System Of Homogeneous Differential Equations Research Articles

WAYS OF INCREASING OF LOADING CAPACITY OF THE REINFORCED CONCRETE SHALLOW SHELLS WITH A FLAT RECTANGULAR CONTOUR

Shallow shells with variable curvature, prestressing contour and fixed above by horizontal reinforcing of two opposite contour elements are considered in the given article. These two offered ways practically produces non-moment state and increase loading capacity of the shallow shells several times. Method of definition of loading capacity of the shallow shells with a flat rectangular contour was worked out, and it's aim consists of that initial moment state before loss of loading capacity is described by nonlinear differential equations of the theory of the shallow shells. And then accentuating the main part of the solution of equations we receive the linear system of homogeneous differential equations with variable coefficients. The solution of the linearizated equations enables to define the forms of losing of loading capacity of shallow shells.

NUMERICAL AND ANALYTICAL INVESTIGATION OF THE J-INTEGRAL VALUE IN A DOUBLE CANTILEVER BEAM SAMPLE WITH AN EXTREMELY THIN ELASTIC-PLASTIC ADHESION LAYER

The loading of a double cantilever beam sample, which is a composition of cantilevers and an adhesive layer, is considered, which leads to deformation of the layer by normal rupture. The basis of the mathematical model describing the equilibrium of the composite is the variational equilibrium equation connecting the stresses averaged over the thickness of the layer and the stress-strain state of the cantilever by a linear parameter. The behavior of the cantilever is considered within the framework of a linearly elastic body, the adhesive layer is considered elastic-plastic without hardening. The composite is considered in a state of plane deformation. Assuming the fulfillment of the condition of complete plasticity of the equality of two principal stresses acting orthogonally to separation and elastic volumetric deformation, the constitutive relations of the behavior of the layer at the stage of elastoplastic deformation are obtained. Using the distribution of the displacement field according to the theory of Mindlin-Reisner plates, a system of ordinary differential equations, conjugation conditions for solutions, and boundary conditions are obtained from the variational formulation of the problem. It is shown that, depending on the ratio of the linear parameter and the height of the console, the characteristic equation of the corresponding homogeneous system of differential equations has both real and complex roots. The corresponding general solutions of the system are found. Based on the consideration of one region of elastoplastic deformations, an iterative procedure for finding the length of the plastic zone is proposed, taking into account the satisfaction of boundary conditions. From the solution of the problem for a given value of the linear parameter and a critical external load, the stress-strain state of the layer is found, on the basis of which the J-integral is calculated. It is shown that as the linear parameter tends to zero, the J-integral asymptotically tends to the value found in the framework of the linearly elastic behavior of the layer material. For a certain range of values of the linear parameter, the obtained solution was compared with the finite element solution of the 2D problem.

МОДЕЛЬ ДВОМАСОВОЇ КОЛИВНОЇ СИСТЕМИ З ПАРАЛЕЛЬНИМИ ПРУЖНИМ І ДЕМПФУЮЧИМ ЕЛЕМЕНТАМИ

Aim. Development of a method of analytical research of a two-mass oscillating system with parallel elastic and damping elements, which makes it possible to expand the design of such systems in various tasks of the functioning of machines and equipment. Method. We will conduct a parametric study of the dynamic oscillation system to assess the effect of the elasticity coefficient and damping on the change in the natural frequency, using the Laplace transform method. A mathematical model of the system with two masses connected by elastic and damping elements placed in parallel is presented. Disturbances are transmitted to the masses through an elastic-damping system. Results. The solution of the system of differential equations through the Laplace transform for two variants of the characteristic equation is given. The biquadratic characteristic equation is solved by the Ferrari method. Both for the roots with complex numbers and for the roots with real numbers, the values of the roots λ1 ... λi are obtained. The value of the roots of the biquadratic equation for the general solution of a homogeneous system of differential equations enables stability for the values of the six roots of the characteristic equation. Unlike previous studies, where applied disturbances were described in the form of harmonic oscillations, we proposed a solution for disturbances in the form of discrete single pulses of different durations and different tracking frequencies, from one pulse to n pulses. Scientific novelty. The analytical solution of the system of differential equations that describe the mutual oscillation of masses connected in parallel by elastic and viscous elements refers to a subspecies of the classical Kelvin-Voigt body. In comparison with other mathematical models, the analytical solution of the system of differential equations describing the movement of the masses of the system will allow to study the system with structural parameters in a wide numerical range of their numerical values. Practical value. The analytical model allows modeling technical systems that work according to this principle. For example, suspensions of cars and other vehicles for various purposes. The mathematical model is analytically solved and allows to optimize suspension designs

