AMM.376.207

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Assessment of perturbation and homotopy-perturbation methods for

solving nonlinear oscillator equations

Maziar Ramezani and Thomas Neitzert

Centre of Advanced Manufacturing Technologies (CAMTEC), Auckland Universit of Technolog,Ne! "ealand#

Email$ maziar#ramezani%aut#ac#nz , thomas#neitzert%aut#ac#nz

Keywords: Artificial &arameter method (A'M) omoto& &ertur*ation method ('M) +an der 'oloscillator#

Abstract: In this paper, two modified perturbation methods, namely, artificial parameter method

(APM) and homotopy perturbation method (HPM) have been successfully implemented to find the

solution of van der Pol nonlinear oscillator equation. ifferent from classical perturbation method,

APM and HPM do not require small parameter and therefore, obtained appro!imate solutions may beuniformly valid for both wea" nonlinear systems and stron# nonlinear systems. $omparison of the

results obtained by the proposed methods reveals that APM and HPM are more effective compared to

classical perturbation method and with only a few terms, appro!imate the e!act solution with a fairly

reasonable error.

Introduction

Most scientific problems and physical phenomena occur nonlinearly. %!cept in a limited number of

these problems, we have difficulty in findin# their e!act analytical solutions. &herefore, there have

been attempts to develop new techniques for obtainin# analytical solutions which reasonably

appro!imate the e!act solutions '. Perturbation method is one of the well*"nown methods to solvethe nonlinear equations which was studied by a lar#e number of researchers such as +ayfeh ' and

-Malley '. /ince there are some limitations with the classical perturbation method, and also

because the basis of the classical perturbation method was upon the e!istence of a small parameter,

developin# the method for different applications is very difficult '0. &herefore, many different new

methods have recently introduced some ways to eliminate the small parameter such as artificial

parameter method introduced by 1iu '2, the homotopy analysis method by 1iao '3 and the homotopy

perturbation method by He '4.

Artificial parameter method (APM) is a modified perturbation method which uses an artificial

parameter as e!pandin# parameter, so the perturbation solutions will not depend upon the small

parameter in the equation. &he obtained appro!imate solutions, therefore, may be uniformly valid for

both wea" nonlinear systems and stron# nonlinear systems '5.

&he essential idea of homotopy perturbation method (HPM) is to introduce a homotopy parameter,

p , which ta"es the values from 6 to . 7hen 6= p , the system of equations usually reduces to a

sufficiently simplified form, which normally admits a rather simple solution. As p #radually

increases to , the system #oes throu#h a sequence of deformation. &he solution of each of which is

close to that at the previous sta#e of deformation. %ventually at = p , the system ta"es the ori#inal

form of equation and the final sta#e of deformation #ives the desired solution. ne of the most

remar"able features of the HPM is that usually only a few perturbation terms are sufficient to obtain a

reasonably accurate solution '8.

ne of the most intensely studied equations in nonlinear dynamics is the equation

),6',6)(

∞∈=+−+ ε ε udt

duu

dt

ud ()

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which 9altha:ar van der Pol '; introduced as a mathematical model of self*sustained oscillations of a

triode circuit with a cubic current*volta#e characteristic. In this model, ε controls the way in which

volta#e flows throu#h the system. &he van der Pol oscillator is one of the systems whose dampin#

forces are nonlinear. &hese nonlinear dampin# forces have a very important property< the dampin#

force will tend to increase the amplitude for small velocities but to decease it for lar#e velocities. It

follows that the state of rest is not stable and that an oscillation will be built up from rest even in theabsence of e!ternal forces, this accounts for the description of these oscillations as self*e!cited or

self*sustained oscillations.

It has a relevant interest, particularly in the e!treme cases when the parameter ε is either small or

very lar#e, which are associated with typical asymptotic behaviors of self*oscillatin# systems it

describes. 7hen ε is small enou#h, one obtains wea"ly nonlinear oscillations, i.e., oscillations which

sli#htly differ from harmonic motion, while when ε is very lar#e, in the limit tendin# to infinity, one

obtains rela!ation oscillations, i.e., stron#ly nonlinear oscillations e!hibitin# sharp periodic =umps.

