System Of Second-order Differential Equations Research Articles

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

Jacobi Stability for T-System

In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the Oz-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point.

Substantiation of the Theory of Forces Calculating in a Steel Winch Cable for Lifting Seed Material to the Upper Tiers of Hydroponic Installations

Hydroponic green fodder (HGF) is a product containing a full range of required nutrients and vitamins. To implement this technology, it is necessary to develop not only basic hydroponic installations, but also additional devices that reduce the labor intensity of production of HGF. The purpose of the research is the theoretical substantiation of the dynamics of the system “winch – steel cable of variable length – end load. The conclusion and solution of a system of second-order differential equations with variable dynamics coefficients of elements of a lifting installation (winch) are given. In addition, dependencies have been compiled to determine the elongation and force in the lifting cable.

Aeolian Liquid Vibrations in Conical Tanks with Baffles under Wind Loading with Fuzzy Parameters

The Aeolian liquid vibrations in conical reservoirs caused by low-velocity, steady winds have been under consideration. Both amplitudes and dominant frequencies of wind loadings have been constantly changed, so to adequately describe the vibration process, fuzzy logic methods have been applied. At the first stage, the crisp initial value problem for conical shells with and without baffles has been considered. The liquid inside the reservoirs has been supposed to be an incompressible and ideal one, and its flow induced by the forced harmonic excitation, has been considered as potential. So, there exists a potential to satisfy the Laplace equation. The impermeable condition has been used at wetted surface boundaries of the shell, whereas the dynamic and kinematic boundary conditions have been set on the free liquid surface. A system of singular integral equations has been obtained for values of the velocity potential and the function describing the free surface rise. Its solution has been gained by boundary element methods. The crisp boundary value problem has been reduced to the second-order system of differential equations. After receiving the crisp solution of this system, the initial data have been fuzzified, involving triangular fuzzy numbers, and the fuzzy initial value problem has been formulated. The numerical solution to this problem with uncertain intervals involved has been obtained and analyzed.

Localization of Nash-type equilibria for systems with partial variational structure

In this paper, we aim to generalize an existing result by obtaining localized solutions within bounded convex sets, while also relaxing specific initial assumptions. To achieve this, we employ an iterative scheme that combines a fixed-point argument based on the Minty-Browder Theorem with a modified version of the Ekeland variational principle for bounded sets. An application to a system of second-order differential equations with Dirichlet boundary conditions is presented.

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. In this approach, we describe the evolution of the Oregonator model in geometric terms, by considering it as a geodesic in a Finsler space. We have found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. We have obtained the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points. Further, a comprehensive examination was conducted to compare the linear stability and Jacobi stability of the Oregonator model at its equilibrium points, and We highlight these instances with a few illustrative examples.

Parametric interaction of modes in the presence of quadratic or cubic nonlinearity

The purpose of this work is a study of the dynamics of the systems of ordinary differential equations of the second order, which is obtained using the Lagrange formalism. These systems describe the parametric interaction of oscillators (modes) in the presence of a general quadratic or cubic nonlinearity. Also, we compare the dynamics of the systems of ordinary differential equations of the second order and dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models in order to determine the possibilities of the latter models when modeling coupled oscillators of the above type. Methods. The study is based on the numerical solution using the methods of the theory of the obtained analytically differential equations. Results. For both systems of second-order differential equations, is was presented a chart of in the parameter plane, a graphs of Lyapunov exponents at the value of the parameter that specifies the dissipation of oscillators, a time dependences of the generalized coordinates of oscillators and its amplitudes, portraits of attractors, a projection of the attractors on a phase planes of oscillators. A comparison with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models is carried out. These models are three-dimensional real approximations of the above systems obtained by the method of slowly varying amplitudes. Conclusion. The study of the constructed systems showed that in the parameter space there are regions corresponding to both various regular regimes, such as the equilibrium position, limit cycle, two-frequency tori, and chaotic regimes. For both systems, it was shown that the transition to chaos occurs as a result of a sequence of period doubling bifurcations of the tori. In addition, a comparison of the dynamics of the constructed systems with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models allows us to assert that if the Vyshkind–Rabinovich model predicts the dynamics of the corresponding initial system well enough, then the Rabinovich–Fabrikant model does not have such a property.

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters.

Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system.

Positive solutions for boundary value problems of a class of second-order differential equation system

Abstract This article discusses the existence of positive solutions for the system of second-order ordinary differential equation boundary value problems − u ″ ( t ) = f ( t , u ( t ) , v ( t ) , u ′ ( t ) ) , t ∈ [ 0 , 1 ] , − v ″ ( t ) = g ( t , u ( t ) , v ( t ) , v ′ ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , \left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(t)=f\left(t,u\left(t),v\left(t),u^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ -{v}^{^{\prime\prime} }\left(t)=g\left(t,u\left(t),v\left(t),v^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ u\left(0)=u\left(1)=0,\hspace{1em}v\left(0)=v\left(1)=0,\end{array}\right. where f , g : [ 0 , 1 ] × R + × R + × R → R + f,g:\left[0,1]\times {{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{+}\times {\mathbb{R}}\to {{\mathbb{R}}}^{+} are continuous. Under the related conditions that the nonlinear terms f ( t , x , y , p ) f\left(t,x,y,p) and g ( t , x , y , q ) g\left(t,x,y,q) may be super-linear growth or sub-linear growth on x , y , p x,y,p , and q q , we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition ( F 3 ) \left(F3) is presented to restrict the growth of f ( t , x , y , p ) f\left(t,x,y,p) and g ( t , x , y , q ) g\left(t,x,y,q) on p p and q q . The super-linear growth or sub-linear growth of the nonlinear terms f f and g g is described by related inequality conditions instead of the usual independent inequality conditions about f f and g g . The discussion is based on the fixed point index theory in cones.

