YuanDepartm

ReceiveReceiveAccepted 8 March 2013 imate solution and the initial condition to eliminate secular terms. For the large e, the

n

;

heman der

e that the van derous syste

indstedtPomethod [20] introduces a transformation of the independent variable and expands the solution in power series of e. Anand Geer [3] expand the frequency in power series of e up to 163 terms by using FORTRAN and have determined tfrequency has two complex-conjugate singularities in the complex e2-plane. Rouald [20] introduces a parqe e1e ; eP 0 to improve LindstedtPoincare method. This method is available not only for small but for all e > 0. Bunomo[4] introduces another parameter qe e21e2 derived from the Shohat transformation and obtains a series expansion for the

0307-904X/$ - see front matter 2013 Elsevier Inc. All rights reserved.

Corresponding author.E-mail addresses: yuanju07@hotmail.com (Y.-J. Huang), hkliu.nccu@gmail.com (H.-K. Liu).

Applied Mathematical Modelling 37 (2013) 81188130

Contents lists available at SciVerse ScienceDirect

Applied Mathematical Modellinghttp://dx.doi.org/10.1016/j.apm.2013.03.033Such a system is called an autonomous system. By LienardLevinsonSmith Theorem, we can concludpol Eq. (1.1) has an essentially unique nontrivial periodic solution. In other words, this equivalent autonomhas a unique closed path in the phase plane [21].

A considerable number of methods [3,4,20] are proposed to nd the approximate solution of (1.1). Lm (1.2)

incaredersenhat theameteras phase plane (see [21]), that is,

_U V ;_V U e 1 U2

V :

8 0

where _U dUdt . This equation is a matvoltage characteristic in [27]. The va1:1

tical model of self-sustained oscillations of a triode electric circuit with currentPol equation can be written in the system of two rst order differential equationsAvailable online 24 March 2013

Keywords:Van der Pol equationVariational iteration methodHes polynomials

numerical results demonstrate that the modication method get an accurate approximateperiod than the other presented methods.

2013 Elsevier Inc. All rights reserved.d 22 May 2012d in revised form 5 December 2012

for solving van der Pol equations. The modication couples the classical variational itera-tion method with Hes polynomials, where the Hes polynomials are applied to the approx-a r t i c l e i n f o

Article history:

a b s t r a c t

In this paper, we provide a new modication of the variational iteration method (MVIM)-Ju Huang, Hsuan-Ku Liu ent of Mathematics and Information Education, National Taipei University of Education, TaiwanA new modication of the variational iteration methodfor van der Pol equations

journal homepage: www.elsevier .com/locate /apm

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frequency. Coefcients in the expansion are computed up to O(e402). Liu [12] applied the modied homotopy perturbationmethod to consider the large parameter e.

In contrast to previous methods, the variational iteration method, which was proposed originally by He [7,9], attractedmuch attention in the past few years as a promising method for solving nonlinear differential equations. The variational iter-ation method has been proved by many authors to be a powerful mathematics tool for dealing with various kind of linear andnonlinear problems [1,2,5,7,9,10,1319,28,29]. Additionally, the homotopy perturbation transform method and Hes polyno-mials are used to solve the fractional partial differential equation [2326]. From their results, it shows that this method iseffective and simple.

So far many modications are provided of the variational iteration method. As the solution is approximated by a polyno-mial, Abassy et al. [1,2] analyze the procedure of the variational iteration solution and introduce the modication to over-come the calculation of unneeded terms. They use it to given an approximate power series solutions for non-linear problems.Shao and Han [22] couple variational iteration method with the homotopy perturbation method to solve the sineGordonequation. In their numerical results, both modications provide an accurate approach to nd an approximation for the non-linear differential equations.

