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

1 | P a g e

CHAPTER ONE

Introduction:

Differential equation is a mathematical tool, which has its applications in many branches

of knowledge of mankind. Numerous physical, mechanical, chemical, biological,

biochemical and many other relations appear mathematically in the form of differential

equations that are linear or non-linear, autonomous or non-autonomous. Generally, in

many physical phenomena, such as spring mass systems, resistor-capacitor-inductor

circuits, bending of beams, chemical reactions, pendulums, the motion of the rotating

mass around another body, etc, the differential equations occur. Also in ecology and

economics, the differential equations are vastly used. Basically, many differential

equations involving physical phenomena are non-linear. Differential equations, which are

linear are comparatively easy to solve and non-linear are laborious and in some cases, it

is impossible to solve them analytically. In such situations, mathematicians convert the

non-linear equations into linear equations by imposing some conditions. The method of

small oscillations is a well-known example of the linearization. But such a linearization is

not always possible and when it is not, then the original nonlinear equation itself must be

used. With the discovery of numerous phenomena of self-extraction of circuits containing

non-linear equations of electricity, like electron tubes gaseous discharge, etc, and in many

cases of non-linear mechanical vibrations of special types, the method of small

oscillations becomes inadequate for their analytical treatment. The knowledge of the non-

linear equations is generally confined to a variety of rather special cases. There exists an

important difference between the phenomena, which oscillate in steady state and the

phenomena governed by linear differential equations with constant coefficients. For

example, oscillations of a pendulum with amplitudes, the amplitude of the ultimate stable

seems to be entirely independent of the initial conditions, where as in oscillations

governed by a linear differential equation with constant coefficients, it depends upon the

initial conditions. Originally the Krylov-Bogoliubov-Mitropolskii (KBM) method was

2 | P a g e

developed for the systems only to obtain the periodic solutions of second order non-linear

differential equations. Now the method is used to obtain oscillatory as well as damped,

over damped, near critically damped, more critically damped oscillatory and non-

oscillatory solutions of second, third, fourth and higher order non-linear differential

equations by imposing some specific conditions to make the solutions uniform. To solve

non-linear differential equations there exist some methods. Among the methods, the

method of perturbations i.e. asymptotic expansions in terms of a small parameter is

foremost. Perturbation methods have recently received much attention as methods for

accurately and quickly computing numerical solutions of dynamic stochastic economic

equilibrium models, both single agent or rational-expectations models and multi-agent or

game-theoretic models. A perturbation method is based on the following aspects: the

equations to be solved are sufficiently smooth or sufficiently differentiable a number of

times in the required regions of variables and parameters. At first Van Der Pol paid

attention to the new oscillations i.e. self-excitations and indicated that their existence is

inherent in the in the non-linearity of the differential equations characterizing the process.

This non-linearity appears thus as the very essence of these phenomena and by

linearizing the differential equation in the sense of the method of small oscillations, one

simply eliminates the possibility of investigating such problems. Thus it is necessary to

deal with the non-linear problems directly instead of evading them by dropping the non-

linear terms. The method of Krylov and Bogoliubov is an asymptotic method in the sense

that ε→0. An asymptotic series itself may not be convergent, but for a fixed number of

terms, the approximate solution tends to be the exact solution as ε→0. It may be noted

that the term asymptotic is frequently used in the theory of oscillations in the sense that

ε→∞. But in this case, the mathematical method is quite different. It is an important

approach to the study of such non-linear oscillations is the small parameter expansion.

Two widely used methods in this theory are mainly used in the literature; one is

averaging asymptotic method of KBM and the other is multi time scale method. Among

the methods used to study non-linear systems with a small non-linearity, the KBM

method is particularly convenient and is the extensively used technique to obtain the

approximate solutions. The method of KBM starts with the solution of linear equation

(sometimes called the general solution of the linear equation), assuming that in the non-

3 | P a g e

linear case, the amplitude and the phase in the solution of the linear differential equation

are time dependent functions rather than constants. This procedure introduces an

additional condition on the first derivative of the assumed solution for determining the

solution of a second order equation. It is customary in the KBM method that correction

terms (i.e., the terms with small parameter) in the solutions do not contain secular terms.

These assumptions are mainly valid for second and third order equations. But for the fifth

order differential equation, the correction terms sometimes contain secular terms,

although the solution is generated but the classical KBM asymptotic method.

Consequently, the traditional solutions failed to explain the proper solutions of the

systems. To remove the presence of secular terms and obtaining the desired results, we

need to impose some special conditions. The main target of this project is b to find out

these limitations and determine the proper solutions under some special conditions. The

method has its use mainly in engineering and technology, notably in mechanics, electrical

circuit theory, and also used in population dynamics, chemistry, control theory, plasma

physics, etc. it may be noted that most of the researchers have tried to find the solutions

of second, third, and fourth order non-linear systems. Although some investigators have

obtained the solutions of fifth order non-linear differential equations, which have not

been studied extensively.

In this work we have chosen a fifth order non-linear differential equation that describes

more critically damped systems with small non-linearities, to solve by the modified KBM

method and the quality of the solution is being tested.

4 | P a g e

CHAPTER TWO

LITERATURE REVIEW

2.1 Damped Oscillatory Nonlinear Systems

Nonlinear differential equations show strange characters, but the mathematical

formulation of a number of physical problems results in differential equations which are

actually nonlinear. In many cases to convert such a nonlinear differential equation into a

related linear equation is possible, which approximates the actual nonlinear equation

closely enough to deliver useful results. This type of a “linearization” is not always

possible; and when it is not, the original nonlinear differential equation must be tried to

solve directly. While the general theory and methods of linear equations are highly

developed but very little of an ordinary character is known about nonlinear equations.

The study of nonlinear equations is generally confined to a variety of rather special cases

and one must resort to various methods of approximations. During last several decades in

the 20th century, some Russian scientists like Mandelstam and Papalexi [38], Krylov and

Bogoliubov [32], Bogoliubov and Mitropolskii [14] jointly worked on the nonlinear

dynamics. To solve nonlinear differential equations there exist some methods, among the

methods, the method of perturbations, i.e., an asymptotic expansion in terms of a small

parameter is the most advanced.

Krylov and Bogoliubov [32] developed a method of finding an approximate solution of

an equation of the form

=+ εεω ,,,22

2

tdt

dxxfx

dt

xd (2.1.1)

where ε is a sufficiently small positive parameter but not equal to zero, so that the

nonlinear term ( )εε ,,, tdtdxxf is relatively small and f is a power series in ε , whose

coefficients are polynomials in ,cos andsin,, ttdtdxx and their proposed solution

procedure is known as Krylov-Bogoliubov (KB) solution. Generally, f does not contain

either ε or t. To depict the nature of nonlinear oscillations by the solutions obtained by

perturbation method, Gylden [29] and Liapounoff [35] discussed only periodic solutions,

5 | P a g e

transients were not considered. Around the year 1830, Poisson initiated nearly accurate

solutions of nonlinear differential equations and Liouville established his method. The

KBM method started with the solution of the linear equation, assuming that in the

nonlinear systems, the amplitude and phase in the solution of the linear equation are time

dependent functions rather than constants. This process introduced an additional

condition on the first derivative of the assumed solution for determining the solution of a

second order ordinary differential equation. Some outstanding works were done by

Stoker [33], McLachlan [39], Minorsky [42], Nayfeh [49], Bellman [12] and elaborative

uses have been made by them. Duffing [28] investigated many significant results on the

periodic solutions of the equation of the form:

322

2

2 xxdt

dxk

dt

xd εω −=++ (2.1.2)

Sometimes different kinds of nonlinear phenomena occur, if the amplitude of the

dependent variable of a dynamical system is less or greater than unity. The damping will

be negative if the amplitude is less than unity and the damping will be positive if the

amplitude is greater than unity. The governing equation of these phenomena is of the

form:

( ) 01 32

2

=+−− xdt

dxx

dt

xd ε (2.1.3)

This equation is called Van der Pol [13] equation. Kruskal [31] has extended the KB

method for solving the completely nonlinear differential equation, which is of the form:

= ε,,2

2

dt

dxxF

dt

xd (2.1.4a)

Cap [27] has examined nonlinear systems of the form:

=+dt

dxxFxf

dt

xd,)(2

2

2

εω (2.1.4b)

Since, f does not contain either ε or t in general, so the equation (2.1.1) becomes

=+dt

dxxfx

dt

xd,2

2

2

εω (2.1.5)

6 | P a g e

As pointed out previously that, in the behavior of nonlinear oscillations by perturbation

method, only periodic solutions were discussed, transients were not considered by

different researchers, where as Krylov and Bogoliubov first discussed transient response.

If ,0=ε then the equation (2.1.5) reduces to a linear equation and its solution is

)cos( θω += tax (2.1.6)

where a and θ are arbitrary constants to be determined by using the initial conditions.

If ,0≠ε but small enough, then Krylov and Bogoliubov considered that the solution of

(2.1.5) is still given by (2.1.6) together with the derivative of the form

)sin( θωω +−= tadt

dx (2.1.7)

where a and θ are functions of t, rather than being constants.