&ypical e!amples of such systems are nearly sinusoidal electronic oscillators and multi*vibrators '6.

%!tensive studies have been done on investi#ation of oscillator problems by appro!imate analytical

methods. 9uonomo '6 presented the periodic solution of the van der Pol equation in the form of aseries conver#in# for all values of the dampin# parameter ε . >or small ε , the series solution reduced

to a perturbation series in powers of ε and was obtained from this series, essentially by the analytical

continuation method, usin# a suitable transformation of the parameter. He '5 presented a modified

perturbation method to search for analytical solutions of nonlinear oscillators without possible small

parameters. In his wor", an artificial perturbation equation was constructed by embeddin# an artificial

parameter, which was used as e!pandin# parameter. 7aluya and van Horssen ' used the

perturbation method based on inte#ratin# factors to appro!imate first inte#rals for a #enerali:ed

non*linear van der Pol oscillator equation. ?afei et al. ' applied variational iteration method to

nonlinear oscillators with discontinuities. &hey illustrated that this method is very effective and

convenient and does not require lineari:ation or small perturbation. :is and @ildirim '0 used

homotopy perturbation method combined with avera#in# to solve van der Pol oscillator with verystron# nonlinearity. &he result revealed that appro!imation obtained by this approach is valid

uniformly even for very lar#e parameters and is more accurate than strai#htforward e!pansion

solution. 1ope: et al. '2 studied the limit cycle of the van der Pol oscillator by applyin# the

homotopy analysis method. &he results reveal that homotopy analysis method for limit cycle of the

van der Pol equation is computationally e!tensive. $hen and 1iu '3 constructed deformation

equations usin# different initial conditions to reduce computational efforts for solvin# van der Pol

equation with homotopy analysis method.

As can be seen above, different techniques are applicable to solve nonlinear oscillator equations, but

there is no clear view that which method is more effective and accurate. In this paper we solve

nonlinear van der Pol oscillator equation usin# three different techniques, i.e., classical perturbationmethod, artificial parameter method and homotopy perturbation method. After chec"in# the equation

results and calculatin# the error differences with the e!act amount, the advanta#es and disadvanta#es

of each method will be discussed.

Classical perturbation method

Many physical systems are described by differential equations consistin# of two parts, one part

containin# linear terms with constant coefficients and a second part containin# nonlinear terms andor

terms with time*dependent coefficients, where the second part is relatively small compared to the

first. 7e refer to the small terms renderin# the system nonlinear andor possessin# time*dependent

coefficients as perturbations '4.7e consider the quasi*harmonic system described by the differential equation

),(6 uu f uu =+ω ()

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where ),( uu f is a sufficiently small nonlinear function of the displacement u and velocity u that it

can be re#arded as a perturbation. &o emphasi:e the fact that ),( uu f represents a perturbation, it is

convenient to rewrite %q. () in the form

),(6 uu f uu ε ω =+ (0)

in which ε is a small parameter. >or 6=ε , %q. (0) reduces to the equation of a harmonic oscillatorwhose solution is well "nown and for =ε , %q. (0) reduces to %q. () whose solution we see". In

essence, the introduction of the parameter ε enables us to effect the transition from the "nown

solution to the desired solution.

It is #enerally assumed that %q. (0) does not possess an e!act solution, so that the interest lies in an

appro!imate solution. /uch a solution must depend on the small parameter ε , in addition to the time

t , and must reduce to the harmonic solution as ε reduces to :ero. 9ecause ε is a small quantity, we

see" a solution of %q. (0) in the form of the power series

+++= )()()(),(

6 t ut ut ut u ε ε ε (2)

where the functions ,...),,6(,)( =it ui are independent of ε . %!pansion (2) permits a solution of

%q. (0) to any desired de#ree of appro!imation. Indeed, )(6 t u is the solution of the equation of the

harmonic oscillator, obtained by lettin# 6=ε in %q. (0) and the rest of the e!pansion (2) are the

solution of the nonlinear part of %q. (0).

