Differential Equations Of Hyperbolic Type Research Articles

Free vibrations of variablesection beams taking rotational and frictional forces into account

Introduction. Beams are widely used in the construction industry as load-bearing structures of bridges, overpasses, coverings, slabs, stairs, equipment platforms, etc. In order to fully utilize the bearing capacity of such structures and to reduce the rate of material consumption, beams of variable cross-section along the length can be used. During operation, such structural elements are subjected to various types of vibrations, which determines the relevance of studying various aspects of vibrational motion.Aim. To apply numerical methods for studying the influence of rotational inertial forces in the presence of viscous frictional forces on free vibrations of variable-section beams. This area of research presents interest, since the calculations are directly related to the determination of frequencies and forms of natural vibrations of structures.Materials and methods. Free vibrations were described by a homogeneous partial differential equation of hyperbolic type. The methods of variable separation and finite differences were used. A discrete domain in the form of a set of uniform grid nodes and a homogeneous system of algebraic equations were introduced. The system of equations in the matrix-vector form was used.Results. The spectra of natural frequencies, damping coefficients, and eigenforms of beam vibrations are determined. It is shown that the coefficient matrix has a banded and pentadiagonal form. The matrix elements are functions of the characteristic index. The damping coefficient and the frequency of free vibrations are determined from a system of two nonlinear equations. The solution of the system of equations is found using the method of coordinate descent. An example of calculation of a welded I-beam is considered. Five elements of the spectra of damping coefficients and natural frequencies are calculated.Conclusions. State-of-the-art MATLAB solvers allows numerical and graphical methods to be combined. In the solved examples, the advantages of these methods were successfully applied to determine the eigenvalues of matrices and eigenfunctions. The high validity and accuracy of the results obtained confirm the simplicity and versatility of the methodology for determining the characteristics of free vibrations of beams of variable cross-section.

STATIONARY MODE OF FLUID FLOW IN THE PIPELINE IN THE PRESENCE OF ASSOCIATED SAMPLING AND PUMPING

Pipeline transportation of oil and petroleum products plays a significant role in the transport system of the Russian Federation. When operating oil pipelines, reliability and energy efficiency are the most important criteria. To achieve these indicators, it is necessary to perform optimal technological modes. When constructing models of technological regimes, a methodology is used in which the movement of oil and petroleum products is considered as steady. Based on this assumption, the throughput, pump operating modes, strength characteristics and other parameters are determined. During the operation of the equipment, almost every day it is necessary to switch from one mode to another, which entails unsteady fluid movement for some time. At the same time, the unsteady mode makes up a small part of the total operating time of the system.The beginning of studies of unsteady fluid processes in pipes dates back to the last quarter of the 19th century. N. E. Zhukovsky made a fundamental contribution to the study of this problem. His classic work on water hammer in water pipes was the beginning for the creation of a large number of works in pipe hydraulics. The problem was solved for an ideal elastic fluid, i.e. without taking into account the friction forces against the pipe walls. The resulting formula, according to which the pressure increment is proportional to the velocity, has received numerous experimental confirmations for cases when it is possible to neglect pressure losses to overcome hydraulic resistances. N. E. Zhukovsky also obtained a formula for determining the sound velocity of a liquid droplet in a pipe with elastic walls. It is proved that this speed depends on the pipe material and its thickness.In the future, a large number of works were carried out to study the flow of liquid and gas in round pipes. The fundamental works of I. A. Charny, who obtained a mathematical model in the form of a system of partial differential equations of hyperbolic type, proved fruitful in this direction. Linearization of the equation, taking into account viscosity and hydraulic resistance allowed the researcher to obtain solutions to a number of engineering problems.The introduction of the Dirac function in the description of selectionpumping at specified points made it possible to describe processes in complex pipelines with one partial differential equation.This article discusses a horizontal section of the pipeline with sampling and pumping points. A stationary process for a drip liquid is considered. The problem is solved using the sine-Fourier transform.

On a non-local problem for system of partial differential equations of hyperbolic type in a specific domain

On a non-local problem for system of partial differential equations of hyperbolic type in a specific domain

On one solution of a periodic boundary value problem for a hyperbolic equations

In a rectangular domain, we consider a boundary value problem periodic in one variable for a system of partial differential equations of hyperbolic type. Introducing a new unknown function, this problem is reduced to an equivalent boundary value problem for an ordinary differential equation with an integral condition. Based on the parametrization method, new approaches to finding an approximate solution to an equivalent problem are proposed and its convergence is proved. This made it possible to establish conditions for the existence of a unique solution of a semiperiodic boundary value problem for a system of second-order hyperbolic equations.

