System Of First-order Differential Equations Research Articles

Spin and quadrupolar effects in the secular evolution of precessing compact binaries with black hole, neutron star, gravastar, or boson star components

We discuss precessing compact binaries on eccentric orbit with gravastar, black hole, neutron star, or boson star components. We derive secular evolution equations to second post-Newtonian--order accuracy, with leading-order spin-orbit, spin-spin, and mass quadrupole-monopole contributions. The emerging closed system of first-order differential equations evolves the pairs of polar and azimuthal angles of the spin and orbital angular momentum vectors together with the periastron angle. The secular dynamics is autonomous. We confirm numerically, that secular evolutions look like smoothed-out instantaneous evolutions over timescales where radiation reaction is negligible. The secular evolution of the spin polar angles and the difference of their azimuthal angles generates a closed subsystem. We study analytically this system for the particular cases of one spin dominating over the other and for black hole - boson star binaries of equal masses. In the first case known large flip-flops of the smaller spin are reproduced, when the larger spin is almost coplanar with the orbit. We also find new, quadrupole-induced flip-flops occurring when the neutron star with dominant spin has a quadrupolar parameter $w_1\approx 3$. Finally we analyze the evolutions of the spin angles numerically by comparing the cases when the black hole companion is either a gravastar, another black hole or a boson star with identical mass. We find that both the amplitude and period of the flip-flop are the largest, when the companion is a black hole. In the case of a boson star companion the frequency of the flip-flop increases significantly. Further, while in the case of gravastars and black holes a swinging-type azimuthal evolution occurs, with the spins of the components periodically overpassing each other, their sequence is conserved when the companion is a boson star.

PHƯƠNG PHÁP LẶP SONG SONG RUNGE-KUTTA HAI BƯỚC

In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

A discontinuous finite element approximation to singular Lane-Emden type equations

In this article, we develop the local discontinuous Galerkin finite element method for the numerical approximations of a class of singular second-order ordinary differential equations known as the Lane-Emden type equations equipped with initial or boundary conditions. These equations have been considered via different models that naturally appear for example in several phenomena in astrophysical science. By converting the governing equations into a first-order systems of differential equations, the approximate solution is sought in a piecewise discontinuous polynomial space while the natural upwind fluxes are used at element interfaces. The existence-uniqueness of the weak formulation is provided and the numerical stability of the method in the L∞ norm is established. Five illustrative test problems are given to demonstrate the applicability and validity of the scheme. Comparisons between the numerical results of the proposed method with existing results are carried out in order to show that the new approximation algorithm provides accurate solutions even near the singular point.

Closed form solutions for an anisotropic composite beam on a two-parameter elastic foundation

Beams resting on elastic foundations are widely used in engineering design such as railroad tracks, pipelines, bridge decks, and automobile frames. Laminated composite beams can be tailored for specific design requirements and offer a desirable design framework for beams resting on elastic foundations. Therefore, the analysis of flexural behaviour of laminated composite beams on elastic foundations is of important consequence. Exact solutions for flexural deflection of composite beams with coupling terms between stretching, shearing, bending and twisting, resting on two-parameter elastic foundations for various types of loading and boundary conditions, are presented for the first time. The proposed new formulation is based on Euler–Bernoulli beam theory having four degrees of freedom, namely bending in two principal directions, axial elongation and twist. Governing equations and boundary conditions are derived from the principle of virtual work and expressed in a compact matrix–vector form. By decoupling bending in both principal directions from twist and axial elongation, the fourth-order differential equation for bending is derived and transformed into a system of first-order differential equations. An exact solution of this system of equations is obtained using a fundamental matrix approach. Fundamental matrices for different configurations of elastic foundation are provided. The ability of the presented mathematical model in predicting flexural behaviour of beams on elastic foundations is verified numerically by comparison with results available in the literature. In addition, the deflection of anisotropic beams is analysed for different types of stacking sequences, boundary and loading conditions. The effect of elastic foundation coefficients on the flexural behaviour is also investigated and discussed.