+ow consider the van der Pol oscillator which is described by %q. (). $onsistent with the

formulation of this section, we rewrite %q. () in the form

uuuu f uu )(),( −==+ ε ε (3)

Bsin# %q. (2) and retainin# small terms in ε throu#h second order, we write

)(')(

')()()(')(),(

66666

6

666

6

6

6

uuuuuuuuuuuu

uuuuuuuuuuuuuuu f

−+−−−+−

+−+−≅++×++−≅−=

ε

ε ε ε ε ε

(4)

&hen, insertin# %qs. (2) and (4) into %q. (3) and separatin# terms of different orders of ma#nitude,we obtain the desired perturbation equations

6<)( 666

=+ uuO ε (5)

6

6 )(<)( uuuuO −=+ε (8)

666

)(<)( uuuuuuuO −+−=+ε (;)

&he solution of %q. (5) with initial conditions 6)6(,)6( 66 == uu is

)cos(6 t u = (6)

/ubstitutin# 6u into %q. (8), we have

)cos(80)(cos)sin(

8)sin(

2 t t t t t u +−−= ()

/ubstitutin# 6 , uu into %q. (;), we have

)(cos548

80)(cos

;

3)cos(

8

0)sin(

34

)(cos)sin(

42

;)cos(

548

60 03

t t t t t t t t t t u −−+−−= ()

7e, therefore, obtain the second*order appro!imation of %q. (3) as

))(cos548

80)(cos

;

3)cos(

8

0)sin(

34

)(cos)sin(

42

;

)cos(548

60())cos(

8

0)(cos)sin(

8

)sin(

2

()cos(),(

03

6

t t t t t t t t t

t t t t t t t uuut u

−−+−−

++−−+=++= ε ε ε ε ε

(0)

Cenerally, the perturbation solutions are uniformly valid as lon# as the nonlinear part of the system

is small. However, we cannot rely fully on the appro!imations, because there is no criterion on which

the small nonlinear term should e!ists. /o, it is necessary to develop new nonlinear analytical

methods which do not require small parameters at all.

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Artificial parameter method (APM)

As mentioned above, classical perturbation technique has many shortcomin#s, because it is based

on small parameter assumption. &his so*called small parameter assumption #reatly restricts

applications of classical perturbation technique in en#ineerin# science since, as it is well "nown, an

overwhelmin# ma=ority of nonlinear problems havin# no small parameters at all. Meanwhile, the

determination of small parameters seems to be a special art requirin# special techniques. An

appropriate choice of small parameters leads to ideal results, however, an unsuitable choice of small

parameters results in bad effects, sometimes seriously '5. &o overcome these difficulties, a modified

perturbation technique dependin# upon an artificial parameter has been proposed by 1iu '2 which

will be discussed in the ne!t section.

Basic idea of Artificial parameter method

&o illustrate the basic idea of 1iuDs artificial perturbation method, we consider the followin#

e!ample<

6)6(,==+ uuu (2)

which has the e!act solution

t

t

exe

eu

−

−

+

−= (3)

In %q. (2) there e!ists no possible small parameter, so the perturbation method cannot be applied to

this simple e!ample. In order to use the perturbation method, 1iu '2 embedded an artificial parameter

in (2), so that he obtained the followin# equation<

,6))(( ==+−+ α α uuu (4)

/upposin# that the solution can be e!panded in powers of α , and processin# in a traditional fashion

of perturbation technique, we obtain the followin# result<

)()( −++−= −−− t eeeu t t t α (5)1et =α in %q. (5), we obtain the appro!imate solution of the ori#inal equation (2)

)()( −++−= −−− t eeeu t t t (8)

$omparison with the e!act solution (3) reveals that the obtained appro!imation (8) is of relatively

hi#h accuracy. n the other hand, if we embed the parameter li"e

,6))(( ==+−+ α α uuu (;)

its perturbation solution is not valid even for small time t . /o, 1iuDs method requires some special

s"ill.