Grid-characteristic methods. 55 years of developing and solving complex dynamic problems

The development of a computational method is not a simple matter and boils down to replacing the differential operator with a difference one. To construct it, it is necessary to correctly set a mathematical problem that is adequate to the physical one under consideration. In addition, the algorithm must meet some other requirements. Therefore, to create a numerical algorithm requires not only ingenuity and imagination, but also a deep understanding of the reasons why these requirements are caused.Systems of partial differential equations of hyperbolic type are used to describe the unsteady behavior of continuous media. To solve these problems, characteristic methods were developed in such a way as to take into account the corresponding properties of hyperbolic equations and to be able to build a so-called characteristic irregular grid adapting to the solution of the problem. Methods of end-to-end counting have been developed that take into account the properties of systems of hyperbolic equations — inverse methods of characteristics or grid-characteristic methods.In grid-characteristic methods, a regular computational grid is used, not a solvable initial system is approximated on it, but compatibility conditions along characteristic lines with interpolation of the desired functions at the points of intersection of characteristics with a coordinate line on which the data is already known. The obtained characteristic form of the gas dynamics equations makes it possible to understand how to set the boundary conditions correctly.The construction of a numerical method is not a simple matter and is not reduced to the formal replacement of derivatives by approximating their difference relations (for example, using finite differences). When developing the method, it is necessary to take into account the physical side of the problem being solved. At the same time, the method must meet certain requirements, the understanding of which is necessary during its development.

Mesoscopic analytical approach in a three state opinion model with continuous internal variable

Analytical approaches in models of opinion formation have been extensively studied either for an opinion represented as a discrete or a continuous variable. In this paper, we analyze a model which combines both approaches. The state of an agent is represented with an internal continuous variable (the leaning or propensity), that leads to a discrete public opinion: pro, against or neutral. This model can be described by a set of master equations which are a nonlinear coupled system of first order differential equations of hyperbolic type including non-local terms and non-local boundary conditions, which cannot be solved analytically. We developed an approximation to tackle this difficulty by deriving a set of master equations for the dynamics of the average leaning of agents with the same opinion, under the hypothesis of a time scale separation in the dynamics of the variables. We show that this simplified model accurately predicts the expected transition between a neutral consensus and a bi-polarized state, and also gives an excellent approximation for the dynamics of the average leaning of agents with the same opinion, even when the time separation scale hypothesis is not completely fulfilled.

Self-Learning: Application of differential equations of hyperbolic type to some engineering problems using information and communication technologies

This article analyzes the issue of organizing students' self-study, which is one of the urgent problems of the credit-modular system. The student independently discusses in detail what to pay attention to in order to solve an engineering problem using information and communication technologies using differential equations with special derivatives of hyperbolic type. It is stated that the initial-boundary value problem of the process is solved by the Fourier method. The construction of a decision graph using the Maple system is explained. The student's attention is focused on the analysis of the results based on the nature of the considered engineering problem.

The hyperbolic equation with fractional derivative in modelling the polymer concrete behaviour under constant load

The paper considers an inhomogeneous partial differential equation of hyperbolic type containing both second order partial derivatives and fractional derivatives of an order lower than the second order on the spatial variable. The fractional derivative is understood in the Riemann-Liouville sense. The paper first part presents analytical solution of the boundary value problem of the first kind with arbitrary initial conditions of the equation in question by the Fourier method (separation of variables). The solution is given in the form of two functions, one of which characterizes the stationary state of the process, and the second is treated as a deviation from the stationary state. The second function is found using eigenfunctions and eigenvalues of the corresponding two-point Dirichlet problem, while the first is written out as a Mittag-Leffler function. In the second part of the paper, the solution application examples of the boundary value problem of the equation under consideration in modelling the behaviour of polymer concrete based on polyester resin (dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) under load are considered. The examples give specific values for the parameters of the equation and graphs of the solutions and the error resulting from replacing a series by its partial sum. All calculations were performed in the MATLAB application package.

Alternative methods for solving a hyperbolic system of equations in a simplified water hammer problem

The paper addresses a mathematical model of the transient process in a pipeline, namely, a water hammer model. Based on this model, a mixed problem for a system of partial differential equations of hyperbolic type is set. In this paper, we obtain formulas for the exact solution of the Cauchy problem for a linear homogeneous hyperbolic system. We compare the effectiveness of explicit and implicit numerical methods for solving a simplified system of water hammer equations.