Thus the solution of (2.1.5) is of the form

))(cos()( tttax θω += (2.1.8)

So the equation (2.1.7) becomes

)}(sin{)( tttadt

dx θωω +−= (2.1.9)

Differentiating the assumed solution (2.1.8) with respect to t, we get

ψθψωψ sinsincosdt

daa

dt

da

dt

dx −−= (2.1.10)

where )(tt θωψ +=

Using the equations (2.1.7) and (2.1.10), we obtain

ψθψ sincosdt

da

dt

da = (2.1.11)

Again differentiating (2.1.9) with respect to t, we have

ψθωψωψω coscossin 22

2

dt

daa

dt

da

dt

xd −−−= (2.1.12)

Substituting the value of 2

2

dt

xd from (2.1.12) into the equation (2.1.5) and using equations

(2.1.8) and (2.1.9), we obtain

( )ψωψεψθωψω sin,coscossin aafdt

da

dt

da −−=+ (2.1.13)

Solving (2.1.11) and (2.1.13), we get

7 | P a g e

( )ψωψψωε

sin,cossin aafdt

da −−= (2.1.14)

( )ψωψψωεθ

sin,coscos aafadt

d −−= (2.1.15)

Thus, we observe that, a second order basic differential equation (2.1.5) in the unknown x

reduces to two first order differential equations (2.1.14) and (2.1.15) in the unknowns a

and θ. It is pointed out that these two first-order equations are written in terms of the

amplitude a and phase θ as dependent variables. From equations (2.1.14) and (2.1.15) it

is clear that dt

da and

dt

dθ both are proportional to the small parameterε . Hence, although

a and θ are functions of time t, they are slowly varying functions of t with a period of

.2

ωπ=T That is, in a time interval of length ,

2

ωπ

a and θ are almost constant, while

θωψ += t increases by approximately .2π Therefore, the right sides of equations

(2.1.14) and (2.1.15) show that both dt

da and

dt

dθ are periodic functions of time T. We

can convert the equations (2.1.14) and (2.1.15) into more convenient form. Now,

expanding ( )ψωψψ sin,cossin aaf − and ( )ψωψψ sin,coscos aaf − in Fourier series

in the total phase ,ψ the first approximate solution of (2.1.5), by averaging (2.1.14) and

(2.1.15) within the period of ωπ2=T is

( )

( )∫

∫

−−=

−−=

π

π

ψψωψψπωεθ

ψψωψψπωε

2

0

2

0

sin,coscos2

sin,cossin2

daafadt

d

daafdt

da

(2.1.16)

where a and θ are independent of time under the integrals.

Krylov and Bogoliubov [32] called their method asymptotic as .0→ε An asymptotic

series itself may not be convergent, but for a fixed number of terms the approximate

solution tends to the exact solution as .0→ε Later, the technique was verified

mathematically by Bogoliubov and Mitropolskii [14], and then extended to nonstationary

vibrations by Mitropolskii [43]. They sought a solution of the nonlinear differential

equation (2.1.5) in the form:

8 | P a g e

( )12

21 ),(),(),(cos ++++++= n

nn Oauauauax εψεψεψεψ ⋯⋯ (2.1.17)

where ),,2,1(, nkuk …= are periodic functions of ψ with a period of π2 and the

quantities a and ψ which are functions of time t satisfy the first order differential

equations

( )

( )12

21

12

21

)()()(

)()()(

+

+

+++++=

++++=

nn

n

nn

n

OaBaBaBdt

d

OaAaAaAdt

da

εεεεωψ

εεεε

⋯⋯

⋯⋯

(2.1.18)

The functions ),,2,1(,and, nkBAu kkk …= are to be chosen in such a way that the

equation (2.1.17), after replacing a and ψ by the functions obtained from equation

(2.1.18); is a solution of the equation (2.1.5).

Since there are no restrictions in the choosing of functions ,and kk BA therefore, there is

arbitrariness in the definition of the functions ku (Bogoliubov and Mitropolskii [14]). For

removing this arbitrariness, the following conditions are imposed

0sin),(

0cos),(

2

0

2

0

=

=

∫

∫π

π

ψψψ

ψψψ

dau

dbu

k

k

(2.1.19)

Absences of the secular terms in all successive approximations are guaranteed by these

conditions. Differentiating (2.1.17) two times with respect to t and using the relations

(2.1.18), then substituting the values of 2

2

and,dt

xd

dt

dxx into the equation (2.1.5) and

finally equating the coefficients of ),,2,1(, nkk…=ε we obtain

)sincos(2),()1(2

22 ψψωψ

ψω kk

kk

k AaBafuu

++=

+

∂∂ − (2.1.20)

9 | P a g e

∂∂+

∂∂∂−

−+

−+−×

∂∂

+−+−=

−=

21

2

11

2

11

111

11

21

1111

)1(

)0(

2sin2

cos)sin,cos(

sincos)sin,cos(),(and

)sin,cos(),(where

ψψωψ

ψψωψ

ψωψψψωψψ

ψωψψ

uB

a

uA

da

dBaABA

da

dAAaBaaf

uaBAaafuaf

aafaf

dt

dx

x

(2.1.21)

Here )1( −kf is a periodic function of ψ with period π2 which depends on the amplitude

a.

Now, expanding )1( −kf and ku in Fourier series, we obtain

{ }

{ }∑

∑∞

=

−−−

∞

=

−−−−

++=

++=

1

)1()1()1(0

1

)1()1()1(0

)1(

sin)(cos)()(),(

sin)(cos)()(),(

n

kn

kn

kk

n

kn

kn

kk

nawnavavau

nahnagagaf

ψψψ

ψψψ (2.1.22)

∫

∫

∫

−=

−=

−=

−−

−−

−−

π

π

π

ψψψωψπ

ψψψωψπ

ψψωψπ

2

0

)1()1(

2

0

)1()1(

2

0

)1()1(0

sin)sin,cos(1

cos)sin,cos(1

)sin,cos(2

1where

dnaafh

dnaafg

daafg

kkn

kkn

kk

(2.1.23)

∫

∫

∫

=

=

=

−

−

−

π

π

π

ψψψπ

ψψψπ

ψψπ

2

0

)1(

2

0

)1(

2

0

)1(0

sin),(1

cos),(1

),(2

1and

dnauw

dnauv

dauv

kk

n

kk

n

kk

(2.1.24)

Here, 0)1(1

)1(1 == −− kk wv for all values of k, as both integrals of (2.1.19) disappear.

Substituting the values of )1( −kf and ku into the equation (2.1.20), we obtain

10 | P a g e

( ){ }{ } { }

{ }∑

∑

∞

=

−−

−−−

∞

=

−−−

++

++++=

+−+

2

)1()1(

)1(1

)1(1

)1(0

2

)1()1(22)1(0

2

sin)(cos)(

sin2)(cos2)()(

sin)(cos)(1)(

n

kn

kn

kk

kkk

n

kn

kn

k

nahnag

naAahnaBagag

nawnavnav

ψψ

ψωψω

ψψωω

(2.1.24)

Now equating the coefficients of the harmonics of the same order, yield

( )

( )

( 1) ( 1)1 1

( 1) ( 1)( 1) ( 1)00 2 2 2

( 1)( 1)

2 2

( ) 2 0, ( ) 2 0,

( ) ( )( ) , ( ) ,

1

( )( ) , 2

1

k kk k

k kk k n

n

kk n

n

g a aB g a aB

g a g av a v a

n

h aw a n

n

ω ω

ω ω

ω

− −

− −− −

−−

+ = + =

= =−

= ≥−

(2.1.25)

These are the sufficient conditions to obtain the desire order of approximation. For the

first order approximation, we have

( )

( )∫

∫

−−=−=

−−=−=

π

π

ψψψωψπωω

ψψψωψπωω

2

0

)0(1

1

2

0

)0(1

1

cossin,cos2

1

2

)(

sinsin,cos2

1

2

)(

daafa

agB

daafah

A

(2.1.26)

Thus the variational equations in (2.1.18) become

( )

( )∫

∫

−−=

−−=

π

π

ψψψωψπωεωψ

ψψψωψπωε

2

0

2

0

cossin,cos2

sinsin,cos2

daafadt

d

daafdt

da

(2.1.27)

It is seen that, the equations of (2.1.27) are similar to the equations of (2.1.16). Hence, the

first order solution obtained by Bogoliubov and Mitropolskii [14] is the same as the

solution obtained by Krylov and Mitropolskii [32]. Higher order solutions can be found

easily. The correction term 1u is obtained from (2.1.22) by using (2.1.25) as:

( )∑∞

= −+

+=2

22

)0()0(

2

)0(0

1 1

sin)(cos)()(

n

nn

n

nahnagagu

ωψψ

ω (2.1.28)

The solution (2.1.17) together with 1u is known as the first order improved solution

where a and ψ are obtained from (2.1.27). If the values of the functions 1A and 1B are

11 | P a g e

substituted from (2.1.26) into the second relation of (2.1.21), then the function )1(f will

be found and in this way, the functions 22 , BA and 2u can be obtained. Therefore, the

determination of the higher order approximation will be completed. The KB method is

very much similar to the Van der Pol’s method and related to it. Van der Pol applied the

method of variation of constants to the basic solution tbtax ωω sincos += of

,022

2

=+ xdt

xd ω on the other hand Krylov-Bogoliubov applied the same method to the

basic solution )cos( θω += tax of the same equation. Thus in the KB method the varied

constants are a and θ , while in the Van der Pol’s method the constants are a and b. The

method of Krylov-Bogoliubov is more exciting from the viewpoint of applications, since

it deals directly with the amplitude and phase of the quasi-harmonic oscillation.