However, we nd that the variational iteration method as well as the modications is not applicable to (1.1). Observingthe variational iteration method, the expressions of the approximated solution are based in the selected linear operator

Lut d2udt2

u and Lut d2udt2. When Lut d2u

dt2 u, the approximation of (1.1) are expressed by the set of basis functions

tm cosnt; tm sinntjmP 0;nP 1f g which breaks down for t P O e2 because of the existence of the secular term as tnsintor tncost. When the linear operator Lut d2u2 , the solution of (1.1) is approximated as the form of polynomials. Obviously,

Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130 8119dt

this approximation tends to positive or negative innity as the time tends to innity for any order of tn. This is because thatthe van der Pol equation has a nontrivial periodic solution.

We nd it has better to express a periodic solution by the set of periodic basis functions. Hence, to nding an accurateapproximation of the van der Pol equations (or other nonlinear differential equations with a periodic solution), we shouldselect a set of periodic basis functions and eliminate the secular terms in each iteration. However, the modication of var-iational iteration method with Hes polynomials that select a set of periodic basis functions and eliminate the secular termshas not been found yet. The methods [1,2,22] do not eliminate the secular terms that makes their approximations inaccuratefor the van der Pol equations (see Figs. 1 and 2 in Section 2). On the other hand, these three methods did not provide thefrequency and the amplitude of the periodic solution directly, where the frequency and the amplitude are the major char-acteristics of a periodic function.

In this paper, the variational iteration method is modied by introducing a transformation such that the solution is ex-pressed by the periodic basis functions. Precisely, we couple the classical variational iteration method with Hes polynomialsand construct a new homotopy to solve (1.1). Moreover, our method expands the solution as well as the frequency and theamplitude of the solution and correct the secular terms by using the expansion of the frequency and the amplitude. Ourmodication proposed in Section 3 extends the variational iteration method with Hes polynomials to select a set of periodicbasis functions and eliminate the secular terms in each iteration. Therefore, our modication provides an accurate approx-imation for the van der Pol equation. This implies that our method provides a new idea of the variational iteration methodwith Hes polynomials for nding an approximation of the nonlinear differential equations with a periodic solution.

Observing the numerical results, the proposed method gets the periodic solution up to 10 terms after 6 iterations and theapproximation is close to the numerical solution when e = 1 and e = 2. We compare our approximation with other ones andthe numerical solution. We nd that our approximation with 6 iterations is more accurate than other high order

t 0.5 1.0 1.5 2.0

1.0

1.5

2.0

2.5

3.0Dashed line: the 5th-order approximation of the MVIM(I)Solid line: the numerical solution

Fig. 1. Comparison of the numerical solution with the 5th-order approximation of MVIM (I) when e = 1 and A = 2.

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8120 Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130t

2 4 6 8 10

2

2

4

Solid line: the numerical solution

Fig. 2. Comparison of the numerical solution with the 6th-order approximation of MVIM (II) when e = 1 and A = 2.appro203 tewhileSectio

Thtencetion 3basis fIn Secmodi

2. Var

Co

Table 1Valuse

e

12345

1020

100

Table 2Error fo

e

12345

1020

1006Dashed line: the 6th-order approximation of the MVIM(II)ximations from perturbation methods where Andersens result uses 82 terms in the expansion; Bunomos result usesrms in the expansion. For example, the error of our period T(e) for e 6 5 is less than 1% and for eP 10 is less than 1.5%,the error of other methods are all large than 1%. The details of the comparison results are displayed in Tables 1 and 2 inn 4.is paper is organized as follows. In Section 2, we introduce the basic idea of variational iteration method and its exis-modications. We apply these methods for the van der Pol equation and indicate they are not applicable. Then, in Sec-, we modify the variational iteration method by introducing a transformation to express the solution by the periodicunctions. We couple the classical variational iteration method with Hes polynomials to solve the van der Pol equation.tion 4, we compare our results with the numerical solution and other ones to show the efciency of our newcation.

iational iteration method and its existence modications

nsider the van der Pol equation as the form

d2udt2

u e 1 u2 dudt 0; t P 0u0 A; u00 0;

2:1

for period T(e).