Sundman transformation and alternative tangent structures

A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity of a change of the tangent structure by means of the function defining the Sundman transformation is shown. Among other applications of such theory we study the linearisability of a system of second-order differential equations and in particular the simplest case of a second-order differential equation. The theory is illustrated with several examples.

Railway Track Ballast Layer Stiffness Dynamic Control Elastic-Linear Model

The paper presents the results of mathematical modeling of the shock-elastic interaction process in the dynamic densimeter for dynamic control of the ballast layer of the railroad track. The installation is used to determine the compaction level and to estimate the carrying capacity of the ballast layer during the inter-repair period of tampering, renewal and track stabilization operations. The effect of exposure on the controlled elastic base is considered within a four-element mathematical model. The time intervals of interaction between the model elements are revealed. Each time interval is described by a system of second-order differential equations, for which analytical solutions were obtained. The model verification was evaluated during tests with a dynamic control unit consisting of a loading plate, a falling weight, and an elastic-damping element located between the weight and the loading plate. Tests were conducted on a polyurethane surface with known stiffness. In the process, signals of vertical acceleration of the loading plate when the load strikes the elastic-damping element were recorded. The acceleration was recorded by an accelerometer sensor with a sampling rate of 42 kHz being connected to a computer with specialized software. The acceleration data were obtained for different values of the stiffness of the elastic-damping element in the installation. To evaluate the simulation results, the obtained signals were compared with the theoretical model by the correlation method. According to the results of modeling, the main amplitude and time informative parameters used to determine the degree of compaction of the base layer have been singled out. The dependence of the values of informative parameters on the installation parameters (load weight, loading plate weight, stiffness of an elastic-damping element) in conditions of uncertainty was studied. The obtained dependences are used to estimate the error of each parameter in calculating the rigidity of the base. Informative parameters have been determined that allow you to determine the stiffness of the base with an error of not more than 7%. A method for selecting the optimal characteristics of the unit design and informative parameters to ensure minimum errors has been developed.

Coupled systems with Ambrosetti-Prodi-type differential equations

<abstract><p>In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters.</p></abstract>

Аффинные преобразования касательного расслоения общего пространства путей

In this paper, we study infinitesimal transformations of the tangent bundle of a common path space. The general path space is a genera­liza­tion space of the affine connectivity. By affine connectivity of the com­mon path space, we construct an affine connection on the tangent bundle. For the infinitesimal transformation of the tangent bundle, a system of in­va­riance equations for the constructed affine connectivity is compiled. This system is a system of second-order differential equations with res­pect to the components of the infinitesimal transformation. The main re­sults of the article are obtained by analyzing this system taking into account the properties of homogeneous functions. It is shown that the comp­lete lift of an infinitesimal transformation of base is an infinitesimal af­fine motion of a tangent bundle if and only if the infinitesimal transfor­ma­tion of base is an affine motion in the general path space. Necessary and sufficient conditions are found that the infinitesimal transformation of a tangent bundle generated by a vertical vector field leaves the affine connec­ti­vity of the tangent bundle invariant. Conditions are given that are neces­sa­ry and sufficient so that the infinitesimal transformation of a tangent bundle with affine connectivity that preserves layers is an affine motion.

‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain

A formally selfadjoint system of second-order differential equations is considered in a three-dimensional domain on small parts of whose boundary an analogue of Steklov spectral conditions is set, while the Neumann boundary conditions are set on the rest of the boundary. Under certain algebraic and geometric conditions an asymptotic expression for the eigenvalues of this problem is presented and a limiting problem is put together, which produces the leading asymptotic terms and involves systems of integro-differential equations in half-spaces, interconnected by means of certain integral characteristics of vector-valued eigenfunctions. One example of a concrete problem in mathematical physics describes surface waves in several ice holes made in the ice cover of a water basin, and the asymptotic formula for eigenfrequencies shows that the local wave processes interact independently of the distance between the holes. Another series of applied problems relates to elastic fixings of bodies along small pieces of their surfaces. Possible generalizations are discussed; a number of related open questions are stated. Bibliography: 41 titles.

A Crank–Nicolson Compact Difference Method for Time-Fractional Damped Plate Vibration Equations

This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats with fourth-order space precision and 3−α(1<α<2)-order time precision. We apply the energy method to analyze the stability and convergence of this difference format. We provide numerical cases, which not only validate the convergence order and feasibility of the given difference format, but also simulate the influence of the damping coefficient on the amplitude of plate vibration.