Asymptotics of the solution of the hyperbolic system with a small parameter

Asymptotic study of singularly perturbed differential equations of hyperbolic type has received relatively little attention from researchers. In this paper, the asymptotic solution of the singularly perturbed Cauchy problem for a hyperbolic system is constructed. In addition, the regularization method for singularly perturbed problems of S. A. Lomov is used for the first time for the asymptotic solution of a hyperbolic system. It is shown that this approach greatly simplifies the construction of the asymptotics of the solution for singularly perturbed differential equations of hyperbolic type.

Nonlinear oscillations of a rod with residual stress in the presence of contact dry friction

Wave processes are studied in an elastic rod interacting with its environment according to the law of dry friction. Oscillations of the system at residual stresses are emerged under the influence of instantaneous unloading. Analytical solution of the partial differential equation of hyperbolic type with a nonlinear mechanism of energy dissipation is obtained. Thus, it makes it possible to describe the oscillatory process at various loading parameters of the system for an arbitrary section of the rod. The property, that for the class of problems under consideration dry friction does not change the natural frequency of the system, is shown additionally. The problem can be considered as a model for describing dynamic processes in rocks under the effect of internal stresses. Results of the modeling problem allow us to conclude: if we consider that rocks resist differently to tension and compression, such as resistance to tension is less than compression’ resistance, it is obvious that in the case of relatively rapid unloading of the system, the residual compressions will create stresses that correspond to the first maximal displacements of the end section, and that exceed the limit of tensile strength as well. In this case, the edge sections undergo the brittle fracture that can be sustained by the tensile stresses of the subsequent maximum displacements.

Existence and smoothness of solutions of a singular differential equation of hyperbolic type

This paper investigates the question of the existence of solutions to the semiperiodic Dirichlet problem for a class of singular differential equations of hyperbolic type. The problem of smoothness of solutions is also considered, i.e., maximum regularity of solutions. Such a problem will be interesting when the coefficients are strongly growing functions at infinity. For the first time, a weighted coercive estimate was obtained for solutions to a differential equation of hyperbolic type with strongly growing coefficients.

On approximate controllability of multi-term time fractional measure differential equations with nonlocal conditions

The purpose of this paper is to investigate the existence of mild solutions and approximate controllability of a class of multi-term time-fractional measure differential equations of hyperbolic type involving nonlocal conditions in Hilbert spaces. The approximate controllability is demonstrated by utilizing fundamental tools, namely: $$(\beta ,\gamma _{k})$$ -resolvent family, measure functional (Henstock-Lebesgue-Stieltjes integral), regulated functions and fixed point techniques. Finally, an example is presented.

Оцінка пластичності поверхневого шару металу при обкочуванні деталей циліндричної форми кулькою

The article deals with the study of the peculiarities of the process of plastic deformation of the surface layer of cylindrical parts during rolling with a ball. The stress-strain state and the regularities of its change, depending on the process parameters in the surface layer of the part, were determined by the slip line method and the finite element method.The vast majority of parts of mechanisms and machines work in contact with the relevant parts, namely various friction pairs, shafts, rollers, gears or tools, product housings, strikers, and so on. Specific requirements for these parts are due to the peculiarities of the operation of the latter. As a rule, it is impossible to meet all the requirements when making parts from one material. This mainly applies to the requirements of combining the high strength characteristics of the material of the internal volume of the part and the significant hardness of the same material on its surface in contact with the corresponding part. In addition, the surface of the parts under certain conditions must have high wear resistance and low coefficient of friction (antifriction parts) or high coefficient of friction (friction parts), or self-changing surface layer, and sometimes the entire surface must have some porosity and sweat during operation .The slip line method can be attributed to the traditional methods for solving problems of the theory of plasticity. This method can be used to solve problems in which the equations of equilibrium and plasticity create a closed system of differential equations of hyperbolic type. Therefore, this method can be used for solving plane problems, and only in some cases for solving axisymmetric problems. The solutions that can be obtained if a certain area of slip lines is directly known are called straight lines. When information about the shape of the initial slip lines is insufficient, inverse and semi-inverse methods are used.A numerical study of the regularities of damage accumulation was carried out, calculations of the accumulated deformation, the stress state index, and the distributions of the used plastic resource over the volume of the plastic zone when rolling with a ball in the environment of a specialized software complex were carried out. The adequacy of the model is checked in comparison with the available experimental data.On the basis of the research carried out in the article, recommendations can be developed for the choice of processing modes that provide the most rational parameters of the state of the surface layer of parts such as bodies of revolution.