Volosov [30] and Museenkov [48] also obtained higher order effects. The solution of the

equation (2.1.4a) is based on recurrent relations and is given as the power series of the

small parameter. Cap [27] solved the equation (2.1.4b) by using elliptical functions in the

sense of Krylov-Bogoliubov (KB) method. Popov [54] extended the KBM method to

damped nonlinear systems:

=++ xdt

dxfx

dt

dxk

dt

xd,2 2

2

2

εω (2.1.29)

where dt

dxk2− is the linear damping force and ω<< k0 . It is notable that, the

importance of the Popov’s method [54] in the physical systems with damping force,

Mendelson [40] and Bojadziev [23] retrieved Popov’s results. In case of damped

nonlinear systems the first equation of (2.1.18) has been replaced by

( )12

21 )()()( ++++++−= n

nn OaAaAaAka

dt

da εεεε ⋯⋯ (2.1.18a)

A simple analytical method was presented by Murty and Deekshatulu [46] for obtaining

the time response of second order nonlinear over-damped systems with small nonlinearity

which is presented by the equation (2.1.29), based on the Krylov-Bogoliubov method of

variation of parameters. Shamsul [53] developed a method to find the solutions of over-

damped nonlinear systems, when one of the roots of the auxiliary equation is much

smaller than the other root. Murty et al. [45] found a hyperbolic type asymptotic solution

12 | P a g e

using the KBM method of an over-damped system presented by the nonlinear differential

equation (2.1.29), i.e., in the case ω>k . They used hyperbolic functions, ϕcosh and

ϕsinh instead of their circular counterpart, which was used by Krylov, Bogoliubov,

Mitropolskii, Popov and Mendelson. In case of oscillatory or damped oscillatory process

ϕcosh may be used arbitrarily for every type of initial conditions. But in case of

nonoscillatory systems ϕcosh or ϕsinh should be used depending on the given set of

initial conditions (Bojadziev and Edwards [24], Murty et al. [45], Murty [46]). Murty

[46] extended a unified KBM method for obtaining the approximate solution of nonlinear

systems presented by the equation (2.1.29), which can face the under-damped, damped

and over-damped cases. Bojadziev and Edwards [24] investigated solutions of oscillatory

and nonoscillatory systems presented by (2.1.29) when k and ω are slowly varying

functions of time t. Arya and Bojadziev [10, 11] examined damped oscillatory systems

and time-dependent oscillating systems with slowly varying parameters and delay. Sattar

[62] investigated an asymptotic method for solving a second order critically damped

nonlinear system presented by (2.1.29). He has found the asymptotic solution of the

system (2.1.29) in the form:

( )12

21 ),(),(),()1( +++++++= n

nn Oauauauax εψεψεψεψ ⋯⋯ (2.1.30)

where a is defined by the equation (2.1.18a) and ψ is defined by

( )12

21 )()()(1 ++++++= n

nn OaCaCaC

dt

d εεεεψ⋯⋯ (2.1.18b)

Osiniskii [51] developed the KBM method to solve a third order nonlinear differential

equation of the form

=+++ x

dt

dx

dt

xdfxc

dt

dxc

dt

xdc

dt

xd,,

2

2

322

2

13

3

ε (2.1.31)

where ε a small positive parameter and f is is a nonlinear function. He considered the

asymptotic solution of (2.1.31) in the form

( )2

1 2

1

cos ( , , ) ( , , )

( , , )n nn

x a b u a b u a b

u a b O

ψ ε ψ ε ψε ψ ε +

= + + +

+ + +⋯⋯

(2.1.32)

13 | P a g e

where each ),,2,1(, nkuk …= are periodic functions of ψ with a period π2 and the

quantities a, b and ψ are functions of time t, defined by

( )

( )

( )12

21

12

21

12

21

)()()(

)()()(

)()()(

+

+

+

+++++=

+++++−=

+++++−=

nn

n

nn

n

nn

n

ObCbCbCdt

d

ObBbBbBbdt

db

OaAaAaAadt

da

εεεεωψ

εεεεµ

εεεελ

⋯⋯

⋯⋯

⋯⋯

(2.1.33)

where ωµλ ±−− , are the eigen-values of the equation (2.1.31) when .0=ε

Following the KBM method, Bojadziev [15] investigated solutions of nonlinear damped

oscillatory systems with small time lag. Bojadziev [20] has also found solutions of

damped forced nonlinear vibrations with small time delay. For solving the problems of

population dynamics, Bojadziev [21], Bojadziev and Chan [22] applied the KBM

method. Bojadziev [23] used the KBM method to investigate solutions of nonlinear

systems arised from biological and biochemical fields. Lin and Khan [36] have also used

the KBM method for some biological problems and Bojadziev et al. [16] have

investigated periodic solutions of nonlinear systems by the KBM and Poincare method

and compared the two solutions. Bojadziev and Lardner [17, 18] have investigated mono

frequent oscillations in mechanical systems including the case of internal resonance,

governed by hyperbolic differential equations with small nonlinearities. Bojadziev and

Lardner [19] have also investigated solution for a certain hyperbolic partial differential

equation with small nonlinearity and large time delay included into both unperturbed

parts of the equation. Bojadziev [25] and Bojadziev and Hung [26] developed a technique

by using the method of KBM to investigate a weakly nonlinear mechanical system with

strong damping. Osiniskii [51] has also extended the KBM method to a third order

nonlinear partial differential equation with initial friction and relaxation. Mulholland [44]

studied nonlinear oscillations governed by a third order differential equation. Lardner and

Bojadziev [34] investigated nonlinear damped oscillations governed by a third order

partial differential equation. They introduced the concept of “couple amplitude” where

the unknown functions kk BA , and kC depend on the both amplitudes a and b.

Bojadziev [25], and Bojadziev and Hung [26] used at least two trial solutions to

investigate time dependent differential systems – one is for resonant case and the other is

14 | P a g e

for the nonresonant case. But Shamsul [47] used only one set of variational equations,

arbitrarily for both resonant and nonresonant cases. Shamsul et al. [37] presented a

general form of the KBM method for solving nonlinear partial differential equations.

Raymond and Cabak [57] examined the effects of internal resonance on impulsive forced

nonlinear systems with two-degree-of-freedom. Later, Shamsul [56, 58] has extended the

method to n-th order nonlinear systems. Shamsul [47, 55] has also extended the KBM

method for certain nonoscillatory nonlinear systems when the eigen values of the

unperturbed equation are real and non-positive. Shamsul [63] has presented a new

perturbation method based on the KBM method to find approximate solutions of second

order nonlinear systems with large damping. Sattar [64] has extended the KBM

asymptotic method for three-dimensional over-damped nonlinear systems. Shamsul et al.

[59] extended the KBM method to certain nonoscillatory nonlinear systems with varying

coefficients. Later, Shamsul [47] has unified the KBM method for solving n-th order

nonlinear differential equation with varying coefficients. Shamsul and Sattar [65] studied

time dependent third order oscillating systems with damping based on an extension of the

asymptotic method of Krylov-Bogoliubov-Mitropolskii. Alam [52, 53] has also

developed a method for obtaining nonoscillatory solution of third order nonlinear

systems. Alam and Sattar [61] presented a unified KBM method, which is not the formal

form of the original KBM method, for solving n-th order nonlinear systems. The solution

contains some unusual variables, yet this solution is very important. Shamsul [41] has

also presented a modified and compact form of the Krylov-Bogoliubov-Mitropolskii

unified method for solving an n-th order nonlinear differential equation. Shamsul [4]

proposed a general formula based on the extended KBM method for obtaining asymptotic

solution of an n-th order time dependent quasi linear differential equation with damping.

An asymptotic method was presented by Ali Akbar et al. [2] based on the KBM method

for solving the fourth order over-damped nonlinear systems. Later, Ali Akbar et al. [2]

extended the KBM method which present in [1] for the forth order damped oscillatory

systems. Ali Akbar et al. [3] also found a simple technique for obtaining certain over-

damped solution of an n-th order nonlinear differential equation. Ali Akbar et al. [4]

presented a unified KBM method for solving n-th order nonlinear systems under some

15 | P a g e

special conditions including the case of internal resonance. Ali Akbar et al. [6] also

extended the theory of perturbation for forth order nonlinear systems with large damping.

2.2 Near Critically Damped Nonlinear Systems

The Krylov-Bogoliubov-Mitropolskii (KBM) method [14, 31] was basically devised for

approximating periodic solutions of second order nonlinear differential systems with

small nonlinearities

−=+dt

dxxfx

dt

xd,2

2

2

εω (2.2.1)

where ε is a very small positive parameter but not equal to zero.

The KBM [14, 31] method is particularly easy to understand and widely used tool to

obtain approximate solution of weakly nonlinear systems. For the physical importance,

Mendelson [40] reproduced Popov’s solutions [54]. Murty et al. [46] and Shamsul [60]

developed the method to nonlinear over-damped systems. However, both over-damped

solutions [46, 60] are not up to the mark for certain damping effects, especially, close to

the critically damped. Shamsul [62] has extended a novel perturbation technique to find

approximate analytical solution of second order both over-damped and critically damped

nonlinear systems. First of all, Shamsul and Sattar [62] presented a method to solve third

order critically damped autonomous nonlinear systems. Again Shamsul developed the

method presented in [62] to find approximate solutions of critically damped nonlinear

systems in the presence of different damping forces by considering different sets of

variational equations. Afterward, he unified the KBM method for solving critically

damped nonlinear systems. Shamsul [55] researched a third order critically damped

nonlinear systems whose unequal eigen values are in integral multiple. Shamsul [55] has

also extended the method to a third order over-damped system when two of the eigen

values are almost equal (i.e., the system is near to the critically damped) and the result is

16 | P a g e

small. In recent times, Shamsul has presented an asymptotic method for certain third

order nonoscillatory nonlinear systems, which gives desired results when the damping

force is close to the critically damping force.

