Krylov–Bogoliubov–Mitropolskii Method Research Articles

A modified Krylov–Bogoliubov–Mitropolskii method for solving damped nonlinear oscillators with large oscillation

A modification of the extended Krylov–Bogoliubov–Mitropolskii​ method (by Popov et al.) has been presented to investigate damping effects of strongly nonlinear oscillators when the amplitude of oscillation is large. The modification is similar to the Lindstedt-Poincare method, which was early presented by Cheung et al. for handling un-damped large oscillations. The determination of solution is quite different from earlier all techniques and it is valid whether linear restring force is present or not. Moreover, the solution approaches toward modified Lindstedt-Poincare’s solution when damped force vanishes.

Free vibration of nonlinear autonomous systems modeled by 4-th order ordinary differential equations including internal resonance

Abstract In this article, the dynamic behavior of nonlinear autonomous system modeled by 4-th order ordinary differential equations is considered. Based on the pioneer work of Krylov-Bogoliubov-Mitropolskii (KBM), a modified KBM method is applied to achieve analytical solutions. Two different cases such as non-resonant and resonant conditions are studied in the presence of quadratic nonlinearities. The utility of the proposed method is justified by numerical result which is generated by Runge-Kutta fourth-order procedure. The analytical solution in each case shows an excellence agreement with numerical results.

Asymptotic Solutions of Fifth Order More Critically Damped Nonlinear Systems in the Case of Four Repeated Roots

In this article, we have modified the Krylov-Bogoliubov-Mitropolskii (KBM) method, which is one of the most widely used methods to delve into the transient behavior of oscillating systems, to find out the solutions of fifth order more critically damped nonlinear systems. In this paper, we have considered the asymptotic solutions of fifth order more critically damped nonlinear systems when the four eigenvalues are equal and another one is distinct. This article suggests that the perturbation solutions obtained by the modified KBM method for both the cases (when repeated eigenvalues are greater than the distinct eigenvalue, and when the distinct eigenvalue is greater than repeated eigenvalues) satisfactorily correspond to the numerical solutions obtained by Mathematica 9.0.

Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems

In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.

A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING

Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an n-th, n = 2, 3, <TEX>${\ldots}$</TEX>, order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Struble's method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Struble's method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.

A Unified KBM Method for Obtaining the Second Approximate Solution of a Third Order Weakly Non-linear Differential System with Strong Damping and Slowly Varying Coefficients

To obtain the second order approximate solution of a third order weakly nonlinear ordinary differential system with strong damping and slowly varying coefficients modeling a damped oscillatory process is considered based on the extension of a unified Krylov-Bogoliubov- Mitropolskii (KBM) method. The asymptotic solution for different initial conditions shows a good coincidence with those obtained by the numerical procedure for obtaining the transient’s response. The method is illustrated by an example.DOI: http://dx.doi.org/10.3329/jbas.v35i1.7973Journal of Bangladesh Academy of Sciences, Vol.35, No.1, 77-89, 2011

Perturbation solutions of fifth order oscillatory nonlinear systems

Oscillatory systems play an important role in the nature. Many engineering problems and physical systems of fifth degrees of freedom are oscillatory and their governing equations are fifth order nonlinear differential equations. To investigate the solution of fifth order weakly nonlinear oscillatory systems, in this article the Krylov–Bogoliubov–Mitropolskii (KBM) method has been extended and desired solution is found. An example is solved to illustrate the method. The results obtain by the extended KBM method show good agreement with those obtained by numerical method.

An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

Combining Krylov–Bogoliubov–Mitropolskii (KBM) and harmonic balance methods, an analytical technique is presented to determine approximate solutions of nonlinear oscillatory systems with damping. The first approximate perturbation solutions in which the unperturbed solutions contain two harmonic terms agree with numerical solutions nicely even if the damping force is significant. With suitable examples it has been shown that the combination of classical KBM and harmonic balance methods sometimes fails to measure satisfactory results; but the combination of extended KBM method (by Popov) and harmonic balance method always give the desired results. The method is illustrated by several examples and the solutions are compared to some existing solutions.

