Krylov Bogoliubov Mitropolsky Method Research Articles

The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

In the present investigation, a quadratically forced damped pendulum-type equation is solved analytically using several effective and highly accurate approaches. Some different types of pendulum oscillators linked to the forced and damped terms, in addition to the power of the damping term, are analyzed. In the first part, the Krylov–Bogoliubov–Mitropolsky (KBM) technique and the ansatz method are carried out for analyzing the quadratically damped pendulum oscillator. In the second part, the two variants of the KBM technique are implemented for investigating the quadratically forced damped pendulum oscillator. Some symmetric approximations are derived and compared with the fourth-order Runge–Kutta numerical approximation. In addition, the maximum distance error is estimated in the whole study domain for ensuring that all obtained approximations are accurate. The obtained results are illustrated through concrete examples.

On the Krýlov–Bogoliúbov-Mitropólsky and Multiple Scales Methods for Analyzing a Time Delay Duffing–Helmholtz Oscillator

This study is divided into two important axes; for the first one, a new symmetric analytical (approximate) solution to the Duffing–Helmholtz oscillatory equation in terms of elementary functions is derived. The obtained solution is compared with the numerical solution using 4th Range–Kutta (RK4) approach and with the exact analytical solution that is obtained using elliptic functions. As for the second axis, we consider the time-delayed version for the same oscillator taking the impact of both forcing and damping terms into consideration. Some analytical approximations for the time delayed Duffing–Helmholtz oscillator are derived using two different perturbation techniques, known as Krylov–Bogoliubov–Mitropolsky method (KBMM) and the multiple scales method (MSM). Moreover, these perturbed approximations are analyzed numerically and compared with the RK4 approximations.

Analytical and Numerical Approximations to Some Coupled Forced Damped Duffing Oscillators

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other.

Analytical Approximant to a Quadratically Damped Duffing Oscillator.

The Duffing oscillator of a system with strong quadratic damping is considered. We give an elementary approximate analytical solution to this oscillator in terms of exponential and trigonometric functions. We compare the analytical approximant with the Runge–Kutta numerical solution. We also solve the oscillator by menas of He's homotopy method and the famous Krylov–Bogoliubov–Mitropolsky method. The approximant allows estimating the points at which the solution crosses the horizontal axis.

3D-Modulational Stability of Envelope Soliton in a Quantum Electron–Ion Plasma—A Generalised Nonlinear Schrödinger Equation

In physical reality, the phenomena of plasma physics is actually a three-dimensional one. On the other hand, a vast majority of theoretical studies only analyze a one-dimensional prototype of the situation. So, in this communication, we tried to treat the quantum electron–ion plasma in a full 3D setup and the modulational stability of envelope soliton was studied in a quantum electron–ion plasma in three dimensions. The Krylov–Bogoliubov–Mitropolsky method was applied to the three-dimensional plasma governing equations. A generalized form of the nonlinear Schrödinger (NLS) equation was obtained, whose dispersive term had a tensorial character, which resulted in the anisotropic behavior of the wave propagation even in absence of a magnetic field. The stability condition was deduced ab initio and the stability zones were plotted as a function of plasma parameters. The modulational stability of such a three-dimensional NLS equation was then studied as a function of plasma parameters. It is interesting to note that the nonlinear excitation of soliton took place again here due to the balance of nonlinearity and dispersion. The zones of contour plots are given in detail.

Some Novel Solutions to a Quadratically Damped Pendulum Oscillator: Analytical and Numerical Approximations

In this paper, some novel analytical and numerical techniques are introduced for solving and analyzing nonlinear second‐order ordinary differential equations (ODEs) that are associated to some strongly nonlinear oscillators such as a quadratically damped pendulum equation. Two different analytical approximations are obtained: for the first approximation, the ansatz method with the help of Chebyshev approximate polynomial is employed to derive an approximation in the form of trigonometric functions. For the second analytical approximation, a novel hybrid homotopy with Krylov–Bogoliubov–Mitropolsky method (HKBMM) is introduced for the first time for analyzing the evolution equation. For the numerical approximation, both the finite difference method (FDM) and Galerkin method (GM) are presented for analyzing the strong nonlinear quadratically damped pendulum equation that arises in real life, such as nonlinear phenomena in plasma physics, engineering, and so on. Several examples are discussed and compared to the Runge–Kutta (RK) numerical approximation to investigate and examine the accuracy of the obtained approximations. Moreover, the accuracy of all obtained approximations is checked by estimating the maximum residual and distance errors.