In this thesis, we desire to develop the KBM method to solve third and fourth order

damped oscillatory and near critically damped nonoscillatory nonlinear systems with

small nonlinearities.

17 | P a g e

CHAPTER THREE

Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for

Fifth Order More Critically Damped Systems

3.1 The Method

We are going to propose a perturbation technique to solve a fifth order non-linear

differential equation of the form

1 2 3 4 5 ( , , , , )v iv ivx k x k x k x k x k x f x x x x xε+ + + + + = −ɺɺɺ ɺɺ ɺ ɺ ɺɺ ɺɺɺ (3.1.1)

where ( )vx and ( )ivx stand for the fifth and fourth derivatives respectively and over dots

are used for the first, second and third derivatives of x with respect to t; 1 2 3 4 5, , , ,k k k k k

are constants, ε is a sufficiently small parameter and ( )( ), , , , ivf x x x x xɺ ɺɺ ɺɺɺ is the given

nonlinear function. As the unperturbed equation of (4.2.1) has five real negative

eigenvalues, where the three eigen values and other two eigen values are assumed to be

the same. Suppose the eigenvalues are , , , and .λ λ λ µ µ− − − − −

When ε = 0, the equation (3.1.1) becomes linear and the solution of the corresponding

linear equation is

20 0 0 0 0( , ) ( ) ( )t tx t o a b t c t e d h t eλ µ− −= + + + + (3.1.2)

where 0a , 0b , 0c , 0d , 0h are constants of integration.

When ε ≠ 0 following Shamsul [16] an asymptotic solution of the equation (3.1.1) is

sought in the form

21( , ) ( ) ( ) ( , , , , , ) ...t tx t a bt ct e d ht e u a b c d h tλ µε ε− −= + + + + + + (3.1.3)

where a, b, c, d, h are the functions of t and they satisfy the first order differential

equations

18 | P a g e

1( , , , , , ) ...a A a b c d h tε= +ɺ

1( , , , , , ) ...b B a b c d h tε= +ɺ

1( , , , , , ) ...c C a b c d h tε= +ɺ (3.1.4)

1( , , , , , ) ...d D a b c d h tε= +ɺ

1( , , , , , ) ...h H a b c d h tε= +ɺ

Now differentiating (3.1.3) five times with respect to t, substituting the value of x and the

derivatives vx , ivx , xɺɺɺ , xɺɺ , xɺ in the original equation (3.1.1) utilizing the relations

presented in (3.1.4) and finally extracting the coefficients of ε, we obtain

2 2 22 21 1 1 1 1

12 2 2

3 3 2 01 11 1

( ) 3 6 6

( ) 2 ( ) ( ) ( , , , , , )

t

t

A B B C Ce D t t t C

t t t t t

D He D H t D D u f a b c d h t

t t

λ

µ

µ λ

λ µ λ µ

−

−

∂ ∂ ∂ ∂ ∂+ − + + + + + + ∂ ∂ ∂ ∂ ∂

∂ ∂ + − + + + + + = − ∂ ∂

(3.1.5)

where (0)( , , , , , ) ( , , , , )ivf a b c d h t f x x x x x= ɺ ɺɺ ɺɺɺ and 20 0 0 0 0( , ) ( ) ( )t tx t o a b t c t e d h t eλ µ− −= + + + +

We have expanded the function(0)f in the Taylor’s series (Sattar, 1986; Shamsul, 2001,

2002b; Shamsul and Sattar, 1996) about the origin in power of t. Therefore, we obtain:

(0) ( )

0 , , 0

( , , , , )q i j t

q i j k

f t a b c d h e λ µ∞ ∞

− +

= =

=

∑ ∑ (3.1.6)

Thus, using (3.1.6), the equation (3.1.5) becomes

2 2 22 21 1 1 1 1

12 2 2

3 3 21 11 1

( )

0 , , 0

( ) 3 6 6

( ) 2 ( ) ( )

( , , , , )

t

t

q i j t

q i j k


t t t t t

D He D H t D D u

t t

t a b c d h e

λ

µ

λ µ

µ λ

λ µ λ µ

−

−

∞ ∞− +

= =

∂ ∂ ∂ ∂ ∂+ − + + + + + + ∂ ∂ ∂ ∂ ∂

∂ ∂ + − + + + + + ∂ ∂

= −

∑ ∑

(3.1.7)

Following the KBM method, Murty and Deekshatulu (1969b), Sattar (1986), Shamsul

(2002b), Shamsul and Sattar (1996, 1997) imposed the condition that u1 does not contain

the fundamental terms (the solution (3.1.2) is called the generating solution and its terms

19 | P a g e

are called the fundamental terms) of f (0). Therefore, Eq. (3.1.7) can be separated for

unknown functions A1, B1, C1, D1, H1 and u1 in the following way:

2 2 22 21 1 1 1 1

12 2 2

13 ( )1 1

10 , , 0

( ) 3 6 6

( ) 2 ( , , , , )

t

t q i j t

q i j k


t t t t t

D He D H t t a b c d h e

t t

λ

µ λ µ

µ λ

λ µ

−

∞− − +

= =

∂ ∂ ∂ ∂ ∂+ − + + + + + + ∂ ∂ ∂ ∂ ∂

∂ ∂ + − + + = − ∂ ∂ ∑ ∑

(3.1.8)

3 2 ( )1

2 , , 0

( ) ( ) ( , , , , )q i j t

q i j k

anb D D u t a b c d h e λ µλ µ∞ ∞

− +

= =

+ + = −

∑ ∑ (3.1.9)

Now equating the coefficients of 0 1 2, ,t t t from equation (3.1.8), we obtain

22 31 1 1

1 12

( )0,

, , 0

( ) 3 6 ( ) 2

( , , , , )

t t

i j tK

i j k

A B De D C e D H

t t t

F a b c d h e

λ µ

λ µ

µ λ λ µ− −

∞− +

=

∂ ∂ ∂ + − + + + + − + ∂ ∂ ∂

= − ∑ (3.1.10)

22 31 1 1

2

( )1,

, , 0

( ) 6 ( )

( , , , , )

t t

i j tK

i j k

B C He D e D

t t t

F a b c d h e

λ µ

λ µ

µ λ λ µ− −

∞− +

=

∂ ∂ ∂+ − + + + − ∂ ∂ ∂

= − ∑ (3.1.11)

22 ( )1

1,2, , 0

( ) ( , , , , )t i j tK

i j k

Ce D F a b c d h e

tλ λ µµ λ

∞− − +

=

∂+ − = −∂ ∑ (3.1.12)

Here, we have only three Eqs. (3.1.10) , (3.1.11) and(3.1.12) for determining the

unknown functions A1, B1, C1 , D1 and H1 Thus, to obtain the unknown functions A1, B1,

C1, D1 and H1 we need to impose some conditions (Shamsul, 2002a, 2002b, 2002c,

2002d, 2003) between the eigen values. Different authors imposed different conditions

according to the behavior of the systems; such as in Shamsul (2002a) imposed the

condition

1 1 2 2 1 1 1 1 2... ( ... )( ... )n n ni i i i i iλ λ λ λ λ λ+ + + ≤ + + + + + + .

In this study, we have investigated solutions for the case µ >> λ. Therefore, we shall be

able to separate the Eq. (3.1.12) for unknown functions C1 and solving them for B1 and

20 | P a g e

H1 substituting the values of C1 into the Eq. (3.1.11) and applying the condition µ >> λ:

we can separate the Eq. (11) for two unknown functions B1 and H1; and solving them for

A1 and D1. Since , , , ,a b c d hɺ ɺ ɺɺ ɺ are proportional to small parameter, they are slowly

varying functions of time t and for first approximate solution, we may consider them as

constants in the right side. This assumption was first made by Murty and Deekshatulu

(1969b). Thus the solutions of the Eq. (3.1.4) become

0 1

0

( , , , , , )t

a a A a b c d h t dtε= + ∫

0 1

0

( , , , , , )t

b b B a b c d h t dtε= + ∫

0 1

0

( , , , , , ) (3.1.13)t

c c C a b c d h t dtε= + ∫

0 1

0

( , , , , , )t

d d D a b c d h t dtε= + ∫

0 1

0

( , , , , , )t

h h H a b c d h t dtε= + ∫

Equation (3.1.9) is a non-homogeneous linear ordinary differential equation; therefore, it

can be solved by the well-known operator method. Substituting the values of a, b, c, d, h

and u1 in the Eq. (3.1.3), we shall get the complete solution of (3.1.1). Therefore, the

determination of the first approximate solution is complete.