(1) Numerical result (2) Our result (3) Lius result (4) Andersens result (5) Bunomos result

6.6628686 6.66219485 6.6490 6.663286859 6.632868597.62987448 7.60337105 7.5535 7.629874480 7.629874478.85909550 8.82517012 8.7271 8.85909604 8.85909549

10.20352369 10.1828653 10.033 10.203911 10.2035237611.61223067 11.6152236 11.412 11.61892 11.6122406119.07836957 19.2259700 18.743 19.446 19.0984672134.6823253 35.0490082 33.981 36.631 35.0603638

162.83707 163.661083 157.79 179.38 169.122629

r period T(e).

(1) Numerical result (2) Our result (%) (3) Lius result (%) (4) Andersens result (%) (5) Bunomos result (%)

6.6628686 0.01 0.21 0.01 0.457.62987448 0.35 1.00 0.00 0.008.85909550 0.38 1.49 0.00 0.00

10.20352369 0.20 1.67 0.00 0.0011.61223067 0.03 1.72 0.06 0.0019.07836957 0.77 1.76 1.93 0.1134.6823253 1.06 2.02 5.62 1.09

162.83707 0.51 3.10 10.16 3.86

Liu [12] applied the LienardLevinsonSmith Theorem to show that the van der Pol Eq. (2.1) has an essentially uniquenontrivial periodic solution; that is, the equivalent autonomous system of (2.1) has a unique closed path in the phase plane.

Lu Nu gt; 2:2wherefuncti

where

ear anAp

Thobtain

For

Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130 81212.2. Modied variational iteration method (I)

Abassy et al. [1,2] proposed the modied method (MVIM) and used it to given an approximate power series solutions fornon-linear problems.

MVIM is used for solving the general nonlinear initial value problem204801536

672 cos t 840 cos 3t 200 cos 5t 35 cos 7t 3 cos 9t 1680t sin te4

:u2 2 cos t 4 4 sin3t e 96 13 cos t 18 cos 3t 5 cos 5t 12t sin t e

120t cos t 165 sin t 162 sin 3t 50 sin 5t 7 sin 7te3u1 2 cos t 3e4 sin t e4sin 3t 2:8

Substituting (2.8) into (2.6), we get

3 sin t 1

1 2u0t A cos t 2:7example, we take A = 2 and obtainAssuming that the initial approximate solution of (2.1) has the form 0e multiplier, therefore, can be identied as k sins t, and the following variational iteration formula can beed:

un1t unt Z t0

sins t d2unsds2

uns e 1 u2ns duns

ds

( )ds 2:6 dunt ksdu0nsjst k0sdunsjst Z t0k00s ksdunjds 2:5dun1t dunt dZ t0kfu00ns uns e1 ~unsu0nsgdswhere ~un is considered as a restricted variation, i.e. d~un 0.We identify ks as an optimal Lagrange-multiplier0 ds ds

un1t unt

Z tk

d2uns2 uns e 1 ~u2ns

d~uns ds; 2:4d nonlinear problems [1,2,5,7,9,10,12,14,28,29].plying (2.3) to (2.1), the correction functional is written down as follows:( )solution, and ~un denotes a restricted variation, i.e. d~un 0. This method has been applied in dealing with various kind of lin-L is a linear operator, N is a nonlinear operator and g(t) is a inhomogeneous term. Then we can construct a correctonal as follows:

un1t unt Z t0kfLunn Nunn gngdn 2:3

k is a general Lagrange multiplier which can be identied optimally via variational theory, un is the nth approximateIn this section, we rst apply the variational iteration method as well as the modications to the van der Pol Eq. (2.1). Andthen, we demonstrate that these methods could not provide an accurate approximation for the periodic solution.