Optimal volume selection of air vessels in long-distance water supply systems

Abstract The use of air vessels is an effective measure to control water hammers, and its volume selection has a certain blindness. This paper aims to reveal the surge wave characteristics and provide design guidelines for air vessels in long-distance water supply systems. First, the analytical formulas of water-level oscillations in the air vessel are derived based on the Krylov–Bogoliubov–Mitropolsky method. Then, an optimization model is constructed for selecting the optimal volume of the air vessels. Finally, the validation of the analytical formulas and the optimization of the model are conducted through two actual projects. The results show that the calculation error of the analytical formulas can be controlled within a very small range, and the process of selecting air vessel volume can be simplified with the provided model. In addition, increasing the air chamber height within its range can reduce the volume of air vessels with the same protection requirements. The optimization analysis results of the air vessel can provide guidance and reference for the design of actual projects.

Subharmonic resonance of single-degree-of-freedom piecewise-smooth nonlinear oscillator

Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example, the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated. The approximate analytical solution of 1/3 subharmonic resonance of the single-degree-of-freedom piecewise-smooth nonlinear oscillator is presented. By changing the solving process of Krylov-Bogoliubov–Mitropolsky (KBM) asymptotic method for subharmonic resonance of smooth nonlinear system, the classical KBM method is extended to piecewise-smooth nonlinear system. The existence conditions of 1/3 subharmonic resonance steady-state solution are achieved, and the stability of the subharmonic resonance steady-state solution is also analyzed. It is found that the clearance affects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stiffness. Through a demonstration example, the accuracy of approximate analytical solution is verified by numerical solution, and they have good consistency. Based on the approximate analytical solution, the influences of clearance on the critical frequency and amplitude-frequency response of 1/3 subharmonic resonance are analyzed in detail. The analysis results show that the KBM method is an effective analytical method for solving the subharmonic resonance of piecewise-smooth nonlinear system. And it provides an effective reference for the study of subharmonic resonance of other piecewise-smooth systems.

Wigner transform and stability of kinetic envelope soliton in an electron depleted dusty plasma with two temperature non-thermal ions

A nonlinear theory of amplitude modulation of electrostatic envelopes in an electron depleted dusty plasma consisting of negatively charged dust grains and two temperature non-thermal ions developed using Vlasov–Poisson equation. By using the Krylov–Bogoliubov–Mitropolsky method a nonlinear Schrödinger equation is derived. The dispersion relation is written in terms of moments of plasma dispersion function ‘’ is analyzed numerically. Spatially periodic unstable modes are studied with the help of Wigner transform defining the spectral density at the initial time. This is actually a statistical approach to modulational stability very relevant due to kinetic formulation of our problem. In this method one can study various form of initial spectral distribution and their stability leading to the various conditions on plasma parameters. Our standard modulational stability is the special case of such a treatment. The explicit solution of the nonlinear Schrödinger equation is seen to be different depending upon the sign of ‘P’ and ‘Q’ (the dispersive and nonlinear coefficient). In one case we get the standard envelope soliton and the other one leads to either to envelope hole or envelope shock solution. It is very interesting to notice that the non-thermal behavior of the ions have a controlling nature on the modulational stability of the system. The advantage of working with the Wigner transformation is that it can be used as a tool for homogenization, that is in a situation with large number of waves. This type of analysis of modulational instability are now-a-days very important because the quantum nature of plasma is gaining importance.

