Oscillation Of Equations Research Articles

Memory-friendly fixed-point iteration method for nonlinear surface mode oscillations of acoustically driven bubbles: from the perspective of high-performance GPU programming

A fixed-point iteration technique is presented to handle the implicit nature of the governing equations of nonlinear surface mode oscillations of acoustically excited microbubbles. The model is adopted from the theoretical work of Shaw [1], where the dynamics of the mean bubble radius and the surface modes are bi-directionally coupled via nonlinear terms. The model comprises a set of second-order ordinary differential equations. It extends the classic Keller–Miksis equation and the linearized dynamical equations for each surface mode. Only the implicit parts (containing the second derivatives) are reevaluated during the iteration process. The performance of the technique is tested at various parameter combinations. The majority of the test cases needs only a single reevaluation to achieve 10-9 error. Although the arithmetic operation count is higher than the Gauss elimination, due to its memory-friendly matrix-free nature, it is a viable alternative for high-performance GPU computations of massive parameter studies.

Coherences in the Dynamics of Physical Systems from a Multifractal Perspective of Motion.

Using an analogy between the multi-fractal Schrödinger equation and the dumped oscillator equation through a special ansatz, Stoler-type coherences in the dynamics of physical systems are highlighted. Such a result implies a Ricatti-type gauge, a process that can be considered a calibration of the difference between the kinetic and potential energy of a Lagrangian, specified as a perfect square in generic coordinates.

Power series approach to nonlinear oscillators

In this article, we introduce a nonlinear oscillator equation containing two strong linear terms. An approximate solution was obtained using power series approach. Furthermore, by introducing a parameter to the original equation, we fined the fixed points of the modified nonlinear oscillator equation and study stability analysis of these fixed points. On the other hand, we simulate the solution of the nonlinear oscillator equation and introduced many plots for different initial conditions. Finally, we make some plots concerning the phase portrait for different cases.

Noether symmetries and conservation laws of a reduced gauged bilayer graphene model

A canonical Hamiltonian is found for a reduced version of the Jackiw-Pi model for bilayer graphene. From the corresponding Lagrangian, the Noether point symmetries and conserved quantities are determined. The Noether symmetry group is the same as the fifth-dimensional group for the time-dependent harmonic oscillator. The realization of the algebra is achieved in terms of just one particular solution g1 of the time-dependent harmonic oscillator equation underlying the reduced Jackiw-Pi model. Some numerical solutions are worked out.

The stable and unstable oscillations of a rod attached to many springs

The oscillation equations of a multi-spring system connected to a rod are developed. Oscillations can be stable or unstable depending on the distribution of the spring constant and the number of springs. The symmetrical spring distribution with respect to the rod center tends to stabilize. The system becomes more stable as the number of springs increases. The equations of motion are identified to satisfy the Mathieu’s equation. The system could be thought of as a rough imitation of the crazy bamboo attraction, a traditional attraction by old civilization that is frequently associated with the mystical phenomenon. One spring represents one performer. The behavior of the performer’s force (opposite to the direction of the bamboo movement and providing a stronger force as the displacement of the bamboo increases) is similar to that of the spring force.

Towards Resonantly Enhanced Acoustic Phonon-Exchange Magnon Interactions at THz Frequencies

Using valid experimental parameters, we quantify the magnitude of resonantly phonon-driven precession of exchange magnons in freestanding ferromagnetic nickel thin films on their thickness L. Analytical solutions of acoustically driven equations for magnon oscillators display a nonmonotonous dependence of the peak magnetization precession on the film thickness. It is explained by different L-dependence of multiple prefactors entering in the expression for the total magnetization dynamics. Depending on the ratio of acoustic and magnetic (Gilbert) damping constants, the magnetization precession is shown to be amplified by a Q-factor of either the phonon or the magnon resonance. The increase in the phonon mode amplitude for thinner membranes is also found to be significant. Focusing on the magnetization dynamics excited by the two first acoustic eigenmodes with p=1 and p=2, we predict the optimum thicknesses of nickel membranes to achieve large amplitude magnetization precession at multi 100 GHz frequencies at reasonably low values of an external magnetic field. By extending the study to the case of Ni-Si bilayers, we show that these resonances are achievable at even higher frequencies, approaching the THz range.

A novel approach for an approximate solution of a nonlinear equation of charged damped oscillator with one degree of freedom

A novel technique is proposed for finding an approximate solution of the strongly nonlinear ordinary differential equation for the charged damped pendulum with one degree of freedom. The method relies on a transformation of the governing nonlinear differential equation that keeps unchanged the order of the highest derivative, in conjunction with a modified homotopy perturbation technique (MHPM). Only quadratic damping is considered for the numerical computations. To validate the used technique, the obtained results are compared to those arising from a numerical solution by Runge–Kutta of the fourth order (RK4) and by finite differences (FD). Good agreement between the two solutions is reached when quadratic damping is suppressed. In the presence of damping, agreement takes place only for a rather limited range of times. Plots of the analytical solutions are provided for both cases. The proposed method may be used to analyze a wide class of nonlinear differential equations.