An extension of the general Struble's method for solving an nth order nonlinear differential equation when the corresponding unperturbed equation has some repeated eigenvalues

Abstract This paper attempts to show the more suitability of the extended general Struble's technique than the unified Krylov–Bogoliubov–Mitropolskii (KBM) method in solving the problems that occur during the critical conditions. Recently a critically damped condition of an n th, n =2,3, … order weakly nonlinear autonomous ordinary differential equation has been investigated by the unified KBM method, in which the corresponding unperturbed equation has some real (negative) repeated eigenvalues. But there are more important critical conditions, which are still untouched. One of them occurs when a pair of complex eigenvalues is equal to another. It is complicated to formulate as well as to utilize the KBM method to investigate this condition. However, the extended general Struble's technique is applicable to both autonomous and non-autonomous systems. Solutions obtained for different critical conditions as well as for different initial conditions show a good agreement with the numerical solutions. The method is illustrated by an example of a fourth-order nonlinear differential equation whose unperturbed equation has repeated complex eigenvalues. A steady-state solution is determined for the non-autonomous equation. Moreover, a critical condition of a fourth-order nonlinear equation is investigated when two real eigenvalues of the unperturbed equation are non-positive and equal.

A general form of Krylov–Bogoliubov–Mitropolskii method for solving nonlinear partial differential equations

A general asymptotic solution can be obtained for a class of partial differential equations with small nonlinearities whose dominant linear part involves an nth order, n = 2 , 3 , … , time derivative. The method used is an extension of the Krylov–Bogoliubov–Mitropolskii (KBM) method. The formulation as well as the determination of the solution is quite easy. Many authors have extended the KBM method to investigate some physical and mechanical oscillating systems, modelled by either second-order hyperbolic type partial differential equations or certain partial differential equations with third-order time derivative. They mainly extended the method to investigate individual problems. On the contrary, the proposed solution covers various types of nonlinear problems modelled by partial differential equations whose linear part involves second-, third-, etc. order time derivative. Substituting n = 2 , 3 into the general formula, it can be shown that the formula readily becomes to those extended by several authors. The method is illustrated with a physical problem whose linear part involves a third-order time derivative.

Unified Krylov–Bogoliubov–Mitropolskii method under a critical condition

A modified and compact form of Krylov–Bogoliubov–Mitropolskii (KBM) unified method is extended to obtain approximate solution of an nth order, n=2,3,…, ordinary differential equation with small nonlinearities when unperturbed equation has some repeated real eigenvalues. The existing unified method is used when the eigenvalues are distinct whether they are purely imaginary or complex or real. The new form is presented generalizing all the previous formulae derived individually for second-, third- and fourth-order equations to obtain undamped, damped, over-damped and critically damped solutions. Therefore, all types of oscillatory and non-oscillatory solutions are determined by suitable substitution of the eigenvalues in a general result. The formulation of the method is very simple and the determination of the solution is easy. The method is illustrated by an example of a fourth-order equation when unperturbed equation has two real and equal eigenvalues. The solution agrees with a numerical solution nicely. Moreover, this solution is useful when the differences between conjugate eigenvalues (real or complex) are small. Thus the method is a complement of the existing modified and compact form of KBM method.

A modified and compact form of Krylov–Bogoliubov–Mitropolskii unified method for solving an nth order non-linear differential equation

A modified and compact form of Krylov–Bogoliubov–Mitropolskii (KBM) (Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, NJ, 1947; Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961) unified method (J. Franklin Inst. 339 (2002) 239) is determined for obtaining the transient response of an nth order ( n⩾2) differential equation with small non-linearities. The formula presented in (J. Franklin Inst. 339 (2002) 239) is a changed form of KBM method. For n=2,3,4, some previous formulas were found separately by several authors in terms of amplitude and phase variables; but the formula of Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is derived in terms of some unusual variables instead of amplitudes and phases. The formula of Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is a general form and used arbitrarily to obtain asymptotic solution for n=2,3,4,…. However, a solution obtained by formula Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is transformed to a formal form replacing the unusual variables by amplitude and phase variables. In the present paper, the formula of Shamsul Alam, J. Franklin Inst. 339 (2002) 239) is itself transformed to a usual form (i.e. in terms of amplitude and phase variables). The later form of the formula is similar to most of the previous formulas found by several authors when n=2,3,4. This form of the formula is also generalized and it is easier than those obtained in all previous papers (extension) and identical to that initiated by original contributors (Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, NJ, 1947; Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961).