3.2 Example:

As an example of the above method, we have considered the Duffing type equation of

fifth order nonlinear differential system:

31 2 3 4 5

v ivx k x k x k x k x k x xε+ + + + + = −ɺɺɺ ɺɺ ɺ (3.2.14)

comparing (3.1.13) and (3.1.1), we obtain 3( , , , , )ivf x x x x x x=ɺ ɺɺ ɺɺɺ

(0) 2 3{( ) ( ) }t tf a bt ct e d ht eλ µ− −= + + + + (3.2.15)

21 | P a g e

For example equation (3.2.14), the equations (3.1.9)-(3.1.12) respectively become

22 31 1 1

1 12

3 3 2 (2 ) 2 ( 2 ) 3 3

( ) 3 6 ( ) 2

3 3

t t

t t t t

A B De D C e D H

t t t

a e a de ad e d e

λ µ

λ λ µ λ µ µ

µ λ λ µ− −

− − + − + −

∂ ∂ ∂ + − + + + + − + ∂ ∂ ∂

= − − − −

(3.2.16)

22 31 1 1

2

2 3 2 (2 ) 2 ( 2 ) 2 3

( ) 6 ( )

3 3(2 ) 3( 2 ) 3

t t

t t t t

B C He D e D

t t t

a be abd a h e bd adh e d he

λ µ

λ λ µ λ µ µ

µ λ λ µ− −

− − + − + −

∂ ∂ ∂+ − + + + − ∂ ∂ ∂

= − − + − + −

(3.2.17)

22 2 2 3 2 (2 )1

2

2 2 ( 2 ) 2 3

( ) 3( ) 3( 2 2 )

3( 2 ) 3

t t t

t t

Ce D ab a c e b d acd abh e

t

cd bdh ah e dh e

λ λ λ µ

λ µ µ

µ λ− − − +

− + −

∂+ − = − + − + +∂

− + + − (3.2.18)

() (

) (

3 2 3 2 2 2 3 3 2 2 3 2 41

2 4 2 5 3 6 3 2 2 2

2 3 2 4 2 2 2 3

3 4 2 5 (2 ) 2 2 2 2

2

and ( ) ( ) 3 3 3 6 3

3 3 3 2

2 2 2

2 2 3

2 2 2

t

t

D D u a a bt ab t b t a ct abct b ct

ac t bc t c t e a d abdt b dt

acdt bcdt c dt a ht abht b ht

acht bcht c ht e ad bd t cd t

adht bdht cd

λ

λ µ

λ µ−

− +

+ + = + + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + )( )

3 2 2 2 3 2 4 ( 2 )

3 2 2 2 3 3 33 3

t

t

ht ah t bh t ch t e

d d ht dh t h t e

λ µ

µ

− +

−

+ + +

+ + + +

(3.2.19)

Solution of equation (3.2.18) is

( ) ( )( )

2 2 2 2 ( )1 1 2

2 2 2 2 (3 )3 4

2 2

2

t t

t t

C l ab a c e l b d acd abh e

l cd bdh ah e l dh e

λ λ µ

µ µ λ

− − +

− − −

= + + + +

+ + + + (3.2.20)

where 1 2 2

3

4 ( 3 )l

λ µ λ= −

−, 2 2 2

3

4 ( )l

λ µ λ= −

+, 3 2 2

3

4 ( )l

µ µ λ= −

+, 4 2 2

3

4 (3 )l

µ µ λ= −

−

Now differentiating Eq.(3.2.20) with respect to t and putting the value of 1C

t

∂∂

in

Eq.(3.2.17) and solving B1 and H1 imposing the condition µ >> λ, we get

2 2 2 2 21 1 2( )t tB r a be r ab a c eλ λ− −= + + (3.2.21)

where 1 2 2

3

4 ( 3 )r

λ µ λ= −

−, 2 3 2

9

4 ( 3 )r

λ µ λ= −

−

and

22 | P a g e

( ) ( )( )

( )

2 2 2 ( )1 1 2

2 2 2 23 4

2 2 ( ) 2 25 6

2 2

2 2

2

t t

t t

t t

H p abd a h e p bd adh e

p d he p b d acd abh e

p cd bdh ah e p dh e

λ λ µ

µ λ

λ µ µ

− − +

− −

− + −

= + + +

+ + + +

+ + + +

(3.2.22)

where 1 3

3

2 ( )p

λ λ µ= −

+, 2 3

3

8 ( )p

µ λ µ= −

+, 3 3

3

2 (3 )p

µ µ λ= −

−,

4 4

9

( )p

λ λ µ= −

+, 5 4

9

8 ( )p

µ λ µ= −

+, 6 4

9

(3 )p

µ µ λ= −

−

Again, putting the value 1B

t

∂∂

, C1 and H1 in Eq. (3.2.16) and solving A1 and D1 and

imposing the condition µ >> λ we get

3 2 2 2 2 2 21 1 2 3 4( )( )t t tA q a e q a be q q a c ab eλ λ λ− − −= + + + + (3.2.23)

where 1 2 2

1

4 ( 3 )q

λ µ λ= −

−, 2 3 2

9

8 ( 3 )q

λ µ λ= −

−,

3 4 2

27

8 ( 3 )q

λ µ λ= −

−, 4 4 2

9

8 ( 3 )q

λ µ λ=

−

2 2 2 ( ) 3 2 2 21 1 2 3 4

2 2 ( ) 2 2 2 25 6 7

2 ( ) 2 2 2 28 9 10

2 2 ( )11

( 2 2 )

( 2 ) (2 )

( 2 ) ( 2 2 )

( 2 )

t t t t

t t t

t t t

t

and D n a de n ad e n d e n b d acd abh e

n cd bdh ah e n dh e n abd a h e

n bd adh e n d he n b d acd abh e

n cd bdh ah e

λ λ µ µ λ

λ µ µ λ

λ µ µ λ

λ µ

− − + − −

− + − −

− + − −

− +

= + + + + +

+ + + + + +

+ + + + + +

+ + + + 2 212

tn dh e µ−

(3.2.24)

where 1 3

3

2 ( )n

λ λ µ= −

+, 2 3

3

8 ( )n

µ λ µ= −

+, 3 3

1

2 (3 )n

µ µ λ= −

−, 4 5

9

( )n

λ λ µ= −

+,

5 5

9

16 ( )n

µ λ µ=

+, 6 5

9

(3 )n

µ µ λ=

−, 7 2 3

3

2 ( )n

λ λ µ= −

+, 8 3 2

3

8 ( )n

µ λ µ= −

+,

9 2 3

3

2 (3 )n

µ µ λ= −

−, 10 2 4

9

( )n

λ λ µ= −

+, 11 4 2

9

4 ( )n

µ λ µ= −

+, 12 2 4

9

(3 )n

µ µ λ= −

−

The solution of the Eq. (3.2.19) for u1 is

23 | P a g e

{

} {

3 3 3 2 2 2 4 31 1 2 3 4 5 6

2 2 5 4 3 2 3 67 8 9 10 11 12 13 14 15 16

5 4 3 2 (2 ) 217 18 19 20 21 22

3 2 223 24 25 26

( 6 )( ) ( )(

) ( ) (

) (2 2 )

( ) (

t

t

u e b abc m t m t m t m b c ac m t m t

m t m t m bc m t m t m t m t m t m c m t

m t m t m t m t m t m e bch ach b h

m t m t m t m c

λ

λ µ

−

− +

= + + + + + + +

+ + + + + + + + + +

+ + + + + + + + +

+ + + +

} {}

4 3 227 28 29 30 31

2 5 4 3 2 ( 2 ) 232 33 34 35 36 37

3 2 2 4 3 238 39 40 41 42 43 44 45 46

3 3 3 247 48 49 50

2 )( )

( ) (2 )

( ) ( )

( )

t

t

d bch m t m t m t m t m

c h m t m t m t m t m t m e cdh bh

m t m t m t m ch m t m t m t m t m

e h m t m t m t m

λ µ

µ

− +

−

+ + + + +

+ + + + + + + +

+ + + + + + + +

+ + + +

(3.2.25)

where

1 3 2

1

8 (3 )m

λ λ µ=

−, 2 3 2

3 3 2

8 (3 ) 2 3m

λ λ µ λ λ µ = + − −

,

3 3 2 2 2

3 3 3 3

4 (3 ) 2 (3 ) (3 )m

λ λ µ λ λ µ λ λ µ

= + + − − − ,

4 3 2 3 3 2 2

3 5 4 9 3

4 (3 ) 4 (3 ) 2 (3 ) (3 )m

λ λ µ λ λ µ λ λ µ λ λ µ

= + + + − − − − ,

5 3 2

3

8 ( 3 )m

λ µ λ=

−, 6 3 2

12 3 2

8 ( 3 ) 2 3m

λ µ λ λ λ µ = + − −

,

7 3 2 2 2

9 3 3 3

2 ( 3 ) 2 (3 ) (3 )m

λ µ λ λ λ µ λ λ µ

= + + − − − ,

8 3 2 3 3 2 2

9 5 4 9 3

( 3 ) 4 (3 ) 2 (3 ) (3 )m

λ µ λ λ λ µ λ λ µ λ λ µ

= + + + − − − − ,

9 3 2 4 4 3 3 2 2

9 15 5 5 6 9

( 3 ) 16 (3 ) 2 (3 ) (3 ) 2 (3 )m

λ µ λ λ λ µ λ λ µ λ λ µ λ λ µ

= + + + + − − − − − ,

10 3 2

3

8 ( 3 )m

λ µ λ=

−, 11 3 2

15 3 2

8 ( 3 ) 2 3m

λ µ λ λ λ µ = + − −

,

12 3 2 2 2

15 3 3 3

2 ( 3 ) 2 (3 ) (3 )m


= + + − − − ,

13 3 2 3 3 2 2

45 5 4 9 3

2 ( 3 ) 4 (3 ) 2 (3 ) (3 )m


= + + + − − − − ,

24 | P a g e

14 3 2 4 4 3 3 2 2

45 15 5 5 6 9

( 3 ) 16 (3 ) 2 (3 ) (3 ) 2 (3 )m


= + + + + − − − − − ,

15 3 2 5 5 4 4

3 2 2 3

45 21 6 15 15

( 3 ) 32 (3 ) 8 (3 ) 2 (3 )