2.1. Basic concept for the variational iteration method (VIM)

Consider the following differential equation:

where

where

where ks s1! is called a general Lagrange multiplier, and Gn(x, t) obtained from NUn(x, t) = Gn(x, t) + O(t ) is apolyn

Ap

and o

where U = 0, U = A, and G (x, t) is a polynomial of degree (2n + 1) in t and is obtained from

Th

and

Th

8122 Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130p : u1x 0 ksLu0s H0 gsds;p2 : u2x u1

R x0 ksLu1s H1ds;

..

.

pn1 : un1 un R x0 ksLuns Hnds

..

.

2:16Let u n0p un and assume that Nu n0p Hn. By comparing the coefcients of the same power of p in Eq. (2.15), weobtain

p0 : u0 u0x1

R xHu;p 1 p u0x u p0ksLu N~u gsds 0 2:15

P1 n P1 nnomials [22].Consider the following homotopy: Z x is modied variational iteration method is made by the classical variational iteration method coupled with Hes poly-U2 U1 t

0s tfRU1 U0 G1 G0gds A At

2

2 At

4

24 16At3e 1

6A3t3e 1

20A3t5e:

2.3. Modied variational iteration method (II)0 2

ZU1 U0 t

s tfRU0 U1 G0 G1gds A A t2 2:14n n

e rst two approximations are obtained by observing (2.12):Z1 0 n

NU x; t G x; t Ot2n1 2:130

Un1 Un

t

s tfRUn Un1 Gn Gn1gds; 2:12dt dt dt

btain the following MVIM iteration formulaZ

Lut d

2u2 ; Rut ut e

du; Nut eut2 du ;omial of degree (s(n + 1) 1) in t (see [2]).plying (2.9) and (2.10) to (2.1), we dene1 0 0 n n nwhich is the Lagrange multiplier and is identied via variational theory; for more details and illustrative examples, see [1].

The modied variational iteration formula is obtained as form:

Un1x; t Unx; t Z t0ksfRUn Un1 Gn Gn1gds 2:11

tss1 s(n + 1)Un1 Un 0kfRUn Un1 Gn Gn1gds 2:10

U = 0, U = f and G (x, t) is a polynomial of degree n in t and is obtained from NU (x, t) = G (x, t) + O(tn + 1) and k 1atives with respect to x and Nu(x, t) is the nonlinear term.Eq. (2.9) (with s = 1) is solved using the MVIM formula:Z tLux; t Rux; t Nux; t 0ux;0 f0x@ux;t

@t jt0 f1x...

@s1ux;t@ts1 jt0 fs1x

2:9

L @s@ts, s = 1, 2, 3, . . . is the highest partial derivative with respect to t, R is a linear operator which has partial deriv-

and get the solution u limp!1P1

i0uipi.

Given u0 = 2 cos t. Consider the following homotopy:Z t

Let u n0p un and assume that Nu n0p Hn. By comparing the coefcients of the same power of p in Eq. (2.17), we

For

Sub

relationships with linear operators (see [11]). From above three methods, we nd that the solutions of Eq. (2.1) have different

types

Lut

20480

Th

whichWh

Thative i

Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130 8123d u

dt2 u 1 u2 du

dt 0 ; u0 A; u00 0:These approximations are valid only of small t, as shown in Figs. 1 and 2 for the following equation (e = 1):

2breaks down for t P O e2

because of the existence of the secular term as tn sin t or tn cos t.en Lut d2u

dt2, the initial approximation u0(t) governed by (2.1) is A and the second-order approximation is

u A At2

2 At

4

24 16At3e 1

6A3t3e 1

20A3t5e

is solution of (2.1) is approximated as the form polynomials. Obviously, this approximation tends to positive or neg-nnity as the time tends to innity for any order of tn.tm cosnt; tm sinntjmP 0;nP 1f g

is approximation of (2.1) are expressed by the set of basis functions 672 cos t 840 cos 3t 200 cos 5t 35 cos 7t 3 cos 9t 1680t sin te4of expressions for the second-order linear differential equations Lut d2udt2