Application of Artificial parameter method

+ow, we consider the van der Pol equation () with initial conditions 6)6(,)6( == uu . In our

study the parameter ε does not require to be small, that is ∞<< ε 6 . 7e construct the followin#

perturbation equation<

),( uu g uu α ω =+ (6)

where

uuuuu g )()(),( −+−= ε ω ()

/upposin# that solution of %q. (6) can be e!pressed in the followin# form<

+++=

6 uuuu α α ()

)( 6

+++= ω α ω α

ω ω

(0)

/ubstitutin# () and (0) into (6), and proceedin# with the same manipulation as in the classical

perturbation method, we have the followin# equations<


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6<)( 6

66

=+ uuO ω α (2)

6

666

)()(<)( uuuuuO −+−=+ ε ω ω α (3)

which can provide the first*order appro!imation of %q. (6). &he solution of %q. (2) with the initial

conditions 6)6(,)6( 66 == uu is

)cos(6 t u ω = (4)

and substitutin# the result into (3) we have

)sin())(cos()cos()(

6

t t t uu ω ω εω ω ω ω −−−=+ (5)

Avoidin# the presence of secular term needs 6 =ω . &hen, we obtain a particular solution for u

with the initial conditions 6)6(,6)6( == uu , which reads

ω

ω ω ω ω ε

ω

ω ε )0)sin()cos(()cos(

8

)sin(

2

t t t t t u

+−+−= (8)

/ubstitutin# 6ω into (0), the an#ular frequency can be identified as =ω . &hus, by usin# %q. ()

and settin# =α

we have the first*order appro!imation of van der Pol equation as)0)sin()cos(()cos(

8

)sin(

2

)cos(),( 6 t t t t t t uut u +−+−=+= ε ε ε (;)

&he hi#her*order appro!imations can be obtained by continuin# this procedure.

Homotopy perturbation method (HPM)

Homotopy perturbation method has been recently developed by Ei*Huan He '4 to overcome the

shortcomin#s of classical perturbation method. &he effectiveness of the new technique is presented in

'5,8. &his approach can ta"e full advanta#e of the classical perturbation technique and homotopy

analysis method.

Basic idea of homotopy perturbation method

HPM is a combination of the classical perturbation technique and homotopy technique. &o e!plain

the basic idea of HPM for solvin# nonlinear differential equations, we consider the followin#

nonlinear differential equation<

Ω∈=− r r f u A ,6)()( (06)

sub=ect to boundary condition

Γ ∈=∂∂ r nuu B ,6),( (0)

where A is a #eneral differential operator, B a boundary operator, )(r f is a "nown analytical

function, Γ is the boundary of domain Ω , and n∂∂ denotes differentiation alon# the normal drawn

outwards from Ω .&he operator A can, #enerally spea"in#, be divided into two parts< a linear part L and a nonlinear

part N . &herefore %q. (06) can be rewritten as follows<

6)()()( =−+ r f u N u L (0)

7e construct a homotopy of %q. (06), ℜ→×Ω ,6'<),( pr v which satisfies<

Ω∈∈=−+−−= r pr f v A pu Lv L p pv H ,,6'6)()(')()()'(),( 6 (00)

which is equivalent to

6)()(')()()(),( 66 =−++−= r f v N pu pLu Lv L pv H (02)

where ,6'∈ p is an embeddin# parameter, and 6u is an initial #uess appro!imation of %q. (06)

which satisfies the boundary conditions. It follows from (00) and (02) that6)()(),(6)()()6,( 6 =−==−= r f v Av H u Lv Lv H (03)


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&hus, the chan#in# process of p from :ero to unity is =ust that of ),( pr v from )(6 r u to )(r u . In

topolo#y, this is called deformation and )()( 6u Lv L − and )()( r f v A − are called homotopic.