15 6

4 (3 ) (3 )

mλ µ λ λ λ µ λ λ µ λ λ µ

λ λ µ λ λ µ

= + + +− − − −

+ + − −

16 3 2

1

8 ( 3 )m

λ µ λ=

−, 17 3 2

3 3 2

4 ( 3 ) 2 3m

λ µ λ λ λ µ = + − −

,

18 3 2 2 2

15 3 3 3

4 ( 3 ) 2 (3 ) (3 )m


= + + − − − ,

19 3 2 3 3 2 2

15 5 4 9 3

( 3 ) 4 (3 ) 2 (3 ) (3 )m


= + + + − − − − ,

20 3 2 4 4 2 2 3 3

45 15 5 9 6 5

( 3 ) 16 (3 ) 2 (3 ) (3 ) 2 (3 )m


= + + + + − − − − − ,

21 3 2 5 5 3 2 2 3

4 4

90 21 6 15 6

( 3 ) 32 (3 ) 4 (3 ) (3 )

15 15

2 (3 ) 8 (3 )

mλ µ λ λ λ µ λ λ µ λ λ µ

λ λ µ λ λ µ

= + + +− − − −

+ + − −

22 3 2 6 6 5 5 4 2

2 4 3 3

90 7 7 21 9 45

( 3 ) 16 (3 ) 16 (3 ) (3 ) 16 (3 )

15 5

2 (3 ) 4 (3 )

mλ µ λ λ λ µ λ λ µ λ λ µ λ λ µ

λ λ µ λ λ µ

= + + + +− − − − −

+ + − −

23 2 3

3

4 ( )m

λ µ λ=

+, 24 2 3

9 1 3

4 ( )m

λ µ λ λ λ µ = + + +

,

25 2 3 2 2

9 3 3 6

2 ( ) 4 ( ) ( )m

λ µ λ λ λ λ µ λ µ

= + + + + + ,

26 2 3 3 2 3 2

9 1 6 10 9

2 ( ) 2 ( ) ( ) 4 ( )m

λ µ λ λ λ λ µ λ µ λ λ µ

= + + + + + + + ,

27 2 3

3

4 ( )m

λ µ λ=

+, 28 2 3

3 1 3

( )m

λ µ λ λ λ µ = + + +

,

25 | P a g e

29 2 3 2 2

9 3 3 6

( ) 4 ( ) ( )m


= + + + + + ,

30 2 3 3 2 3 2

18 1 6 10 9

( ) 2 ( ) ( ) 4 ( )m


= + + + + + + + ,

31 2 3 4 3 4 3 2 2

18 15 10 15 3 9

( ) 16 ( ) ( ) 2 ( ) 2 ( )m

λ µ λ λ λ λ µ λ µ λ λ µ λ λ µ

= + + + + + + + + +

32 2 3

3

4 ( )m

λ µ λ=

+, 33 2 3

15 1 3

4 ( )m

λ µ λ λ λ µ = + + +

34 2 3 2 2

15 3 3 6

( ) 4 ( ) ( )m


= + + + + +

35 2 3 3 2 3 2

45 1 6 10 9

( ) 2 ( ) ( ) 4 ( )m


= + + + + + + +

36 2 3 4 3 4 3 2 2

90 5 10 15 3 9

( ) 16 ( ) ( ) 2 ( ) 2 ( )m

λ µ λ λ λ λ µ λ µ λ λ µ λ λ µ

= + + + + + + + + +

37 2 3 5 4 4 5

2 3 3 2

90 3 15 15 21

( ) 16 ( ) 16 ( ) ( )

15 3

2 ( ) ( )

mλ µ λ λ λ λ µ λ λ µ λ µ

λ λ µ λ λ µ

= + + ++ + + +

+ + + +

38 3 2

3

8 ( )m

µ µ λ=

+, 39 3 2

9 3 2

8 ( ) 2m

µ µ λ λ λ µ = + + +

40 3 2 2 2

9 3 3 3

4 ( ) 2 ( ) ( )m

µ µ λ λ λ µ λ λ µ

= + + + + +

41 3 2 3 3 2 2

9 5 4 3 9

4 ( ) 4 ( ) ( ) 2 ( )m

µ µ λ λ λ µ λ λ µ λ λ µ

= + + + + + + +

42 3 2

3

8 ( )m

µ µ λ=

+

43 3 2

3 3 2

2 ( ) 2m

µ µ λ λ λ µ = + + +

26 | P a g e

44 3 2 2 2

9 3 3 3

2 ( ) 2 ( ) ( )m

µ µ λ λ λ µ λ λ µ

= + + + + +

45 3 2 3 3 2 2

9 5 4 3 9

( ) 4 ( ) ( ) 2 ( )m

µ µ λ λ λ µ λ λ µ λ λ µ

= + + + + + + +

46 3 2 4 4 3 3 2 2

9 15 5 5 6 9

( ) 16 ( ) 2 ( ) ( ) 2 ( )m

µ µ λ λ λ µ λ λ µ λ λ µ λ λ µ

= + + + + + + + + +

47 2 3

1

4 (3 )m

µ µ λ=

−

48 2 3

3 3 1

4 (3 ) 3m

µ µ λ µ λ µ = + − −

49 2 3 2 2

3 6 3 3

2 (3 ) (3 ) 4 (3 )m

µ µ λ µ λ µ µ µ λ

= + + − − −

50 2 3 3 3 2 2

3 10 1 9 6

2 (3 ) (3 ) 2 4 (3 ) (3 )m

µ µ λ µ λ µ µ µ λ µ µ λ

= + + + − − − −

substituting the values of A1, B1, C1, D1 and H1 and from Eq. (3.2.23), (3.2.21), (3.2.20),

(3.2.24) and (3.2.22) into Eq. (3.1.4), we obtain:

{ }3 2 2 2 2 2 21 2 3 4( )( )t t ta q a e q a be q q a c ab eλ λ λε − − −= + + + +ɺ

{ }2 2 2 2 21 2( )t tb r a be r ab a c eλ λε − −= + +ɺ

{}

2 2 2 2 ( )1 2

2 2 2 2 (3 )3 4

( ) ( 2 2 )

( 2 )

t t

t t

c l ab a c e l b d acd abh e

l cd bdh ah e l dh e

λ λ µ

µ µ λ

ε − − +

− − −

= + + + +

+ + + +

ɺ

{ 2 2 2 ( ) 3 21 2 3

2 2 2 2 ( )4 5

2 2 2 2 2 ( )6 7 8

2 2 2 29 10

2 2 ( )11 1

( 2 2 ) ( 2 )

(2 ) ( 2 )

( 2 2 )

( 2 )

t t t

t t

t t t

t t

t

d n a de n ad e n d e

n b d acd abh e n cd bdh ah e

n dh e n abd a h e n bd adh e

n d he n b d acd abh e

n cd bdh ah e n

λ λ µ µ

λ λ µ

µ λ λ µ

µ λ

λ µ

ε − − + −

− − +

− − − +

− −

− +

= + +

+ + + + + +

+ + + + +

+ + + +

+ + + +

ɺ

}2 22

tdh e µ−

(3.2.26)

27 | P a g e

{

}

2 2 2 ( )1 2

2 2 2 23 4

2 2 ( ) 2 25 6

(2 ) ( 2 )

( 2 2 )

( 2 )

t t

t t

t t

h p abd a h e p bd adh e

p d he p b d acd abh e

p cd bdh ah e p dh e

λ λ µ

µ λ

λ µ µ

ε − − +

− −

− + −

= + + +

+ + + +

+ + + +

ɺ

Here all of the Eqs. of (3.2.26) have no exact solutions, but since , , , ,a b c d hɺ ɺ ɺɺ ɺ are

proportional to the small parameterε , so they are slowly varying functions of time t.

Consequently, it is possible to replace a, b, c, d, h by their respective values obtained in

linear case (i.e., the values of a, b, c, d, h obtained whenε = 0) in the right hand side of

Eq. (3.2.26). This type of replacement was first introduced by Murty and Deekshatulu

(1969a, b) to solve similar type of nonlinear equations.

Therefore, the solution of (3.2.26) is:

2 2 23 2 2 2

0 1 0 2 0 0 3 4 0 0 0 0

1 1 1( )( )

2 2 2

t t te e ea a q a q a b q q a c a b

λ λ λ

ελ λ λ

− − − − − −= + + + + +

2 22 2 2

0 1 0 0 2 0 0 0 0

1 1( )

2 2

t te eb b r a b r a b a c

λ λ

ελ λ

− − − −= + + +

2 ( )2 2 2

0 1 0 0 0 0 2 0 0 0 0 0 0 0 0

2 (3 )2 2 2

3 0 0 0 0 0 0 0 4 0 0

1 1( ) ( 2 2 )

2

1 1( 2 )

2 3

t t

t t

e ec c l a b a c l b d a c d a b h

e el c d b d h a h l d h

λ λ µ

µ µ λ

ελ λ µ

µ µ λ

− − +

− − −

− −= + + + + + +

− −+ + + + −

(3.2.27)

2 ( ) 22 2 3

0 1 0 0 2 0 0 3 0

2 ( )2 2 2

4 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0

2 2 (2 2 2

6 0 0 7 0 0 0 0 0 8 0 0 0 0 0

1 1 1

2 2

1 1( 2 2 ) ( 2 )

2

1 1 1(2 ) ( 2 )

2 2

t t t

t t

t t

e e ed d n a d n a d n d

e en b d a c d a b h n c d b d h a h

e e en d h n a b d a h n b d a d h

λ λ µ µ

λ λ µ

µ λ

ελ λ µ µ

λ λ µ

µ λ

− − + −

− − +

− − −

− − −= + + + +

− −+ + + + + ++

− − −+ + + + +)

2 22 2

9 0 0 10 0 0 0 0 0 0 0 0

( ) 22 2 2

11 0 0 0 0 0 0 0 12 0 0

1 1( 2 2 )

2 2

1 1( 2 )

2

t

t t

t t

e en d h n b d a c d a b h

e en c d b d h a h n d h

λ µ

µ λ

λ µ µ

λ µ

µ λ

λ µ µ

+

− −

− + −

+− −+ + + +

− −+ + + + +

28 | P a g e

2 ( )2 2

0 1 0 0 0 0 0 2 0 0 0 0 0

2 22 2

3 0 0 4 0 0 0 0 0 0 0 0

( ) 22 2 2

5 0 0 0 0 0 0 0 6 0 0

1 1(2 ) ( 2 )

2

1 1( 2 2 )

2 2

1 1( 2 )

2

t t

t t

t t

e eh h p a b d a h p b d a d h

e ep d h p b d a c d a b h

e ep c d b d h a h p d h

λ λ µ

µ λ

λ µ µ

ελ λ µ

µ λ

λ µ µ

− − +

− −

− + −

− −= + + + + +

− −+ + + +

− −+ + + + +

Hence, we obtain the first approximate solution of the Eq. (3.2.14) as:

21( , ) ( ) ( )t tx t a bt ct e d ht e uλ µε ε− −= + + + + + (3.2.28)

where a, b, c, d and h are given by the Eq. (3.2.27) and 1u is given by (3.2.25).