u and Lut d2udt2. For example, when

d2udt2

u, the initial approximation u0(t) governed by (2.1) is A cos t and the second-order approximation is

u 2 cos t 3 sin t4

14sin 3t

e 1

9613 cos t 18 cos 3t 5 cos 5t 12t sin te2

120t cos t 165 sin t 162 sin3t 50 sin5t 7 sin7te3

15360

196

13e2 cos t 18e2 cos 3t 5e2 cos 5t 72e sin t 12te2 sin t 24e sin 3t

2.4. The set of basis functions for van der Pol equations

It is well known that a real function can be expressed by some different basis functions and basic functions have closestituting (2.19) into (2.18), we obtain the rst two approximations

u1 Z t0

sins t2 sin 3sds 3e4

sin t e4sin 3t 2:20

u2 u1 Z t

sins t Lu1s H1 dsH0 e u20 1

u00 2e sin 3t 2:19the nonlinear item Nut e u2n 1

u0n, the rst Hes polynomial is given as..

.

pn1 : un1 un R t0 sins tLuns Hnds1 0 0 0

p2 : u2 u1 R t0 sins tLu1s H1ds 2:18obtain the following recurrence formula

p0 : u0 u0tp1 : u R t sins tLu s H dsHu; p 1 p u0t u p0

sins tLu Nuds 0 2:17P1 n P1 n

From the physical points of view, the solution of van der Pol equations is periodic and the rigorous proof for their peri-odicitvan d

In

He

fcosnh; sinnhjnP 0g

3. A n

conce

3.1. Ba

as:

i0

If p

u limv v v v 3:8

He

8124 Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130Hnu0; . . . ;un n! @pn Ni0

piuip0

; n 0;1;2; . . . 3:101 @n Xn !i0i 0 1 2

where Hns are the so-called Hes polynomials [6]re, the nonlinear item N(u) above is considered as the decomposed form

Nu X1

piH H pH p2H 3:9p!1 0 1 2? 1, we get an approximation solution of Eq. (3.1):where u0 is an initial approximation of (2.8), which satises the boundary conditions. Thus, we have

Hv ;0 Lv Lu0 0; 3:5

Hv ;1 Av f r 0; 3:6We use the embedding parameter p as a small parameter, and assume that v in Eq. (3.4) can be written as a power series

of p:

v X1

v ipi v0 pv1 p2v2 3:7Hv ;p 1 pLv Lu0 pAv f r 0; p 2 0;1; r 2 K 3:4

According to HPM, we construct the a homotopyvr; p : K 0;1 ! R which satises:Lu Nu f r 0 3:3B u;@n

0; r 2 C 3:2

where A is a general differential operator, B is a boundary operator, f(r) is a known analytical function and C is the boundaryof domainK. Generally, the operator A can be divided into a linear part L(u) and a nonlinear part N(u). Eq. (3.1) can be written@u with the boundary conditions of:Au f r 0; r 2 K 3:1

Reviewing the basic concept of HPM [8], we consider the following function:pts of the homotopy method as well as Hes polynomials. Our new modication will be illustrated in Section 3.2.

sic concept of the homotopy perturbation methodOur modication is coupled the classical variational iteration method with Hes polynomials. Especially, we expandA P1n0pnAn and x P1n0pnxn to correct the secular term as en sin h and en cos h. In Section 3.1, we introduce the basicew modication of the variational iteration methodre, u(h) can be expressed by the periodic basis functionsxdh2

uxe 1 udh

0Section 3, we apply the transformation h =xt to (2.1) and obtain

2 d2u 2 duwhere x = 2p/T is the frequency, and T is the period of the solution.y can be found in Liu [12]. This implies that the above approximate solutions do not match the principal character ofer Pol equations. Therefore, it might be better to express a periodic solution by the set of periodic basis functions

fcosnxt; sinnxtjnP 0g;

3.2. The modication of the variational iteration method

To illustrate our modication, we rst introduce u P1n0pnun and give Nu P1n0pnHn.Acc