Here the embeddin# parameter is introduced much more naturally and unaffected by artificial

factorsF furthermore, it can be considered as a small parameter for 6 ≤≤ p . /o it is very natural to

assume that the solution of (00) and (02) can be e!pressed as<+++=

6 v p pvvv (04)

&he appro!imate solution of %q. (06), therefore, can be readily obtained<

lim 6

→

+++==

p

vvvvu (05)

&he conver#ence of series (05) has been discussed in '4.

Application of homotopy perturbation method

+ow, we shall illustrate the solution of van der Pol oscillator #iven in %q. () with the initial

conditions 6)6(,)6( == uu usin# homotopy perturbation method. Accordin# to the homotopy

perturbation, we construct the followin# simple homotopy

,6'6)()()(),(

66 ∈=++−= pvdt

dv pu pLu Lv L pv H ε (08)

where udt

du

dt

ud u L +−= ε

)( . /o<

6)(

66

6

66

6

=++−+−+−+− vdt

dv pu

dt

du

dt

ud pu

dt

du

dt

ud v

dt

dv

dt

vd ε ε ε ε (0;)

/ubstitutin# ++= 6 pvvv into %q. (0;) and equatin# the terms with identical powers, yields the

set of the initial value problems, i.e.,6)6(,)6(,6)()(<)( 6666

6=′==− vvu Lv L pO (26)

6)6()6(,6)()(<)(

66

6

=′==++ vvv

dt

dvv Lv L pO ε (2)

or

6<)(

66

66

6

=++−++− vdt

dvv

dt

dv

dt

vd v

dt

dv

dt

vd pO ε ε ε (2)

7e now, set the initial appro!imation of %q. (26) as )cos()()( 66 t t ut v == . &herefore, from %q. (2)

we have

6))(cos())sin(()sin(

=−+++− t t t vdt dv

dt vd ε ε ε (20)

&he solution of %q. (20) with initial conditions 6)6()6( =′= vv reads<

)42;(

)sin()cos();28()(cos)sin(8)(cos0'

)42;)(2(

0)2(42)2('

)42;)(2(0

)42;)(2(

0)2(42)2('0)(

0

02

)))2(

((

20)))2(

((

+

+−−+−+

+−

−−++−+

×+−

++−

−−−++−−=

−−−+

ε

ε ε ε ε

ε ε

ε ε ε ε ε ε

ε ε ε ε

ε ε ε ε ε ε

ε ε ε ε

t t t t t

eet v

t t

(22)

7e therefore, obtain the first order appro!imate solution usin# %q. (05) by settin# =

p , whichresults into


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)cos()42;(

)sin()cos();28()(cos)sin(8)(cos0')42;)(2(

0)2(42)2('0

)42;)(2(

0)2(42)2('0)()(),(

0

02)))2(

((

20)))2(

((

6

t t t t t t

e

et vt vt u

t

t

++

+−−+−+

+−

−−++−++

+−

−−−++−−=+=

−−

−+

ε

ε ε ε ε

ε ε

ε ε ε ε ε ε

ε ε

ε ε ε ε ε ε

ε

ε ε

ε ε

(23)

&he solutions obtained followin# the above calculations usin# classical perturbation method,

artificial parameter method (APM) and homotopy perturbation method (HPM) are tabulated in &ables

G2 for different values of ε . >or computations, Maple Pac"a#e has been used. As can be seen from

&ables G0, as the values of ε increases, classical perturbation method fails to obtain accurate results.

%ven at .6=ε , the proposed APM and HPM (with first order appro!imations) produce more

accurate results compared to classical perturbation method (with second order appro!imation).

$omparison of the results obtained from APM and HPM and those of the e!act solution clearly reveal

the hi#h accuracy of the calculations of APM and HPM at all values of ε . As can be seen from thetables, at stron#er nonlinearities (e.#., =ε ), APM provides more accurate results, however, the

error differences are not remar"able.