Dust Charge Polarity Effect on Dust-Ion Acoustic Modulational Instability in a Nonthermal Dusty Plasma With Adiabatic Ions

The effects of dust charge polarity, ion temperature, and nonthermal electrons on dust-ion acoustic modulational instability and on the region (i.e., modulated wave number) for the formation of bright and dark envelope soliton structures are investigated. The dust particles are considered to be stationary and have positive and negative charges, while ions are taken dynamic and warm, and hot nonthermal electrons are assumed to be inertialess and kappa distributed in a dusty plasma. The Krylov–Bogoliubov–Mitropolsky method is used to derive the nonlinear Schrodinger equation (NLSE) for nonlinear amplitude modulation of dust-ion acoustic wave (DIAW) in plasmas. The dispersive and nonlinear coefficients of NLSE are obtained for DIAW, which depends on ion temperature, spectral indices of the nonthermal distributed electrons, dust density, and charge polarity of stationary dust particles. The modulationally stable and unstable regions of DIAW and its growth rate are investigated numerically. It is found that finite ion temperature, dust density, dust charge polarity, and spectral index of kappa for nonthermal electrons play a significant role to unstable the modulated DIAW modulated wave. The existence of positively and negatively charged dust particles and their observation in different regions of space plasma and in laboratory experiments is also pointed out.

Analytical Study of Periodic Solutions on Perturbed Equatorial Two-Body Problem

This paper presents analytical derivations to study periodic solutions for the two-body problem perturbed by the first zonal harmonic parameter. In particular, three different semianalytical approaches to solve this problem have been studied: (1) the classic perturbation theory, (2) the Lindstedt–Poincaré technique, and (3) the Krylov–Bogoliubov–Mitropolsky method. In addition, the numerical integration by Runge–Kutta algorithm is established. However, the numerical comparison tests show that by increasing the value of angular momentum the solutions provided by Lindstedt–Poincaré and Krylov–Bogoliubov–Mitropolsky methods become similar, and they provide almost identical results using a smaller value for the perturbed parameter which quantify the dynamical flattening of the main body, the Krylov–Bogoliubov–Mitropolsky provides more accurate results to design elliptical periodic solutions than Lindstedt–Poincaré technique when the perturbed parameter has a relatively large value, regardless of the value of angular momentum. This study can be applied to equatorial orbits to obtain closed-form analytical solutions.

Effect of ion temperature on modulational instability and envelope solitons of ion acoustic waves in nonthermal electron–positron–ion plasmas

The modulational instability and envelope solitons of ion acoustic waves in an unmagnetized nonthermal electron–positron–ion (epi) plasma are investigated. The ions are taken to be dynamic and warm while electrons and positrons are assumed to be inertialess and hot which follow the kappa (or Generalized Lorentzian) distribution. The Krylov–Bogoliubov–Mitropolsky method is used to derive the nonlinear Schrödinger equation for nonlinear amplitude modulation of ion acoustic waves in nonthermal epi plasmas with warm ions. The dispersive and nonlinear coefficients are obtained for ion acoustic waves in nonthermal epi plasmas which depend on spectral indices of kappa distributed electrons and positrons, ion temperature and positron density. The modulationally stable and unstable regions are studied for a wide range of wave numbers and it is found that the finite ion temperature, positron density and spectral indices of kappa distributed electrons and positrons play a significant role in the formation of bright and dark envelope solitons in nonthermal epi plasmas with adiabatically heated ions. Our findings are applicable to explain some aspects of nonlinear propagation of envelope solitons in astrophysical plasma situations such as neutron stars or pulsars where nonthermal epi plasmas with warm ions can exist (Arons J 2009 Astrophys. Space Sci. Library 357 373, Blasi P and Amato E 2011 Astrophys. Space Sci. Proc. pp 623–41, Asif and Saeed 2011 Plasma Phys. Control. Fusion 53 095006).

Modulation of whistler waves in nonthermal plasmas

The modulation of whistler waves in nonthermal plasmas is investigated. The dynamics of the magnetized plasma is described by the fluid equations and the electron velocity distribution function is modeled via a nonthermal κ distribution. A multiscale perturbation analysis based on the Krylov–Bogoliubov–Mitropolsky method is carried out and the nonlinear Schrödinger equation governing the modulation of the high-frequency whistler is obtained. The effect of the superthermal electrons on the stability of the wave envelope and soliton formation is discussed and a comparison with previous results is presented.

LIE GROUP ANALYSIS FOR MULTI-SCALE PLASMA DYNAMICS

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov–Bogoliubov–Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory demonstrate the potentialities of the suggested approach. Several examples of invariant solutions for the system of the Vlasov-Maxwell equations for the two-component (electron-ion) plasma are presented.