Th

Sub R

then w

Note tIn

n 2 3 n

Z h Z h

and o

That i

Thus,

Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130 8125Hu; p 1 pu0h u p0ksLu Nu gsds 0we construct the following homotopy:Z h1 pu0 u p0ksLu Nu gsds 00 00

s, Z h

u u pu u p

Z hksLu Nu gsds; u0 p0kfLu0s H0s gsgds u1p2 p

0kfLu1sp H1spgds

u2p3 pZ h0kfLu2sp2 H2sp2gds

un1pn p

Z h0kfLun1spn1 Hn1spn1gds

. . .

btainuh;p n0

p un u0 u1p u2p u3p unp u1 h0 kfLu1s Nu0 u1s Nu0sgds

u1 R h0 kfLu1s H1sgds..

.

un R h0 kfL

Xn2i0

uis Lun1s NXn1i0

ui

!s N

Xn2i0

ui

!s N

Xn2i0

ui

!s gsgds

un1 R h0 kLun1s N

Xn1i0

ui

!s N

Xn2i0

ui

!s

( )ds

un1 R h0 kLun1 Hn1sf gds

hat N(u0 + u1) = H0 + H1, N(u0) = H0 and NPn

i0ui Pn

i0Hi.this manner, we nd

X1u1 0 kfLu0s H0s gsgdsu2

R h0 kfLu0s Lu1 Nu0 u1s Nu0s Nu0s gsgdsRu0 u1 u2 u0 u1 R h0 kfLu0 u1s Nu0 u1s gsgds..

.

Xni0

ui Xn1i0

ui R h0 kfL

Xn1i0

ui

!s N

Xn1i0

ui

!s gsgds

e get

R hu0 u1 u0 h0 kfLu0s Nu0s gsgds;i0

stituting (3.12) into (3.11), from rst approximate solution, we haveUn Xn

ui 3:12e nth approximate solution is also defended asUn1h Unh 0kfLUns NUns gsgds; n 0;1; . . . 3:11ording to the variational iteration method, the nth approximate solution dended asZ h

3.3. Ap

Fir

whereEq

Consid

where

Nv

Thfollow

If p?

where

Substi

p : v A cos h

8126 Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130pn1 : vn1 vn An An1 cos h 0 sins hLuns Hnds n 1;2; . . .For the nonlinear item Nv x2 1 d2v

dh2xev2 1 dvdh, the rst Hes polynomials is given as

H0 e v20 1

v 00 ex0A0 4 A20

4

sin h A0 1x20

cos h ex0A30

4sin 3h.. R h0 0

p1 : v1 A1 cos hR h0 sins hLv0s H0ds

. 3:20tuting (3.16)(3.19) into (3.15) and comparing the coefcients of the same power of p, we obtainoHnv0; . . . ;vn n! @pn Ni0

p v ip0

; n 0;1;2; :::Hns are the so-called Hes polynomials

1 @n Xn i !Ae limp!1

Ae; p A0 A1 A2

Here, the nonlinear item N(v) above is considered as the following decomposed form

Nv X1i0

piHi H0 pH1 p2H2 3:19Ae;p X1n0

Anepn 3:18

1, we get an approximation solution of Eq. (3.14):

uh; e limp!1

vh; e; p v0 v1 v2

xe limp!1

xe; p x0 x1 x2 n0e solution as well as the frequency and the amplitude, u, x and A, are expanded, respectively in power series of p ass:

vh; e;p X1n0

vnh; epn 3:16

xe;p X1

xnepn 3:17h is an initial approximation of (3.14).p 2 [0, 1]. Because we want to nd the periodic approximate solution, we assume that Lv d2vdh2

v ,x2 1 d2v

dh2xe v2 1 dvdh, so that we can identify k sins h as an optimal Lagrange-multiplier and u0 = A cosHv ;p 1 pu0h v p0ksLv Nvds 0 3:15er the following homotopy: Z h

x2

d u

dh2 uxe 1 u2 du

dh 0 3:14st, we let

h xet 3:13x(e) = 2p/T(e) is the frequency. Note x(0) = 1.