RESEARCH OF SENSITIVITY TO EXTERNAL PARAMETERS OF THE LEARNING PROCESS WITH THE HELP OF ERGONOMICS IN THE ACQUISITION OF KNOWLEDGE

This research work is devoted to the development and creation of a mathematical model of the learning process based on ergonomics, to study the quality of education in the acquisition of knowledge. And also one of the types of qualitative indicators of the theory of sensitivityand roughness to the used parameters of different types of technical means is proposed, the process, which is described by a system of first-order differential equations, and its function is determined using partial derivatives.Key words:ergonomics, mathematical modeling of the learning process, quality of teaching of educational systems, mathematical research methods, technical teaching aids

An elastoplastic solution to undrained expansion of a cylindrical cavity in SANICLAY under plane stress condition

Although cylindrical cavity expansion under plane stress condition is commonly encountered in geotechnical problems, most currently available solutions have been developed for cavity expansion under plane strain condition. This paper develops a novel elastoplastic solution for undrained expansion of a cylindrical cavity in SANICLAY under plane stress condition. The SANICLAY model, which could well represent the mechanical behaviour of the anisotropic soil and overconsolidated soil, is employed in the present solution to model the responses of the soil around the expanded cavity. The problem is formulated as a system of first-order differential equations with the unknown variables as the functions of an auxiliary coordinate, which are solved as an initial value problem. The expansion responses under plane stress condition are comprehensively compared with those under plane stress condition to highlight the unique expansion responses under plane stress condition. The results show that the present solution could well reflect the unique expansion responses under plane stress condition, which are totally different from those under plane strain condition. It is expected the proposed solution could provide a reasonable approach to interpret the pressuremeter test and model pile installation effects near the surface of the natural anisotropic clays.

Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones.

A studyof vibrations of a cylinder with a viscoelastic coating

The paper presents the problem of steady-state longitudinal-radial vibrations of an elastic hollow cylinder with a viscoelastic coating. The properties of the coating change along the radial coordinate only and are described by the variable Lame parameters and density. The ends of the cylinder are under sliding interface conditions, and the outer side surface is subject to periodic loads. Within the framework of the model of a standard viscoelastic body and following the principle of correspondence, the variable Lame parameters are replaced by the complex functions of the radial coordinate and vibration frequency. The solution was obtained using two approaches. Within the framework of the first approach, the solution is constructed using the method of separation of variables and is reduced to solving a set of boundary value problems for canonical systems of first-order differential equations with variable coefficients. The solution to each of these problems is obtained numerically using the shooting method. The second approach is based on the finite element method implemented in the FlexPDE package. We compared the obtained solutions for the given laws of variation of the Lame parameters and a fixed frequency by analyzing the graphs of the real and imaginary parts of radial displacement and stress components. The convergence of the solution found using the finite element method is shown as a function of the number of nodes for the values of the radial displacement functions measured at three points. The graphs of the amplitude-frequency characteristics measured on the outer surface are plotted for different values of the relaxation time. The effect of variable properties of the coating on the displacement function is estimated. The advantages of each approach are described, and the areas of their practical application are revealed.

A semi-analytical solution to spherical cavity expansion in unsaturated soils

This paper presents a rigorous solution for spherical cavity expansion in unsaturated soils under constant suction condition. The hydraulic behavior that describes the saturation-suction relationship is modeled by a void ratio-dependent soil-water characteristic curve, which allows the hydraulic behavior to fully couple with the mechanical behavior that is described by an extended critical state soil model for unsaturated soil through the specific volume. Considering the boundary condition and introducing an auxiliary coordinate, the problem is formulated to a system of first-order differential equations with three principal stress components and suction as basic unknowns, which is solved as an initial value problem. Parameter analyses are conducted to investigate the effects of suction and the overconsolidation ratio on the overall expansion responses, including the pressure-expansion response, the distribution of the stress components around the cavity, and the stress path of the soil during cavity expansion. The results reveal that the expansion pressures and the distribution of the stress components in unsaturated soils are generally higher than those in saturated soils due to the existence of suction.