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

In this paper, an analytical approximation of damped oscillations of some strongly non-linear, planar Hamiltonian systems is considered. To apply the Krylov–Bogoliubov–Mitropolsky method in this strongly non-linear case, we mainly provide the formal and exact solutions of the homogeneous part of the variational equations with periodic coefficients resulting from the Hamiltonian systems. It is shown that these are simply expressed in terms of the partial derivatives of the solutions, written in action-angle variables, of the Hamiltonian systems. Two examples, including a non-linear harmonic oscillator and the Morse oscillator, are presented to illustrate this extension of the method. The approximate first order solution obtained in each case is observed to be quite satisfactory.

A Generalized KBM Method for Strongly Nonlinear Oscillators with Slowly Varying Parameters

A generalized Krylov-Bogoliubov-Mitropolsky (KBM) method is extended for the study of strongly nonlinear oscillators with slowly varying parameters. The asymptotic amplitude and phase are derived and then the asymptotic solutions of arbitrary order are obtained theoretically. Cubic nonlinear oscillators with polynomial damping are studied in detail. Three examples are considered: a generalized Van der Pol oscillator, a Rayleigh equation and a pendulum with variable length. Comparisons are also made with numerical solutions to show the efficiency and accuracy of the present method.

On the analytical approximation of damped oscillations of autonomous single degree of freedom oscillators

In this paper, we discuss the generalization of a recently developed perturbation method for conservative single degree of freedom systems to the cases of damping forces. We show that by explicitly considering the solution as a function of the amplitude parameter, frequency and phase of oscillation, one can merge the classical Krylov–Bogoliubov–Mitropolsky method and a modified Lindstedt–Poincaré method to derive better accurate slow-flow for damped oscillation of single degree of freedom systems.

Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters

Two versions of multiple scales and Krylov–Bogoliubov–Mitropolsky (KBM) methods are extended to obtain the asymptotic solutions of a class of strongly nonlinear oscillators with slowly varying parameters. A comparison of the two methods is made to show that the multiple scales method is equivalent to the KBM method for the first order approximation. An example of pendulum with slowly varying length is given to illustrate the comparison of the two methods.

THE KBM DERIVATIVE EXPANSION METHOD IS EQUIVALENT TO THE MULTIPLE-TIME-SCALES METHOD

Several perturbation methods are commonly used to predict the free and forced response of weakly non-linear oscillators. The Krylov-Bogoliubov-Mitropolsky (KBM) and multiple-time-scales (MTS) methods use expansions of dependent variables, ordinary time derivatives, and some system parameters to convert the equations of motion into a set of first order differential equations. Each of these equations represents the slow time modulations of the amplitude and phase of the zeroth order solution(s).In this paper, a simple correspondence between the expansions of ordinary time derivatives employed in these two methods is used to show that, except for notation, these two methods are identical in the sense that to any order of approximation these two methods will provide identical results when they use the same parameter expansions and identical additional constraints. The KBM method attempts to reduce unneeded algebraic calculations by tailoring the derivative expansions to the simplest applicable form. This, however, requires some experience or a trial and error approach to establish the intermediate expansion variables and the implicit and explicit dependence of perturbation solutions on the different time scales. This relation depends not only on the problem at hand but also on the parameter expansions used in the solution procedure. By using the most general expansion for the time derivatives, the MTS method establishes this dependence as a part of the analysis. In this method, the algebraic details are hidden by using a compact derivative operator type notation. However, these operators do not commute in general.

A High Order Generalized Method of Averaging

We develop a high order generalized perturbation technique that extends the Krylov–Bogoliubov–Mitropolsky method of averaging to vector systems written in normal form with multiple angular components. An algorithm is presented that iteratively gives the terms in the asymptotic approximation. A nonresonance condition is assumed that guarantees the smoothness of the terms. The main result establishes that the absolute error between the unaveraged normal system and its Nth order approximation is of the order of the Nth power of the perturbation parameter for a time interval of length the order of the reciprocal of the perturbation parameter. The high order algorithm is applied to a coupled van der Pol oscillator system. Some numerical results are given to show that the main result reflects actual computational experience.