Thermal Model of a Parabolic Trough Solar Field With a Closed-Loop Operation During Sunrise Period

Abstract This work presents a numerical simulation of a thermal model for a solar loop with parabolic trough collectors (PTCs) considering fluid recirculation at closed-loop (CL) operation during sunrise. At the beginning of the day, the heat transfer fluid (HTF) is recirculated in a CL to obtain the inlet loop operating temperature without resorting to additional preheating energy. Energy balances are carried out on the HTF, the absorber tube, and the glass envelope as a function of optical and thermo-physical parameters of the heat collector element (HCE). A system of second-order differential equations was established, and mathematical model was resolved by finite difference and Newton–Raphson methods for solution. This model has been well validated by comparing the results with the existing experimental and numerical data. Three typical days of winter, spring, and summer were simulated for the solar loop operation considering a CL fluid recirculation at start-up conditions. Results show a more flexible CL operation at relatively large flowrates compared to the open-loop (OL) operation, which requires substantial preheating energy at the same conditions; the start-up solar field using the CL strategy allows us both operational autonomy and significant energy savings. Solar loop thermal and optical powers gained and lost are plotted for the typical days considered; we observe that maximum thermal efficiency of 66.53% is achieved at 2.27 p.m. for the summer day.

Определение оптимальной силы и скорости копания грунта одноковшового гидравлического экскаватора с ковшом сферической формы

Introduction. The paper describes the scope and purpose of single-bucket hydraulic excavators. The method of digging a single-bucket excavator with a bucket of spherical shape, which allows to increase productivity, is considered. The urgency of increasing the productivity of these machines is formulated. The purpose of the work: to determine the optimal strength and speed of digging the soil of a single-bucket hydraulic excavator with a spherical bucket. Research methodology. To confirm the effectiveness of the proposed bucket design, a diagram of the forces acting on it during digging is presented. The rheological model of the process of digging with a bucket of spherical shape is considered. A mathematical model in the form of a system of second-order differential equations has been compiled on the basis of the force scheme and the rheological model. Research results. A mathematical model has been identified that makes it possible to calculate the dependence of the speed and optimal force when digging the soil with a spherical bucket of a single-bucket hydraulic excavator, depending on the design parameters of the bucket and the parameters of the soil. The dependence for determining the friction force of the soil inside the bucket arising during digging is given. The dependencies for determining the weight of the soil in the bucket and the bucket itself are described. The dependence for calculating the lateral pressure of the soil using a stabilometer is given. The dependences of the calculation of the friction force of the external soil on the bucket and the friction force of the external soil on the soil in front of the bucket edge are presented. A dependence for determining the normal force from the external soil to the compacted soil in front of the bucket edge is proposed. The concept of a "compacted core of the soil" in front of the bucket edge is given and the dependence for determining its depth is given, however, a more precise definition of the latter requires additional research. A dependence is proposed for determining the mass of soil located in front of the bucket edge. The conditions for the possibility of digging the soil are accepted, in which the digging speed is greater than zero, and the digging force is minimal. An algorithm is described that allows you to select the minimum force for the digging process based on the dependencies given in the work. A block diagram of the algorithm for selecting the minimum force for the process of digging with a spherical bucket is given. The selection of the minimum digging force will confirm the effectiveness of a single-bucket hydraulic excavator with a spherical bucket.

Mathematical model of soil digging spherical excavator bucket

Introduction. The article presents a new design of a bucket of a single bucket hydraulic excavator, which allows you to increase the performance of these machines. Increased efficiency is achieved due to the installation of buckets of larger volume without changing characteristics of hydraulic drive. The purpose of the presented work is to select the volume of the spherical ladle for the digging process. The task of research is to obtain the dependencies of ladle and soil speeds on the parameters of the digging process. Selecting the volume of the spherical ladle will determine how much it can be increased compared to the volume of the serial ladle to carry out the digging process.Materials and methods. The process of digging a new bucket was considered in the form of a two-scale KelvinVoigt rheology model. This model was described by a system of second-order differential equations. The system solved the Cauchy problem, which made it possible to determine the general solution of the equation satisfying the two equations of the system. As a result, derivatives from ladle and soil movements were determined, which are nothing more than ladle and soil speeds.Results. The solution of the above system made it possible to obtain dependencies of ladle speeds and soil masses on parameters of the digging process. The digging process is possible at positive speed values. When substituting depending on the process parameters under the condition of velocity positivity, you can determine the volume of the spherical ladle.Practical significance. To simplify and put into practice the obtained dependencies, an algorithm was compiled to calculate the volume of the spherical ladle. The presented algorithm can be implemented in a computer program.

ИСПОЛЬЗОВАНИЕ PDE TOOLBOX ДЛЯ РЕШЕНИЯ ПЛОСКОЙ ЗАДАЧИ ТЕОРИИ УПРУГОСТИ

A PDE Toolbox for solving a system of second-order differential equations is considered. This package implements two model problems with a known exact solution describing the bending of a thin plate with different conditions for fixing the edges of the plate. Graphical representations of solutions and errors of these problems are given.