морского дна. This work is devoted to the construction of a mathematical model of the influence of currents emerging from the depths on the field of surface wind waves. By analogy with the work [1], an integro-differential equation is derived for the amount of deviation of the free surface of water from the equilibrium position. Next, an asymptotic analysis of this integro-differential equation is carried out based on the assumption that the velocity of the disturbance of the flow field is small compared to the velocity of the moving stream, which, interacting with the irregularities of the bottom, causes a disturbance of the flow. Asymptotic analysis using the Fourier transform leads to an explicit formula for deforming the spectrum of a wind wave by a perturbed flow that has surfaced. The spectrum of surface waves is deformed in a wide range from millimeter waves to meter waves, the deformation zone takes a stable shape, and signs of seabed irregularities can be seen in the deformation zone.

We consider the initial-boundary value problem describing the dynamics of a two-phase medium with periodic structure. The phases of the medium are an isotropic viscoelastic material with memory and an incompressible viscoelastic Kelvin–Voigt viscoelastic material. We derive the corresponding homogenized problem describing the dynamics of a homogenous viscoelastic medium with memory. We find explicit analytic expressions for coefficients and convolution kernels of the homogenized equations corresponding to the case of a layered medium.

The paper considers the problem of determining properties of an underwater moving source by analyzing the perturbation that it creates in electromagnetic or hydrodynamic fields. The computer program has been developed to simulate the spatial propagation of gravitational waves from the mass source moving along an arbitrary trajectory in a stratified fluid. The calculation results are in good agreement with analytical results obtained in the far-field approximation and with the results of experiments on the flow around underwater obstacles, moreover, the proposed technique allows to simulate any arbitrary motion of the source. Two new approaches to solving the inverse problem of determining the characteristics of the source of disturbances are proposed. The first one is based on the analysis of the signal received by radio-sensors, that scan the surface of the ocean. The second one uses data obtained from sensors installed directly in the water column.

We will consider the exact controllability of the distributed system, governed by string equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case, the memory kernel is a linear combination of exponentials.

The article deals with the problems of constructing effective characteristics of a layered composite material, the layers of which are viscoelastic material. In this case, the hereditary property of viscosity is modeled using the Rabotnov functions, which have singularities at zero. It is proved that the average properties of a material can be analytically described in the form of a homogeneous viscoelastic material with memory, which is also specified using kernels, which are also specified analytically by the Rabotnov functions. It is of interest to study the natural vibrations of such a material. The easiest way is to obtain analytical formulas for the natural frequencies of oscillations transverse with respect to the layers. Such analytical expressions for natural frequencies are obtained in the present work. The spectra are compared for the case of composites made of elastic materials, materials in which nonlocal terms are specified with the help of exponential functions and with the help of Rabotnov functions.

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in finite time.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021