AUTOMORPHISM OF SOLUTIONS TO RAMANUJAN'S DIFFERENTIAL EQUATIONS AND OTHER RESULTS

In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. We also investigate possible applications of the result. In part two we prove a similar theorem about the automorphism of solutions to the first-order system of differential equations associated to the generalised Chazy equation with parameter $k=\frac{3}{2}$.

Numerical Investigation on System of Ordinary Differential Equations Absolute Time Inference with Mathematica®

The purpose of this research is to perform a comparative numerical analysis of an efficient numerical methods for second-order ordinary differential equations, by reducing the second-order ODE to a system of first-order differential equations. Then we obtain approximate solutions to the system of ODE. To validate the accuracy of the algorithm, a comparison between Euler’s method and the Runge-Kutta method or order four was carried out and an exact solution was found to verify the efficiency, accuracy of the methods. Graphical representations of the parametric plots were also presented. Time inference analysis is taken to check the time taken to executes the algorithm in Mathematica®12.2.0. The obtained approximate solution using the algorithm shows that the Runge-Kutta method of order four is more efficient for solving system of linear ordinary differential equations.

Undrained cylindrical cavity expansion in modified cam-clay soil: A semi-analytical solution considering biaxial in-situ stresses

This paper presents a semi-analytical solution for undrained cylindrical cavity expansion in modified Cam-clay soils under biaxial in-situ stresses based on the small strain and large strain assumptions in the elastic and plastic regions, respectively. The cylindrical cavity expansion problem is treated as a boundary value problem and formulated by solving a system of first-order differential equations with four stress components as basic unknown variables. The validity of the proposed solution is examined by comparing with the existing solution, where the in-situ stress is uniform. Extensive parametric studies are conducted to investigate the effects of biaxial in-situ stresses on the stress distribution, the expansion process and the shape and size of elastic-plastic boundary. The results indicate that the elastic-plastic boundary is in the shape of an ellipse due to the shear stress induced by biaxial in-situ stresses. The major axis of the elliptical elastic-plastic boundary is in accordance with the direction of the larger in-situ stress. The present solution is free of the limitation that the cavity expands under the uniform in-situ stress, therefore it is expected to provide a more general method for practical geotechnical and petroleum problems.

A Differential Game Problem of Many Pursuers and One Evader in the Hilbert Space $$\ell_2$$

In this paper, we investigate a differential game problem of multiple number of pursuers and a single evader with motions governed by a certain system of first-order differential equations. The problem is formulated in the Hilbert space $$\ell_2,$$ with control functions of players subject to integral constraints. Avoidance of contact is guaranteed if the geometric position of the evader and that of any of the pursuers fails to coincide for all time t. On the other hand, pursuit is said to be completed if the geometric position of at least one of the pursuers coincides with that of the evader. We obtain sufficient conditions that guarantees avoidance of contact and construct evader’s strategy. Moreover, we prove completion of pursuit subject to some sufficient conditions. Finally, we demonstrate our results with some illustrative examples.

Calculation of the reliability of frame systems under seismic impact and survivability assessment during the formation of plastic hinges in the nodes

Aim . The research aims to develop mathematical models for evaluating the reliability of two-story frame systems under the seismic impact, covering all possible ways of transition to the mechanism state due to the formation of plastic hinges in the frame nodes. Methods . The study is based on the application of the theory of Markov random processes in continuous time and their discrete states to describe the transition of a system into a different state. Such states include faults resulting from the formation of plastic hinges in the frame nodes. Results . Possible schemes of forming plastic hinges in two types of statically indeterminate frames under horizontal seismic impact are described: in the frame of a 2-story industrial building with a large-span upper floor and with the support of the roof truss on the heads of columns and a two-story two-span frame of a simple regular structure with rigid nodes. Possible states are noted and transition graphs are compiled, which are typical for the frames under consideration in the case of rigid disks of floor slabs during seismic impact, represented as a stationary random process. By representing the transition of systems to the states under consideration caused by the formation of plastic hinges in one or another sequence under seismic action in the form of a Markov random process in continuous time and discrete states, systems of first-order differential equations describing these transitions are obtained. Having solved them by applying the Laplace transform, analytical expressions for estimating the probabilities of finding frame systems in the states under consideration and expressions for estimating the probabilities of failure-free operation of these systems are obtained. Conclusion . By constructing transition graphs covering all possible states of frame systems during the formation of plastic hinges in nodes under seismic action and describing the process of frame transitions from one state to another by a Markov random process in continuous time and discrete states, analytical expressions for evaluating the probabilities of these system states, as well as their reliability and survivability under seismic action, can be obtained.