Problem of the Boundary Control of Oscillations of a Sample of a Layered Two-Phase Composite Material

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

Inverse Problems for the Equation of Vibrations of a Canister Beam to Find the Source

For the beam vibration equation, inverse problems are studied to find the right side, i.e. vibration source. Solutions of the problems by methods of spectral analysis and Volterra integral equations are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem by determining the factor of the right-hand side, which depends on the spatial coordinate, the problem of small denominators arises. In this regard, estimates of the denominators are established that guarantee their separation from zero, with an indication of the corresponding asymptotics. On the basis of these estimates, the convergence of the series in the class of regular solutions of the beam oscillation equation is substantiated.

On Some Non-local Boundary Value and Internal Boundary Value Problems for the String Oscillation Equation

The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.

Generalized initial-boundary problem for the wave equation with mixed derivative

We study an initial-boundary problem for a second-order inhomogeneous hyperbolic equation in a half-strip of the plane containing a mixed derivative with constant coefficients and zero or nonzero potential. This equation is the equation of transverse oscillations of a moving finite string. The case of zero initial velocity and fixed ends (Dirichlet conditions) is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on opposite sides of the origin. The classical solution of the initial-boundary problem is determined. In the case of zero potential, a uniqueness theorem for the classical solution is formulated and a formula for the solution is given in the form of a series consisting of contour integrals containing the initial data of the problem. Based on this formula, the concepts of a generalized initial-boundary value problem and a generalized solution are introduced. The main theorems on finite formulas for the generalized solution in the case of homogeneous and inhomogeneous problems are formulated. To prove these theorems, we apply an approach that uses the theory of divergent series in the sense of Euler, proposed by A.P. Khromov (axiomatic approach). Using this approach, on the basis of formulas for solutions in the form of a series, the formulated main theorems are proved. Further, as an application of the main theorems obtained, we prove a theorem on the existence and uniqueness of a generalized solution of the initial-boundary problem in the presence of a nonzero summable potential and give a formula for the solution in the form of an exponentially convergent series.

Boundary value problems of integrodifferential equations under boundary conditions taking into account physical nonlinearity

When solving integrodifferential equations under boundary conditions, taking into account physical nonlinearity, a broad class of boundary-value problems of oscillations arises associated with various boundary conditions at the edges of a flat element. When taking into account non-stationary external influences, the main parameters is the frequency of natural vibrations of a flat component, taking into account temperature, prestressing, and other factors. The study of such problems, taking into account complicating factors, reduces to solving rather complex problems. The difficulty of solving these problems is due to both the type of equations and the variety. We analyze the results of previous works on the boundary problems of vibrations of plane elements. Possible boundary conditions at the edges of a flat element and the necessary initial conditions for solving particular problems of self-oscillation and forced vibrations, and other problems are considered. The set of equations, boundaries, and initial conditions make it possible to formulate and solve various boundary value problems of vibrations for a flat element. The oscillation equations for a flat element in the form of a plate given in this paper contain viscoelastic operators that describe the viscous behavior of the materials of a flat component. In studying oscillations and wave processes, it is advisable to take the kernels of viscoelastic operators regularly, since only such operators describe instantaneous elasticity and then viscous flow.

Method of calculating of the information in social networks protection depending on the number of communities

A mathematical model (linear system of differential equations) was developed and a research of the model of personal data protection against the number of communities and the intensity of data transfer in social networks was conducted. The linear system of information protection in social networks in the mathematical sense of this term is considered. When described by linear models, the object should be linear, at least approximately. This approach makes it quite simple to consider mathematical models. If such a thing is not noticed, it is necessary to examine the security system for linearity. Such dependecies has been studied: the dependence of the amount of information flow in the social network on the components of information protection, the amount of personal data, and the speed of the data flow; the security of the system on the size of the system (as well as on the amount of personal data); information security threats on the number of communities, and also calculated: – coefficient representing the impact of information protection measures; – coefficient representing the impact of data leakage rate; – the coefficient representing the influence of the amount of data on its leakage; – the coefficient representing the influence of the system size on system security; – coefficient representing the impact of system security on data leakage; – the number of connections in the social networks; – number of vertices in the social networks; – the parameter can be used to configure the network partitioning algorithm. The solution has been obtained - the harmonic oscillator equation, which breaks down into three cases: pre-resonance zone, resonance zone and post-resonance zone. So, the impact of the parameters of the number of communities on the parameters of the social network security system was investigated. Such a study is useful and important from the point of view of information protection in the network, since the parameters of the number of communities has significant influence, up to 100%, per protection indicator. As the result of research, it was established that social network security systems are non-linear.