Table 1 $omparison of the appro!imate solutions with e!act solution for .6=ε .

time %!act

solution

Perturbation %rror APM %rror HPM %rror

6.6 6 6 6

6. 6.;;3662 6.6636 6.;;3662 .5%*65 6.;;3662 .38%*65

6. 6.;86643 6.;; 6.6605 6.;86643 2.0%*65 6.;86644 3.4%*65

6.0 6.;3303 6.;5 6.6304 6.;3302 3.%*65 6.;3302 3.45%*65

6.2 6.;6 6.;2 6.6645 6.;6 .05%*68 6.;6 .45%*65

6.3 6.855204 6.;62 6.6500 6.855205 5.8%*65 6.855203 4.32%*65

6.4 6.82;8 6.830 6.6028 6.82;8 .3%*64 6.82;5; .85%*64

6.5 6.542 6.5;40 6.6232 6.54260 0.2%*64 6.5426;2 8.%*64

6.8 6.4;302 6.508 6.63380 6.4;305 .;%*63 6.4;306 .4%*63

6.; 6.4;63 6.43805 6.6403 6.4; .3%*63 6.4;83 0.3%*63

.6 6.30428 6.358;; 6.65;042 6.30423 4.0;%*63 6.304088 3.3;%*63

Table 2 $omparison of the appro!imate solutions with e!act solution for 3.6=ε .

time %!act

solution


6.6 6 6 6

6. 6.;;3662 .6 6.63 6.;;3662 .80%*65 6.;;3662 .4%*65

6. 6.;8663; .60 6.636;35 6.;8663; 0.;4%*65 6.;8663; .54%*65

6.0 6.;3355 .60 6.658 6.;3355 2.5%*65 6.;3353 .;%*64

6.2 6.;683 .6 6.6553 6.;684 .6%*64 6.;6865 8.8%*64

6.3 6.854823 6.;;45 6.04488 6.85483 4.40%*64 6.8548; .;8%*63

6.4 6.80322 6.;425 6.52 6.80344 .4%*63 6.80250 8.45%*63

6.5 6.54653 6.;0 6.43 6.543 6.6666 6.546;2 6.666

6.8 6.48;382 6.848;0 6.4665; 6.48;555 6.6668 6.48;53 6.666225

6.; 6.46;62 6.86363 6.0258 6.46;45 6.666542 6.468458 6.666842.6 6.3664; 6.505 6.265800 6.365 6.66;; 6.3;38 6.66338


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Table $omparison of the appro!imate solutions with e!act solution for =ε .

time %!act

solution


6.6 6 6 6

6. 6.;;3662 .62 6.623 6.;;3662 %*65 6.;;3662 .63%*65

6. 6.;8663 .68 6.6;82 6.;8663 0.85%*65 6.;8663 .28%*656.0 6.;335 . 6.335 6.;335 .2%*65 6.;33 4.4%*64

6.2 6.;6344 . 6.6358 6.;635 2.%*64 6.;6303 0.08%*63

6.3 6.8546;5 . 6.580;5 6.8540 .;0%*63 6.853;85 6.6664

6.4 6.8565 . 6.008454 6.85;4 6.66668 6.8262 6.66604;

6.5 6.53535 .68 6.24088 6.53523 6.66608; 6.53425 6.666;65

6.8 6.48622 .62 6.328; 6.48835 6.66; 6.486560 6.66;44

6.; 6.3;3583 6.;853; 6.43548 6.3;552 6.6608 6.3;024 6.660;6

.6 6.2;544 6.;5083 6.82034 6.3682 6.6682;0 6.2;0;8 6.665062

Table ! $omparison of the appro!imate solutions with e!act solution for =ε

.time %!act

solution

APM %rror HPM %rror

6.6 6 6

6. 6.;;3660 6.;;3660 .0%*65 6.;;3660 .;%*65

6. 6.;86603 6.;86603 0.50%*65 6.;86600 .8%*64

6.0 6.;336;8 6.;336;; .64%*64 6.;33650 .4%*63

6.2 6.;6642 6.;665; .45%*63 6.;;;0 6.66623

6.3 6.852345 6.852430 ;.8%*63 6.85268 6.666334

6.4 6.8588; 6.8834 6.666228 6.8435 6.66443

6.5 6.528800 6.53664 6.6642 6.523440 6.66203

6.8 6.44334 6.44;665 6.66358 6.43;45 6.66;4656.; 6.34308 6.35084 6.62;;8 6.330;2 6.668