. (1.1) becomes

2Besides, in order to obtain a uniform expansion for the solution of (1.1), we give the modication of variational iterationmethod for (1.1).plying the modication to the van der Pol equation

In order to ensure that no secular term appears in the next iteration, we equate the coefcients of sin h and cos h equal tozero. T

which

Substi

Hence

Inzero a

which

Assin h a

Insubstiobtainmathe

For

Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130 81272211840

1215 cos h 10170 cos 3h 19825 cos 5h 10745 cos 7h 1647 cos 9he

6 194565 sin h 582975 sin 3h 1322125 sin5h66355200

1106210 sin 7h 402678 sin 9h 49797 sin11he66355200552960

523043285 cos h 16920 cos 3h 21700 cos 5h 10745 cos 7h 1647 cos 9he44 4 96

79 sin h 27 sin 3h 20 sin 5h 4 sin 7he3ue; h 2 cos h 3 sin h 1 sin 3h

e 1 12 cos h 18 cos 3h 5 cos 5he2the process of calculations, we focus on nding out the coefcients of sin h and cos h to eliminate the secular terms. Bytuting v0, v1, . . ., vn into (3.20), we nd that there are no sin h and cos h in Ln but in the Hes polynomials Hn. From Hn, wexn and An iteratively.In this manner, the rest of components of the iteration formula (3.20) will be obtained by thematical soft wares, such as Mathematica in this paper.example, in the 6-th order approximationx2 3e4

512andA2 e

2

96the same reason, that is in order to ensure that no secular terms appear in the next iteration, equating the coefcient ofnd cos h equal to zero yields768x2e sin 3h 216e3 sin 3h 560e3 sin 5h 224e3 sin 7hH2

1384

1536x2 cos h 9e4 cos h 72e4 cos 3h 60e4 cos 5h 768A2e sin h 8e3 sin h 1152A2e sin 3hHence, the next Hes polynomials is given asv2 v1 A2 A1 cos hZ h0

sins hLv1 H1ds

A2 cos h 1396 e2 cos h 3

16e2 cos 3h 5

96e2 cos 5h 3

64e3 sin h 1

64e3 sin 3hSubstituting (3.23) into (3.20), we obtainimply

H1 32 e2 cos 3h 5

4e2 cos 5h 1

8e3 sin 3h 3:23x1 e2

16andA1 0order to ensure that no secular term appears in the next iteration, we equate the coefcients of sin h and cos h equal togain. Thus we requireH1 @@p

x0 x1p 2 1

d2v0dt2

d2v1dt2

p

! ex0 x1p v0 v1p 2 1

dv0

dt dv1

dtp

" #p0

4x1 14 e2

cos h 3

2e2 cos 3h 5e

2

4cos 5h 2A1e sin h 3A1 2x1e sin 3hhus we require

x0 1 and A0 2imply

H0 2e sin 3h 3:21tuting (3.21) into (3.20), we obtain

v1 A1 cos hZ h0

sins h2e sin 3hds A1 cos h 3e4 sin he4sin 3h: 3:22

, the next Hes polynomials is given as

515 sin h 163 sin3h 140 sin 5h 28 sin7he7

393216 12 cos h 18 cos 3h 5 cos 5he

8

49152

105 sin h2097152

35 sin 3h2097152

e9 Oe10

with the amplitude A and the frequency x given by

A 2 e2

96 1033e

4

552960 329e

6

1105920 e

8

49152 Oe9

and

x 1 e2

16 17e

4

3072 35e

6

884736 35e

8

786432 63e

10

8388608 Oe11

Besides, when e is large, we introduce a new parameter q, dened by the transformation qe e1e. Then we expand ourresults in the power series of q.