Побудова асимптотики розв’язку для системи сингулярно збурених рівнянь методом істотно особливих функцій

Singularly perturbed problems with turning points arise as mathematical models for various physical phenomena. The internal turning point problem is a one-dimensional version of the steady-state convection-diffusion problem with a dominant convective term and a velocity field that changes sign in the reservoir. Boundary turning point problems, on the other hand, arise in geophysics and in the modeling of thermal boundary layers in laminar flow. The paper analyzes the results from the asymptotic analysis of singularly perturbed problems with turning points. For a homogeneous system of singularly perturbed differential equations with a small parameter at the highest derivative and a turning point, the conditions for constructing a uniform asymptotic solution are obtained. We consider the case when the spectrum of the limit operator contains multiple and identically zero elements. The asymptotics are constructed by the method of essentially singular functions, which allows using the Airy model operator in the vicinity of the turning point. The construction of asymptotic solutions contains arbitrary constants, which are determined uniquely during the solution of the iterative equations. At the same time, the conditions for the existence of a solution of a system of differentials with a small parameter for the highest derivative and for the presence of a turning point are obtained, provided that the turning point is located on the interval [0; l]. An example of constructing the asymptotic of a homogeneous system of differential equations is given.

Representation Formula for General Solution of a Homogeneous System of Differential Equations

We consider the stationary oscillation case of the theory of linear thermoelasticity with microtemperatures of materials. The representation formula of a general solution of the homogeneous system of differential equations obtained in the paper is expressed by means of seven metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrate applications of these formulas to the Dirichlet- and Neumann-type boundary-value problems for a ball. Uniqueness theorems are proved. We construct explicit solutions in the form of absolutely and uniformly convergent series.

Explicit Solutions of Boundary-Value Problems of Thermoelasticity with Microtemperatures for a Half-Space

We consider the statics case of the theory of linear thermoelasticity with microtemperatures materials. The representation formula of a general solution of the homogeneous system of differential equations obtained in this paper is expressed by means of four harmonic and three metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrare an application of these formulas to the III type boundary value problem for a half-space. Uniqueness theorems are proved. Solutions are obtained in quadratures.

Lyapunov Equivalence of Systems with Unbounded Coefficients

It is shown that any linear system of homogeneous differential equations is Lyapunov equivalent to a system of the same order with piecewise constant coefficients, while a system with a uniformly small perturbation is Lyapunov equivalent to the same system with a piecewise constant perturbation of the same small magnitude.

Bohl Exponents and Baire Classes of Functions

Bohl exponents of a linear homogeneous system of differential equations are considered as functions of a parameter on its right-hand side which continuously depends on this parameter. It is shown that the Bohl exponents can be of the second or the fourth Baire class, according to the type of exponent. A similar result is established for Bohl exponents of diffeomorphisms of a Riemannian manifold.

Macroscopic model of diamond crystal growth

Elementary atomic structures of carbon are selected via the deposition of methyl radicals from the gas phase, and a kinetic scheme of the diamond crystal growth process is proposed. A macroscopic model of the growth process is developed in an average field approximation that corresponds to the kinetic scheme. The model is based on a system of homogeneous differential equations of high dimensionality that describe changes in the surface concentrations of elementary carbon structures in every crystal layer. Experimental data are in good agreement with results from simulations. The influence of different parameters and elementary stages of the process on the dynamics of crystal growth is investigated. The rate of one of the most important stages, downward migration, is determined.

Basic Boundary-Value Problems for a Solid Body with Double Porosity and Two Nonintersecting Spherical Cavities

In this paper, we examine the basic boundary-value problems for a three-dimensional space filled with a solid body having double porosity and two nonintersecting spherical cavities. We search for a solution of the problem by using the representation of a general solution of a system of homogeneous differential equations of statics, which is expressed in terms of four harmonic and one metaharmonic functions. The solution of the problem is reduced to the investigation of an infinite system of linear algebraic equations. It is shown that the obtained system is of normal type. Solutions of the considered problems are obtained in the form of absolutely and uniformly convergent series.