29 | P a g e

CHAPTER FOUR

4.1 Results and Discussion

In order to test the accuracy of an analytical approximate solution obtained by a certain

perturbation method, we compare the approximate solution to the numerical solution

(considered to be exact). With regard to such a comparison concerning the presented

technique of this article, we refer the work of Murty et al. [11]. In the present article, for

different sets of initial conditions as well as different sets of eigenvalues we have

compared the results obtained by perturbation solution (3.2.28) to those obtained by the

fourth order Runge–Kutta method. Beside this, we have computed the Pearson

correlation between the perturbation results and the corresponding numerical results.

From figures, we observed that the perturbation results from equation (3.2.28) show good

coincidence with the numerical results. First of all, for λ =0.97,µ = 3:1,ε = 0:01 and x(t,ε)

has been computed (3.2.28), in which a, b, c, d, h by the equations (3.2.27) with initial

conditions

0 0 0 0 00.005, 0.005, 0.03, 0.4 0.3a b c d and h= = = = =

i.e,

( )( )

x 0 0.00500, (0) 0.013683, (0) 0.01000,

(0) 0.039641, 0.04000iv

x x

x x

= = − =

= = −

ɺ ɺɺ

ɺɺɺ

In these cases, the perturbation results obtained by the solution (3.2.28) and the

corresponding numerical results computed by a fourth order Runge–Kutta method with a

small time increment ∆t = 0:05, are plotted in the Fig. 1. The correlation between these

two results has also been calculated: which is 0:999999956.

30 | P a g e

Fig. 1. Perturbation solution plotted by dot line and numerical solution plotted by

continuous line.

Secondly, we have considered λ= 0.98, µ = 3.2, and ε = 0.01 with initial

Conditions

0 0 0 0 00.05, 0.05, 0.02, 0.3 0.3a b c d and h= = = = =

i.e,

( )( )

x 0 0.541421, (0) 0.0683469, (0) 0.540695,

(0) 1.801215, 2.264117iv

x x

x x

= = − = −

= =

ɺ ɺɺ

ɺɺɺ

In these cases, the perturbation results obtained by the solution (3.2.28), and the

corresponding numerical results computed by a fourth-order Runge–Kutta method with a

small time increment ∆t = 0:05, are plotted in the Fig. 2. The correlation between these

two results has also been calculated which is 0.99992479.

31 | P a g e

Fig.2. Perturbation results are plotted by dotted line and numerical results are plotted by

continuous line.

32 | P a g e

Finally, it can readily be seen that for λ= 0.98, µ = 3.2, and ε = 0.01 with initial

conditions

0 0 0 0 00.5, 0.05, 0.0001, 0.3 0.003a b c d and h= = = = =

( )( )

x 0 0.55, (0) 0.22147, (0) 2.79745,

(0) 0.000866, 21.313951.iv

x x

x x

= = − = −

= =

ɺ ɺɺ

ɺɺɺ

In these cases, the perturbation results obtained by the solution (25), and the

corresponding numerical results computed by a fourth-order Runge–Kutta method with a

small time increment ∆t = 0:05 are plotted in Fig. 3. The correlation between these two

results has also been calculated which is 0:999879695.

33 | P a g e

Fig.3. Perturbation solution plotted by dot line and numerical solution plotted by

Continuous line.

34 | P a g e

5.1 Conclusion:

An analytical approximate solution based on the theory of KBM for fifth order more

critically damped nonlinear systems is investigated in this article. The correlation

between the results of the perturbation solution and the corresponding numerical solution

obtain by a fourth-order Runge–Kutta method have been calculated, which shows that

these two results are strongly-correlated. The result obtained for different sets of initial

conditions as well as different set of triply equal and a set of paired eigenvalues, show a

excellent coincidence with corresponding numerical results.

35 | P a g e

References

1. Ali Akbar, M., Paul A. C. and Sattar M. A., An Asymptotic Method of Krylov-

Bogoliubov for Fourth Order Over-damped Nonlinear Systems, Ganit, J. Bangladesh

Math. Soc., Vol. 22, pp. 83-96, 2002.

2. Ali Akbar, M., Shamsul Alam M. and Sattar M. A., Asymptotic Method for Fourth

Order Damped Nonlinear Systems, Ganit, J. Bangladesh Math. Soc. Vol. 23, pp. 41-

49, 2003.

3. Ali Akbar, M., Shamsul Alam M. and M. Sattar M., A Simple Technique for

Obtaining Certain Over-damped Solutions of an n-th Order Nonlinear Differential

Equation, Soochow Journal of Mathematics Vol. 31(2), pp. 291-299, 2005.

4. Ali Akbar, M., Shamsul Alam M. and Sattar M. A., Krylov-Bogoliubov-Mitropolskii

Unified Method for Solving n-th Order Nonlinear Differential Equations Under Some

Special Conditions Including the Case of Internal Resonance, Int. J. Non-linear Mech,

Vol. 41, pp. 26-42, 2006.

5. Ali Akbar, M., M. Sharif Uddin, Rokibul Islam M. and Afroza Ali Soma, Krylov-

Bogoliubov-Mitropolskii (KBM) Method for Fourth Order More Critically Damped

Nonlinear Systems, J. Mech. of Continua and Math. Sciences, Vol. 2(1), pp. 91-107,

2007.

6. Ali Akbar, M., M. Shamsul Alam, S. S. Shanta, M. Sharif Uddin and M. Samsuzzoha,

"Perturbation Method for Fourth Order Nonlinear Systems with Large

Damping”, Bulletin of Calcutta Math. Soc., Vol. 100(1), pp. 85-92, 2008.

7. Ali Akbar, M., M. Sharif Uddin and Mo.Rokibul Islam, “An Extension of the

Krylov-Bogoliubov-Mitropolskii (KBM) Method for Third Order Critically Damped

Nonlinear Systems”, J. Mechanics of continua and Mathematical Sciences, Vol. 3(1),

pp. 264-279, 2008.

8. Ali Akbar, M. and M. Sharif Uddin “On Asymptotic Solutions of Fourth Order Over

Damped Nonlinear Systems in Presence on Certain Damping Forces”, Journal of

Physical Sciences, (in press).

9. Ali Akbar, M., Anup Kumar Dutta and M. Eliyas Ali, Asymptotic Solution of a

Fourth Order Symmetrical Nonlinera Systems, J. Mech. of Continua and Math.

Sciences, (accepted for publication in Vol. 2(1), 2009).

36 | P a g e

10. Arya, J. C. and Bojadziev G. N., Damped Oscillating Systems Modeled by

Hyperbolic Differential Equations with Slowly Varying Coefficients, Acta

Mechanica, Vol. 35, pp. 215-221, 1980.

11. Arya, J. C. and Bojadziev G. N., Time Dependent Oscillating Systems with Damping,

Slowly Varying Parameters, and Delay, Acta Mechanica, Vol. 41, pp. 109-119, 1981.

12. Bellman, R., Perturbation Techniques in Mathematics, Physics and Engineering, Holt,

Rinehart and Winston, New York, 1966.

13. Van der Pol, B., On Relaxation Oscillations, Philosophical Magazine, 7-th series,

Vol. 2, 1926.

14. Bogoliubov, N. N. and Mitropolskii Yu., Asymptotic Methods in the Theory of

Nonlinear Oscillations, Gordan and Breach, New York, 1961.

15. Bojadziev, G. N., On Asymptotic Solutions of Nonlinear Differential Equations with

Time Lag, Delay and Functional Differential Equations and Their Applications

(edited by K. Schmit), 299-307, New York and London: Academic Press, 1972.

16. Bojadziev, G. N., Lardner R. W. and Arya J. C., On the Periodic Solutions of

Differential Equations Obtained by the Method of Poincare and Krylov-Bogoliubov,

J. Utilitas Mathematica, Vol. 3, pp. 49-64, 1973.

17. Bojadziev, G. N. and Lardner R. W., Monofrequent Oscillations in Mechanical

Systems Governed by Hyperbolic Differential Equation with Small Nonlinearities,

Int. J. Nonlinear Mech. Vol. 8, pp. 289-302, 1973.

18. Bojadziev, G. N. and Lardner R. W., Second Order Hyperbolic Equations with Small

Nonlinearities in the Case of Internal Resonance, Int. J. Nonlinear Mech. Vol. 9, pp.