Unified Krylov–Bogoliubov–Mitropolskii method for solving nth order non-linear systems with slowly varying coefficients

The unified Krylov–Bogoliubov-Mitropolskii (KBM) method is extended for obtaining the transient response of an n-th order ( n⩾2) non-linear system with slowly varying coefficients. The method is a generalization of KBM method and covers all the three cases when the eigenvalues of the unperturbed equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the eigenvalues in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.

A unified Krylov–Bogoliubov–Mitropolskii method for solving nth order nonlinear systems

A unified theory is presented for obtaining the transient response of nth order nonlinear systems with small nonlinearities by Krylov–Bogoliubov–Mitropolskii method. The method is a generalization of Bogoliubov's asymptotic method and covers all three cases when the roots of the corresponding linear equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the roots in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.

Damped oscillating processes in biological and biochemical systems

Damped nonlinear oscillations in biological and biochemical systems are investigated by the extended Krylov-Bogoliubov-Mitropolskii (KBM) method. A review on the extension made by Popov to the KBM method is given and also further improvements are presented. Applications are made to models of oscillating chemical reactions ( Lefever and Nicolis, 1971 ), FitzHugh (1961) equations, and population dynamics ( Gatto and Rinaldi, 1977 ). Comparison to damped oscillating physical and engineering systems is made.

Asymptotic solutions of partial differential equations with damping and delay

Certain hyperbolic partial differential equations with small nonlinearities involving significant damping as well as time delay are investigated. The Krylov—Bogoliubov—Mitropolskii method is extended and applied. An application is given to longitudinal vibrations of a nonlinear elastic rod.

Nonlinear effects of two modified ordinary monochromatic waves

Quantum field theory is used to investigate the nonlinear effects of two modified ordinary monochromatic waves propagating across a uniform magnetic field. Amplitude-dependent wavelength and frequency shifts are derived. Unlike previous work that employs the Krylov–Bogoliubov–Mitropolskii method, the investigation uses the diagrammatic technique of field theory. The calculations show that increase of the above-mentioned physical parameters due to nonlinear interaction and drifting electrons depends upon the ratio of drifting to phase velocity.

Second order hyperbolic equations with small non-linearities in the case of internal resonance

The monofrequent solutions of certain autonomous second order hyperbolic differential equations with weak non-linearities are found in the case when some of the natural frequencies of the generating equation are in integral ratio. The approach use is a development of the KrylovBogoliubov-Mitropolskii method. The solution found is applied to the case of the longitudinal vibrations of a rod for which the stress-strain relation contains a small non-linear term and to the case of the vibrations of a rod with small inhomogeneities of density and elastic modulus and with small damping.

Monofrequent oscillations in mechanical systems governed by second order hyperbolic differential equations with small non-linearities

The solution to certain second order hyperbolic differential equations with weak non-linearities is found by using the Krylov Bogoliubov-Mitropolskii method. Both autonomous and non-autonomous equations are considered, and in the latter case, both the non-resonant and resonant problems are solved. The general methods are applied to the case of the longitudinal vibrations of a rod in which non-linear elastic behavior, nonlinear viscoelastic behavior, and viscous damping occur.

Approximate Solution of a Fourth Order Weakly Non-Linear Differential System with Strong Damping and Slowly Varying Coefficients by Unified KBM Method

The unified Krylov-Bogoliubov-Mitropolskii (KBM) method is used for determining theanalytical approximate solution of a fourth order weakly nonlinear differential system with strongdamping and slowly varying coefficients when a pair of eigen-values of the unperturbed equationis a multiple (approximately or perfectly) of the other pair or pairs. In a damped case, one of thenatural frequencies of the linearized equation may be a multiple of the other. The analytical firstorder approximate solution for different initial conditions shows a good coincidence with thoseobtained by the numerical procedure. The method is illustrated by an example.Key words: Perturbation method; Weak nonlinearity; Oscillatory process; Strong damping; Varying coefficientsDOI: 10.3329/jbas.v34i1.5493Journal of Bangladesh Academy of Sciences, Vol.34, No.1, 71-82, 2010