.6 6.2222; 6.24008 6.62360 6.23;3 6.62548

Conclusions

Artificial parameter method and homotopy perturbation method are employed successfully to study

van der Pol nonlinear oscillator differential equation. &he absolute error, %!act and numerical results

are presented and compared with each other in tables for some values of ε . In conclusion, APM and

HPM provide hi#hly accurate numerical solutions for nonlinear problems in comparison with

classical perturbation method. Moreover, the methods do not require small perturbation. &he study

showed that the APM and HPM are simple and easy to use. Moreover, they minimi:e the

computational calculus and supplies quantitatively reliable results even with the first order

appro!imation.

"eferences

' +ayfeh, A.H.< Introduction to perturbation techniques. 7iley, +ew @or" (;8)

' -Malley, ?.%.< Introduction to /in#ular Perturbation. Academic, +ew @or" (;52)

'0 Can=i, .., ?a=abi, A.< Assessment of homotopyperturbation and perturbation methods in heat

radiation equations. Int. $ommun. Heat Mass &ransfer 00, 0;266 (664)

'2 1iu, C.1.< +ew research directions in sin#ular perturbation theory< artificial parameter approachand inverse*perturbation technique. $onference of 5th Modern Mathematics and Mechanics,

/han#hai, pp. 2530 (;;5)


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'3 1iao, /.E.< An appro!imate solution technique not dependin# on small parameters< a special

e!ample. Int. E. +on*1inear Mech. 060, 05086 (;;3)

'4 He, E.H.< Homotopy perturbation technique. $omput. Methods Appl. Mech. %n#. 58, 354

(;;;)

'5 He, E.H.< A modified perturbation technique dependin# upon an artificial parameter. Meccanica03, ;;0 (666)

'8 9arari, A., midvar, M., Chotbi, A.?., Can=i, ..< Application of homotopy perturbation

method and variational iteration method to nonlinear oscillator differential equations. Acta Appl.

Math. 62, 45 (668)

'; van der Pol, 9.< n rela!ation*oscillations. Philosophical Ma#. 5(), ;58;; (;4)

'6 9uonomo, A.< &he periodic solution of van der Pol-s equation. /IAM E. Appl. Math. 3;(),

345 (;;8)

' 7aluya, /.9., van Horssen, 7.&.< n the periodic solutions of a #enerali:ed non*linear an der

Pol oscillator. E. /ound ib. 48, 6;3 (660)

' ?afei, M., Can=i, .., aniali, H., Pashaei, H.< &he variational iteration method for nonlinear

oscillators with discontinuities. E. /ound ib. 063, 4246 (665)

'0 :is, &., @JldJrJm, A.< A note on HeDs homotopy perturbation method for van der Pol oscillator

with very stron# nonlinearity. $haos /olit. >ract. 02, ;8;;; (665)

'2 1ope:, E.1., Abbasbandy, /., 1ope:*?ui:, ?.< >ormulas for the amplitude of the van der Pol

limit cycle throu#h the homotopy analysis method. /cholarly ?esearch %!chan#e (66;)

doi<6.08266;832646.

'3 $hen, @.M., 1iu, E.K.< A study of homotopy analysis method for limit cycle of van der Pol

equation. $ommun. +onlinear /ci. +umer. /imulat. 2, 848 (66;)

'4 Meirovitch, 1.< >undamentals of vibrations. McCraw Hill, +ew @or" (66)

'5 He, E.H.< Application of homotopy perturbation method to nonlinear wave equations. $haos

/olit. >ract. 4, 4;3566 (663)

'8 He, E.H.< Homotopy perturbation method for solvin# boundary value problems, Phys. 1ett. A

036, 8588 (664)


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