Boundary-Value Problems of Statics in the Two-Temperature Elastic Mixture Theory for a Half-Space

The static case of the two-temperature elastic mixture theory is considered when partial displacements of the elastic components of the mixture are equal to each other. The formula obtained for the representation of a general solution of a homogeneous system of differential equations is expressed in terms of four harmonic functions and one metaharmonic function. The uniqueness theorem for a solution is proved. Solutions are obtained in quadratures by means of boundary functions.

Boundary value problems of stationary oscillation of thermoelasticity of microstretch materials with microtemperatures

Abstract We consider the stationary oscillation case of the theory of linear thermoelasticity with microtemperatures of microstretch materials. A representation formula of a general solution of a homogeneous system of differential equations is written in terms of eight metaharmonic functions. Such formulas are very convenient and useful in many specific problems of concrete geometry. We demonstrate an application of these formulas to Dirichlet and Neumann type boundary value problems in a ball. Explicit solutions in the form of absolutely and uniformly convergent series are constructed.

Построение интерактивной обучающей модели метода решения нормальной однородной системы дифференциальных уравнений n-го порядка

Построение интерактивной обучающей модели метода решения нормальной однородной системы дифференциальных уравнений n-го порядка

Existence of unbounded solutions of a linear homogenous system of differential equations with two delays

Behavior of solutions of a linear homogeneous system of differential equations with deviating arguments in the form \begin{equation*} \dot y(t)=\beta(t)\left[y(t-\delta)-y(t-\tau)\right] \end{equation*} is discussed for $t\to\infty$. It is assumed that $y$ is an $n$-dimensional column vector, $n\geq 1$ is an integer, $\delta,\tau\in{\mathbb{R}}$, $\tau>\delta>0$, and $\beta(t)$ is an $n\times n$ matrix defined for $t\geq t_{0}$, $t_{0}\in\mathbb{R}$, and such that its elements are nonnegative, continuous functions and in every row of this matrix at least one element is nonzero. The existence of solutions in an exponential form under certain assumptions is proved. Sufficient conditions for the existence of unbounded solutions and the estimations for a solution are derived. A comparison with the known results and an illustrative example are given.

On exponentially separated differential systems

Exponentially separated linear homogeneous system of ordinarydifferential equations with continuous limited coe¢ cients in critical cases ofLyapunov exponents is considered. The generalized exponentially separatedlinear system of differential equations with regard to a monotonically increasingfunction is defined.It is established that if a linear homogeneous systemof differential equations is generalized exponentially separated, Lyapunov’sgeneralized exponents are stable in a class of small perturbations

Mellin–Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions

We argue that the Mellin–Barnes representations of Feynman diagrams can be used for obtaining linear systems of homogeneous differential equations for the original Feynman diagrams with arbitrary powers of propagators without recourse to the integration-by-parts technique. These systems of differential equations can be used (i) for the differential reductions to sets of basic functions and (ii) for counting the numbers of master integrals.

On the asymptotic stability of solutions of nonlinear systems with delay

Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin’s approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.

Boundary value problems of statics of the elastic mixture theory

For a homogeneous system of differential equations of statics of the elastic mixture theory, we solve the basic boundary value problems using Papkovich–Neuber representations and Fourier integral transforms when the displacement vector or stress vector values are given on the boundary . The solution uniqueness theorem is proved for the problems posed. Solutions are obtained in quadratures.

Residual $${\bar\partial}$$ -cohomology and the complex Radon transform on subvarieties of $${\mathbb C P^n}$$

We show that the complex Radon transform realizes an isomorphism between the quotient-space of residual \({\bar\partial}\) -cohomologies of a locally complete intersection algebraic subvariety in a linearly concave domain of \({{{\mathbb C}}P^n}\) and the space of holomorphic solutions of the associated homogeneous system of differential equations with constant coefficients in the dual domain in \({({{\mathbb C}}P^n)^*}\) .

Lagrangian asymptotic behaviour of solutions of inhomogeneous systems of ordinary differential equations

The problem under consideration concerns when a system of ordinary differential equations reducible to a weakly non-linear system has solutions with the same asymptotic behaviour as solutions of the corresponding homogeneous system. The existence and uniqueness of global solutions to the inhomogeneous system is established in the form of a solution to a homogeneous system of differential equations, in the case when asymptotic initial data is prescribed at the singular points of these systems.