397-407, 1974.

19. Bojadziev, G. N. and Lardner R. W., Asymptotic Solution of a Nonlinear Second

Order Hyperbolic Differential Equation with Large Time Delay, J. Inst. Math.

Applics Vol. 14, pp. 203-210, 1974.

20. Bojadziev, G. N., Damped Forced Nonlinear Vibrations of Systems with Delay, J.

Sound and Vibration, Vol. 46, pp. 113-120, 1976.

21. Bojadziev, G. N., The Krylov-Bogoliubov-Mitropolskii Method Applied to Models of

Population Dynamics, Bulletin of Mathematical Biology, Vol. 40, pp. 335-345, 1978.

37 | P a g e

22. Bojadziev, G. N. and Chan S., Asymptotic Solutions of Differential Equations with

Time Delay in Population Dynamics, Bull. Math. Biol. Vol. 41, pp. 325-342, 1979.

23. Bojadziev, G. N., Damped Oscillating Processes in Biological and Biochemical

Systems, Bull. Math. Biol. Vol. 42, pp. 701-717, 1980.

24. Bojadziev, G. N. and Edwards J., On Some Method for Non-oscillatory and

Oscillatory Processes, J. Nonlinear Vibration Probs. Vol. 20, pp. 69-79, 1981.

25. Bojadziev, G. N., Damped Nonlinear Oscillations Modeled by a 3-dimensional

Differential System, Acta Mechanica, Vol. 48, pp. 193-201, 1983.

26. Bojadziev, G. N. and Hung C. K., Damped Oscillations Modeled by a 3-dimensional

Time Dependent Differential Systems, Acta Mechanica, Vol. 53, pp. 101-114, 1984.

27. Cap, F. F., Averaging Method for the Solution of Nonlinear Differential Equations

with Periodic Non-harmonic Solutions, Int. J. Nonlinear Mech. Vol. 9, pp. 441-450,

1974.

28. Duffing, G., Erzwungene Schwingungen bei Veranderlicher Eigen Frequenz und Ihre

Technische Bedeutung, Ph. D. Thesis (Sammlung Vieweg, Braunchweig) 1918.

29. Gylden, Differentialgleichungen der Storungs Theorie (Differential Equations of the

Theory of Perturbation), Petersbourg, Vol. 31, 1883.

30. Zasmin Haque, M. Ali Akbar, Mo. Rokibul Islam and Mashbah Uddin, “Asymptotic

Solutions of Third Order Over-damped Non-linear Systems with Special Conditions”,

Journal of Interdisciplinary Mathematics, Vol. 11(3), pp. 357-371, 2008.

31. Kruskal, M., Asymptotic Theory of Hamiltonian and Other Systems with all

Situations Nearly Periodic, J. Math. Phys., Vol. 3, pp. 806-828, 1962.

32. Krylov, N. N. and Bogoliubov N. N., Introduction to Nonlinear Mechanics, Princeton

University Press, New Jersey, 1947.

33. Stoker, J. J., Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience,

New York, 1950.

34. Lardner, R. W. and Bojadziev G. N., Asymptotic Solutions for Third Order Partial

Differential Equations with Small Nonlinearities, Meccanica, Vol. 14, pp. 249-256,

1979.

38 | P a g e

35. Liapounoff, M. A., Probleme General de la Stabilite du Mouvement (General

Problems of Stability of Motion), Annales de la Faculte des Sciences de Toulouse,

Paris, Vol. 9, 1907.

36. Lin, J. and Khan P. B., Averaging Methods in Prey-Predator Systems and Related

Biological Models, J. Theor. Biol. Vol. 57, pp. 73-102, 1974.

37. Shamsul Alam, M., Ali Akbar M., Zahurul Islam M., A General Form of Krylov-

Bogoliubov-Mitropolskii Method for Solving Non-linear Partial Differential

Equations, Journal of Sound and Vibration, Vol. 285, pp. 173-185, 2005.

38. Mandelstam, L. and Papalexi N., Expose des Recherches Recentes sur les

Oscillations Non-lineaires (Outline of Recent Research on Non-linear Oscillations),

Journal of Technical physics, USSR, 1934.

39. McLachlan, N. W., Ordinary Nonlinear Differential Equations in Engineering and

Physical Science, Clarendon Press, Oxford, 1950.

40. Mendelson, K. S., Perturbation Theory for Damped Nonlinear Oscillations, J. Math.

Physics, Vol. 2, pp. 3413-3415, 1970.

41. Shamsul Alam M., A Modified and Compact Form of Krylov-Bogoliubov-

Mitropolskii Unified Method for an n-th Order Nonlinear Differential Equations, Int.

J. Nonlinear Mechanics, Vol. 39, pp. 1343-1357, 2004.

42. Minorsky, N., Nonlinear oscillations, Van Nostrand, Princeton, N. J., pp. 375, 1962.

43. Mitropolskii, Yu., Problems on Asymptotic Methods of Non-stationary Oscillations

(in Russian), Izdat, Nauka, Moscow, 1964.

44. Mulholland, R. J., Nonlinear Oscillations of Third Order Differential Equation, Int. J.

Nonlinear Mechanics, Vol. 6, pp. 279-294, 1971.

45. Murty, I. S. N., and Deekshatulu B. L., Method of Variation of Parameters for Over-

Damped Nonlinear Systems, J. Control, Vol. 9, no. 3, pp. 259-266,1969.

46. Murty, I. S. N., Deekshatulu B. L. and Krishna G., On an Asymptotic Method of

Krylov-Bogoliubov for Over-damped Nonlinear Systems, J. Frank. Inst., Vol. 288,

pp. 49-65, 1969.

47. Shamsul Alam M., A Unified KBM Method for Solving n-th Order Nonlinear

Differential Equation with Varying Coefficients, J. Sound and Vibration, Vol. 265,

pp. 987-1002, 2003.

39 | P a g e

48. Museenkov, P., On the Higher Order Effects in the Methods of Krylov-Bogoliubov

and Poincare, J. Astron. Sci. Vol. 12, pp. 129-134, 1965.

49. Nayfeh, A. H., Perturbation Methods, John Wiley and Sons, New York, 1973.

50. Shamsul Alam, M. and Hossain M. B., On Some Special Conditions of n-th Order

Non-oscillatory Nonlinear Systems, Communication of Korean Math. Soc. Vol. 18,

pp. 755-765, 2003.

51. Osiniskii, Z., Longitudinal, Torsional and Bending Vibrations of a Uniform Bar with

Nonlinear Internal Friction and Relaxation, Nonlinear Vibration Problems, Vol. 4, pp.

159-166, 1962.

52. Osiniskii, Z., Vibration of a One Degree Freedom System with Nonlinear Internal

Friction and Relaxation, Proceedings of International Symposium of Nonlinear

Vibrations, Vol. 111, pp. 314-325, Kiev, Izadt, Akad, Nauk USSR, 1963.

53. Shamsul Alam, M., Method of Solution to the n-th Order Over-damped Nonlinear

Systems Under Some Special Conditions, Bull. Cal. Math. Soc., Vol. 94, pp. 437-440,

2002.

54. Popov, I. P., A Generalization of the Bogoliubov Asymptotic Method in the Theory

of Nonlinear Oscillations (in Russian), Dokl. Akad. USSR Vol. 3, pp. 308-310, 1956.

55. Shamsul Alam, M., Bogoliubov's Method for Third Order Critically Damped

Nonlinear Systems, Soochow J. Math. Vol. 28, pp. 65-80, 2002.

56. Shamsul Alam, M., Perturbation Theory for n-th Order Nonlinear Systems with Large

Damping, Indian J. pure appl. Math. Vol. 33, pp. 1677-1684, 2002.

57. Raymond P. Vito and Cabak G., The Effects of Internal Resonance on Impulsively

Forced Nonlinear Systems with Two Degree of Freedom, Int. J. Nonlinear

Mechanics, Vol. 14, pp. 93-99, 1979.

58. Shamsul Alam, M., On Some Special Conditions of Over-damped Nonlinear

Systems, Soochow J. Math. Vol. 29, pp. 181-190, 2003.

59. Shamsul Alam M, KBM Method for Certain Non-oscillatory Non-linear System,

Journal of Bangladesh Academy of Science, Vol. 25, No. 2, 193-202, 2001.

60. Shamsul Alam, M., Asymptotic Methods for Second Order Over-damped and

Critically Damped Nonlinear Systems, Soochow Journal of Math. Vol. 27, pp. 187-

200, 2001.

40 | P a g e

61. Shamsul Alam, M. and Sattar M. A., A Unified Krylov-Bogoliubov-Mitropolskii

Method for Solving Third Order Nonlinear Systems, Indian J. pure appl. Math. Vol.

28, pp. 151-167, 1997.

62. Shamsul Alam, M. and Sattar M. A., An Asymptotic Method for Third Order

Critically Damped Nonlinear Equations, J. Mathematical and Physical Sciences, Vol.

30, pp. 291-298, 1996.

63. Sethna, P. R., Coupling in Certain Classes of Weakly Nonlinear Vibrating Systems,

Int. Symp. Nonlinear Differential Equation and Nonlinear Mechanics, pp. 58-70,

Academic Press, New York, 1963.

64. Shamsul Alam, M., Generalized Asymptotic method of KBM non-linear system,

Pure Appl. Math. Sci. Vol. 51, pp. 43-55, 2000.

65. Shamsul Alam M. and Sattar M. A., Time Dependent Third-order Oscillating

Systems with Damping, J. Acta Ciencia Indica, Vol. 27, pp. 463-466, 2001.

Documents

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