M. Shamsul Alam

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

Abstract 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 Bogoliubovs 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.

An analytical technique for solving a class of strongly nonlinear conservative systems

Abstract Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.

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

Abstract 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.

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 Strubles 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 Strubles 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 unified Krylov-Bogoliubov-Mitropolskii method for solving hyperbolic-type nonlinear partial differential systems

A general asymptotic solution is presented fo r invest igating t he transient response of nonlinear systems modeled by hyperbolic-type part ia l different ia l equat ions wit h sma ll nonlineari t ies. T he method covers all t he cases when eigen-values of t he corresponding unper t urbed systems are real, complex conjugate , or purely imaginary. It is shown t hat by suitable substit ution for t he eigen-values in t he general resul t t hat t he solut ion correspo nding to each of t he t hree cases can be obtained. T he method is an extension of t he unified Kry lov-Bogoliubov-Mitropolskii met hod , whi ch was ini t ially developed for un-da rn ped , under-clamped and over-clamped cases of t he second order ordinary different ia l equat ion . T he methods a lso cover a specia l condit ion of the over-damped case in which t he general solut ion is useless.

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

Abstract 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.

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, , order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Strubles 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 Strubles 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.

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.

On a special condition of over-damped nonlinear system with slowly varying coefficients

Abstract An asymptotic solution for certain second-order over-damped nonlinear system with slowly varying coefficients is found. The solution shows a good coincidence with numerical solution for different initial conditions. The method is an extension of Krylov-Bogoliubov-Mitropolskii method.