We expand ex e 1 e216 17e4

3072 35e6

884736 35e8

786432 63e10

8388608

in power series of q and obtain

ex q q2 15q3

16 13q

4

16 1937q

5

3072 1237q

6

3072 128771q

7

884736 104875q

8

884736 2558501q

9

7077888 1295279q

10

2359296 Oq11

which shall be compared in Section 4.

e = 100. This implies that our proposed method is more accurate than others. In summary, our modication uses less itera-

Fig. 3.line: th

8128 Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 811881302

Comparison of the numerical solution with the 6 iterations approximation of modied variational iteration method when e = 1 and A = 2. (1) Dashede 6th-order approximation and (2) solid line: the numerical solution.110 20 30 40 50124. Numerical comparison

At rst the numerical solutions are compared with the approximate solutions of our modied variational iteration meth-od in the cases of e = 1, A = 2 and e = 2, A = 2, where the numerical solutions and the approximated results are obtained byusing Mathematica 7. Observing Figs. 3 and 4, we nd that our 6-th iteration approximation is close to the numerical solu-tion. In the same case of e = 1, A = 2 we demonstrated that the variational iteration method as well as other modications areonly valid of small t in Figs. 1 and 2. As a result, our modication provided a better way of approximating the solution for thevan der Pol equation when parameters are small.

We then consider the cases of large parameters. In Table 1, we compare the period of (1.1) with Andersens result [3],Bunomos result [4] and Lius result [12] for the cases of e = 1, 2, 3, 4, 5, 10, 20, 100. We get the numerical values of othermethods from Andersen [3]. These values are also used by Bunomo [4] and Liu [12]. Our result is using 10 terms in theexpansion, Andersens result is using 82 terms, Lius result is using 8 terms and Bunomos result using 203 terms. Here,we get the period up to 10 terms after 6 iterations while Liu [12] gets 8 terms after 8 iterations. By comparing with thenumerical results, we nd that our 6-th iteration approximation is more close to the numerical result than others. To makeit clearly, we indicate the error of period in Table 2. The error in this paper is dened by the formula jTrTn jTn 100%, where Tr isthe approximate result and Tn is the numerical result. Then, we nd that our error of period T(e) for e 6 5 is less than 1%.When eP 10, we nd that our error of period T(e) is less than 1.5% but other methods are all greater than ours. Especially,our error of period T(e) is 0.51% while Lius error is 3.10%, Andersens error is 10.16% and Bunomos error is 3.86% when

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(1982) 678693.[4] A. Buonomo, The periodic solution of van der Pols equation, Soc. Ind. Appl. Math. 59 (1998) 156171.In this paper, we nd the variational iteration method is not applicable for the van der Pol equation because of the exis-tence of the secular terms. We have then introduce a simple transformation to overcome this disadvantage and modify thevariational iteration method by coupling the classical variational iteration method with Hes polynomials. We nd that our 6iterations are more accurate than other high-order approximations from perturbation methods. That is, our modied vari-ational iteration method is more effective. We also get an accurate approximate period of (1.1) for the large e by our proposedmethod and show that our approximation is close to the numerical solution when e = 1 and e = 2. In the future research, itwill be interesting to extend our proposed method to solve the familiar van der Pol Dufng equation.

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5. Conclusion

10 20 30 40 50

2

1

1

2

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8130 Y.-J. Huang, H.-K. Liu / Applied Mathematical Modelling 37 (2013) 81188130

A new modification of the variational iteration method for van der Pol equations1 Introduction2 Variational iteration method and its existence modifications2.1 Basic concept for the variational iteration method (VIM)2.2 Modified variational iteration method (I)2.3 Modified variational iteration method (II)2.4 The set of basis functions for van der Pol equations

3 A new modification of the variational iteration method3.1 Basic concept of the homotopy perturbation method3.2 The modification of the variational iteration method3.3 Applying the modification to the van der Pol equation

4 Numerical comparison5 ConclusionReferences