Out-of-plane stability of concrete-filled steel tubular arches at elevated temperatures

This paper investigates the flexural-torsional buckling behaviour of concrete-filled steel tubular circular arches under mechanical and thermal loading. A thermo-elastic pre-buckling analysis is first conducted by employing the principle of virtual work to derive the non-linear equations of equilibrium. The governing geometrical, equilibrium and constitutive material relations are numerically solved as a system of first-order differential equations with boundary conditions of pinned or fixed ends. The prebuckling analysis is then generalised to consider basic creep strain which is found to have a negligible impact on the prebuckling response under short-term heating. Subsequently, an elastic out-of-plane buckling analysis is performed using energy methods and the influence of thermal loading on buckling loads is examined. The results show that stability boundaries decrease with an increase in thermal loading, and that the rate of reduction is independent of the type of end-supports. Additionally, a Finite Element (FE) model is developed to analyse the inelastic lateral buckling strength of CFST arches under both uniform thermal and fire loading. The FE analysis is validated by comparison to the numerical method derived herein for the elastic buckling analysis.

MODELLING OF DYNAMIC MODES OF AN INDUCTION ELECTRIC DRIVE AT PERIODIC LOAD

Goal. Development of methods and mathematical models, based on them, for the calculation of transients and steady-state modes of induction electric drives operating in periodic load mode. Methodology. The developed algorithms are based on a mathematical model of an induction motor, which takes into account the saturation of the magnetic core and the displacement of current in the rotor bars. The processes are described by a system of nonlinear differential equations in the orthogonal axes x, y, which enables the results to be obtained with the smallest amount of calculations. The magnetization characteristics by the main magnetic flux and the leakage fluxes are used to calculate the electromagnetic parameters of the motor. To account for the current displacement in the rotor bars, the short-circuited winding is considered as a multilayer structure formed by dividing the bars in height by several elements. Results. Due to the variable load on the motor shaft, electromagnetic processes in both transient and steady state modes of the electric drive in any coordinate system are described by a system of nonlinear differential equations. The result of the calculation of the transients is obtained as a result of their integration time dependencies of coordinates (currents, electromagnetic torque, etc.) at a given law of change of the moment of loading. The proposed method of calculating steady-state mode is based on algebraization of differential equations on the mesh of nodes of the process cyclicity period and allows to obtain periodic dependencies in the time domain. Originality. The problem of calculating a steady-state periodic mode is solved as a boundary problem for a system of first-order differential equations with periodic boundary conditions, which allows to obtain instantaneous dependences during the period of currents, electromagnetic torque, capacities and other coordinates. Practical significance. Using the developed algorithm, it is possible to calculate the static characteristics of periodic processes as dependencies on different parameters of the cycle of periodic load or other coordinates, which is the basis for the choice of the motor for overload, power, heating, etc., as well as to detect the possibility of resonance.