Vibration of a Liquid Crystal Elastomer Spring Oscillator under Periodic Electrothermal Drive.

The oscillations of electrically actuated thermally-responsive liquid crystal elastomer (LCE) microfibers under cyclic electric actuation have been discovered in recent experiments. Periodic electric actuation is a common method of active control with potential applications in the fields of micro-actuators. In this paper, the vibration behavior of LCE spring oscillator under periodic electrothermal drive is studied theoretically. Based on the dynamic LCE model, the dynamic governing equation of the LCE spring oscillator is established, and the time history curves of the vibration are obtained by numerical calculations. The results show that the periodic electrothermal drive can cause periodic vibration of the LCE spring oscillator. With the increase of time rate, the vibration amplitude increases first and then decreases. In a small damping system, there exist optimal sets of electrothermal drive period and electrothermal drive time rate to maximize the system amplitude. For the optimum periodic mode, the vibration amplitude of the spring oscillator is affected by the current heat, damping coefficient, gravital acceleration, spring constant and shrinkage coefficient, but not by the initial velocity. The application examples of LCE materials show that periodic electrothermally driven LCEs have promising applications. The results of this study are instructive for the design of soft robots and LCE-based electric locomotives.

Аналитический анализ виброзащитной системы человека-оператора

The task of mathematical description and study of vibration isolation systems for seats of mobile machine operators is relevant, since operators of many construction, road and other mobile machines are exposed to significant vibration effects. An original analytical solution was proposed for the differential equation of forced oscillations of a linear oscillator with kinematic excitation, which describes the vertical oscillations of the seat with the operator, for given sinusoidal oscillations of the seat base. Analytical differentiation in time of the expression for the absolute displacement of the vibration-protected mass of the seat with the operator made it possible to obtain an analytical expression for the absolute velocity of the mass, the simplification of which made it possible to obtain a compact expression for the first and then the zero derivative of the absolute coordinate in the steady state oscillation mode. From the expression for the absolute displacement of the vibration-protected mass using a trigonometric transformation, an analytical expression for its amplitude was obtained, from which, in turn, an analytical expression for the transmission coefficient of the vibration protection system was obtained. The equation based on the analytical expression of the derivative of the transfer coefficient with respect to the circular frequency was solved analytically, which made it possible to obtain analytical expressions for the resonant amplitude of absolute displacements and the transfer coefficient. Examples of functional dependencies obtained using the derived analytical expressions are given. The obtained analytical expressions make it possible to conduct studies of vibration protection systems of seats with maximum accuracy.

Fluid pulsation modes and tidal deformability of anisotropic strange stars in light of the GW170817 event

The effects of anisotropy on the fluid pulsation modes when adopting the so-called Cowling approximation and the tidal deformability of strange-quark stars are investigated by numerically integrating the hydrostatic equilibrium, nonradial oscillation, and tidal deformability equations, all of which are modified from their standard form to include the anisotropic effects. The fluid matter inside compact stars is described by the MIT bag model equation of state. For the anisotropy profile, we consider a local anisotropy that is both regular at the center and null at the star's surface. We find that the effect of anisotropy is reflected in the fluid pulsation modes and tidal deformability. Finally, we analyze the correlation between the tidal deformability of the GW170817 event and anisotropy.

Numerical Solutions of Duffing Van der Pol Equations on the Basis of Hybrid Functions

In the present work, a new approximated method for solving the nonlinear Duffing-Van der Pol (D-VdP) oscillator equation is suggested. The approximate solution of this equation is introduced with two separate techniques. First, we convert nonlinear D-VdP equation to a nonlinear Volterra integral equation of the second kind (VIESK) using integration, and then, we approximate it with the hybrid Legendre polynomials and block-pulse function (HLBPFs). The next technique is to convert this equation into a system of ordinary differential equation of the first order (SODE) and solve it according to the proposed approximate method. The main goal of the presented technique is to transform these problems into a nonlinear system of algebraic equations using the operational matrix obtained from the integration, which can be solved by a proper numerical method; thus, the solution procedures are either reduced or simplified accordingly. The benefit of the hybrid functions is that they can be adjusted for different values of n and m , in addition to being capable of yield greater correct numerical answers than the piecewise constant orthogonal function, for the results of integral equations. Resolved governance equation using the Runge-Kutta fourth order algorithm with the stepping time 0.01 s via numerical solution. The approximate results obtained from the proposed method show that this method is effective. The evaluation has been proven that the proposed technique is in good agreement with the numerical results of other methods.

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with p-Laplacian Operator

The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved via the methods of reducing and topological degree in cone, which extend and enrich some previous results.

One-parameter Darboux-deformed Fibonacci numbers

One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (noninteger) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov–Lewis invariants for these sequences are also discussed.