Singular perturbation results for linear partial differential–algebraic equations of hyperbolic type

We consider constrained partial differential equations of hyperbolic type with a small parameter ε>0, which turn parabolic in the limit case, i.e., for ε=0. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter ε. For the analysis, we consider the system equations as partial differential–algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order estimates are proven and their optimality is shown numerically.

Dynamic first order wave systems with drift term on Riemannian manifolds

An abstract first order differential equation of hyperbolic type with drift term on a Riemannian manifold is considered. For proving its well-posedness, transmutator and commutator relations are needed, which are studied in a general functional analytic setting.

МОДЕЛЮВАННЯ ДИНАМІЧНИХ ПРОЦЕСІВ МЕТОДОМ ГІБРИДНОГО ІНТЕГРАЛЬНОГО ПЕРЕТВОРЕННЯ ТИПУ ЕЙЛЕРА-БЕССЕЛЯ НА СЕГМЕНТІ

На сучасному етапі науково-технічного прогресу, особливо у зв'язку з широким використанням композитних матеріалів, існує нагальна потреба у вивченні фізико-технічних характеристик таких матеріалів, що знаходяться в різних умовах експлуатації, що математично призводить до задачі розв’язування сепаратної системи рівнянь з чатинними похідними другого порядку на кусково-однорідному сегменті з відповідними початковими та крайовими умовами, зокрема, задача динаміки математично призводить до побудови розв’язку сепаратної системи диференціальних рівнянь з частинними похідними гіперболічного типу. Одним із ефективних методів побудови інтегральних зображень аналітичних розв’язків алгоритмічного характеру задач математичної фізики є метод гібридних інтегральних перетворень. У цій роботі побудовано розв’язок задачі динаміки на двоскладовому сегменті полярної осі r ∈[0;R2 ] з точкою спряження методом гібридного інтегрального перетворення Ейлера-Бесселя. Задача динаміки на двоскладовoму сегменті полярної осі математично призводить до побудови обмеженого розв’язку сепаратної системи двох диференціальних рівнянь з частинними похідними гіперболічного типу з відповідними початковими умовами, умовами спряження та крайовими умовами. Застосувавши до цієї крайової задачі гібридне інтегральне перетворення Ейлера-Бесселя, отримаємо задачу Коші. Знайшовши розв’язок задачі Коші, ми застосовуємо до нього обернене гібридне інтегральне перетворення Ейлера-Бесселя. Пряме інтегральне перетворення Ейлера-Бесселя на сегменті полярної осі з точкою спряження записується у вигляді матриці-рядка. Вихідна система та початкові умови записуються в матричній формі, і ми застосовуємо операторну матрицю-рядок до заданої задачі за правилом множення матриць. В результаті отримуємо задачу Коші для звичайного диференціального рівняння другого порядку. Обернене перетворення Ейлера-Бесселя записується у вигляді операторної матриці-стовпця, і ми застосовуємо його до побудованого розв’язку задачі Коші. Після здійснення певних перетворень ми отримуємо єдиний розв’язок вихідної задачі. Побудовані розв’язки крайових задач мають алгоритмічний характер, що дозволяє використовувати їх як у теоретичних дослідженнях, так і в числових розрахунках.

ЗАДАЧА УПРАВЛЕНИЯ ИСТОЧНИКАМИ В КРАЕВЫХ УСЛОВИЯХ ГИПЕРБОЛИЧЕСКОЙ СИСТЕМЫ СЕТЕВОЙ СТРУКТУРЫ

The solution to the optimal control problem by power of external and internal sources acting on the multilink system in nonlocal boundary conditions is investigated. Each arc of the system is an object with distributed parameters, described by a differential equation of hyperbolic type and related only by boundary values, and in an arbitrary way. Due to the long duration of the object's functioning, the exact values of the initial conditions are not known, but a set of their possible values is given. Based on the results of additional measurements of the state of the process at the input or output ends of the arcs (which are not internal vertices), a target functional is constructed, for which minimization a formula for its gradient is obtained.

Представление решения задачи Гурса для линейных стохастических гиперболических дифференциальных уравнений второго порядка

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

Classical solutions for a class of nonlinear wave equations

We study a class of initial value problems subject to nonlinear partial differential equations of hyperbolic type. A new topological approach is applied to prove the existence of nontrivial nonnegative solutions. More precisely, we propose a new integral representation of the solutions for the considered initial value problems and using this integral representation we establish existence of classical solutions for the considered classes of nonlinear wave equations.