Modelling of Dynamic Modes of An Induction Electric Drive at Periodic Load

Goal: Development of methods and mathematical models, based on them, for the calculation of transients and steady-state modes of induction electric drives operating in periodic load mode. Methodology: The developed algorithms are based on a mathematical model of an induction motor, which takes into account the saturation of the magnetic core and the displacement of current in the rotor bars. The processes are described by a system of nonlinear differential equations in the orthogonal axes x, y, which enables the results to be obtained with the smallest amount of calculations. The magnetization characteristics by the main magnetic flux and the leakage fluxes are used to calculate the electromagnetic parameters of the motor. To account for the current displacement in the rotor bars, the short-circuited winding is considered as a multilayer structure formed by dividing the bars in height by several elements. Results: Due to the variable load on the motor shaft, electromagnetic processes in both transient and steady state modes of the electric drive in any coordinate system are described by a system of nonlinear differential equations. The result of the calculation of the transients is obtained as a result of their integration time dependencies of coordinates (currents, electromagnetic torque, etc.) at a given law of change of the moment of loading. The proposed method of calculating steady-state mode is based on algebraization of differential equations on the mesh of nodes of the process cyclicity period and allows to obtain periodic dependencies in the time domain. Originality: The problem of calculating a steady-state periodic mode is solved as a boundary problem for a system of first-order differential equations with periodic boundary conditions, which allows to obtain instantaneous dependences during the period of currents, electromagnetic torque, capacities and other coordinates. Practical significance: Using the developed algorithm, it is possible to calculate the static characteristics of periodic processes as dependencies on different parameters of the cycle of periodic load or other coordinates, which is the basis for the choice of the motor for overload, power, heating, etc., as well as to detect the possibility of resonance.

Fungal Tip Growth Arising Through a Codimension-1 Global Bifurcation

Tip growth is a growth stage which occurs in fungal cells. During tip growth, the cell exhibits continuous extreme lengthwise growth while its shape remains qualitatively the same. A model for single celled fungal tip growth is given by the Ballistic Aging Thin viscous Sheet (BATS) model, which consists of a five-dimensional system of first-order differential equations. The solutions of the BATS model that correspond to fungal tip growth arise through a codimension-1 global bifurcation in a two-parameter family of solutions. In this paper we derive a toy model from the BATS model. The toy model is given by two-dimensional system of first-order differential equations which depend on a single parameter. The main achievement of this paper is a proof that the toy model exhibits an analogue of the codimension-1 global bifurcation in the BATS model. An important ingredient of the proof is a topological method which enables the identification of the bifurcation points. Finally, we discuss how the proof may be generalized to the BATS model.

Elastoplastic solutions for cylindrical cavity expansion in unsaturated soils

Elastoplastic solutions are developed in this paper for cylindrical cavity expansion in unsaturated soils, aiming to provide a theoretical model to analyze the pressuremeter test and pile installation in unsaturated soils. The novelty of the solution lies in the subtle coupling of hydraulic behaviour and mechanical behaviour of the soil in the governing equations. An elastoplastic critical state model for unsaturated soils is used to derive the elastoplastic constitutive matrix for both the undrained (constant gravimetric water contents) and drained (constant suction) expansion of a cylindrical cavity in unsaturated soils, which are finally formulated to a system of first-order differential equations. The governing differential equations are solved as an initial value problem with the initial conditions determined by the yield function of the unsaturated soil model and continuous condition. Parametric studies are performed to investigate the expansion responses of the surrounding soil and particular attention is given to the responses at the cavity wall. The results indicate that suction has significant effects on both the mechanical and hydraulic responses of the surrounding soil.

On Traveling Waves in Compressible Euler Equations with Thermal Conductivity

Heat conduction plays an important role in fluid dynamics. However, the modeling of thermal conductivity involves higher order derivatives which causes a tough obstacle for the study of traveling waves. In this work, we propose a modified term for the thermal conductivity coefficient in viscous–capillary compressible Euler equations. By approximation, which is crucial in any mathematical modeling, the heat conduction may be assumed to depend only on the specific volume. Then, we can derive a $$2\times 2$$ system of first-order differential equations for traveling waves of the given model, whose equilibria can be shown to admit a stable–saddle connection for 1-shocks and a saddle–stable connection for 3-shocks. This establishes the existence of a traveling wave of the viscous–capillary Euler equations with the presence of a modified thermal conductivity effect.