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SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS

International Journal of Modern Physics B

April 2006
20(10):1141-1199

DOI:10.1142/S0217979206033796

Authors:

Ji-Huan He

Soochow University (PRC)

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this field in China so the references, therefore, are not exhaustive.

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April 13, 2006 10:4 WSPC/140-IJMPB 03379

International Journal of Modern Physics B

Vol. 20, No. 10 (2006) 1141–1199

 World Scientiﬁc Publishing Company

SOME ASYMPTOTIC METHODS FOR STRONGLY

NONLINEAR EQUATIONS

JI-HUAN HE

College of ience, Donghua University,

Shanghai 200051, People’s Republic of China

jhhe@dhu.edu.cn

Received 28 February 2006

This paper features a survey of some recent developments in asymptotic techniques,

which are valid not only for weakly nonlinear equations, but also for strongly ones.

Further, the obtained approximate analytical solutions are valid for the whole solution

domain. The limitations of traditional perturbation methods are illustrated, various

modiﬁed perturbation techniques are proposed, and some mathematical tools such as

variational theory, homotopy technology, and iteration technique are introduced to over-

come the shortcomings.

In this paper the following categories of asymptotic methods are emphasized:

(1) variational approaches, (2) parameter-expanding methods, (3) parameterized pertur-

bation method, (4) homotopy perturbation method (5) iteration perturbation method,

and ancient Chinese methods.

The emphasis of this article is put mainly on the developments in this ﬁeld in China

so the references, therefore, are not exhaustive.

Keywords: Analytical solution; nonlinear equation; perturbation method; variational

theory; variational iteration method; homotopy perturbation; ancient Chinese mathe-

matics.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142

2. Variational Approaches . . . . . . . . . . . . . . . . . . . . . . . 1144

2.1. Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144

2.2. Soliton solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147

2.3. Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149

2.4. Limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150

2.5. Variational iteration method . . . . . . . . . . . . . . . . . . . . . . . 1153

3. Parameter-Expanding Methods . . . . . . . . . . . . . . . . . . . . 1156

3.1. Modiﬁcations of Lindstedt–Poincare method . . . . . . . . . . . . . . . 1156

3.2. Bookkeeping parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 1161

4. Parametrized Perturbation Method . . . . . . . . . . . . . . . . . . 1165

5. Homotopy Perturbation Method . . . . . . . . . . . . . . . . . . . 1171

5.1. Periodic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173

5.2. Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178

1141

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1142 J.-H. He

6. Iteration Perturbation Method . . . . . . . . . . . . . . . . . . . . 1179

7. Ancient Chinese Methods . . . . . . . . . . . . . . . . . . . . . . 1185

7.1. Chinese method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185

7.2. He Chengtian’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 1188

8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193

1. Introduction

With the rapid development of nonlinear science, there appears an ever-increasing

interest of scientists and engineers in the analytical asymptotic techniques for non-

linear problems. Though it is very easy for us now to ﬁnd the solutions of linear

systems by means of computer, it is, however, still very diﬃcult to solve nonlinear

problems either numerically or theoretically. This is possibly due to the fact that

the various discredited methods or numerical simulations apply iteration techniques

to ﬁnd their numerical solutions of nonlinear problems, and nearly all iterative

methods are sensitive to initial solutions,

14,27,40,81,101,102,107

so it is very diﬃcult

to obtain converged results in cases of strong nonlinearity. In addition, the most

important information, such as the natural circular frequency of a nonlinear oscil-

lation depends on the initial conditions (i.e. amplitude of oscillation) will be lost

during the procedure of numerical simulation.

Perturbation methods

8,11,15,18,22,94,98

–

100

provide the most versatile tools avail-

able in nonlinear analysis of engineering problems, and they are constantly being

developed and applied to ever more complex problems. But, like other nonlinear

asymptotic techniques, perturbation methods have their own limitations:

(1) Almost all perturbation methods are based on such an assumption that a small

parameter must exist in an equation. This so-called small parameter assumption

greatly restricts applications of perturbation techniques, as is well known, an

overwhelming majority of nonlinear problems, especially those having strong

nonlinearity, have no small parameters at all.

(2) It is even more diﬃcult to determine the so-called small parameter, which seems

to be a special art requiring special techniques. An appropriate choice of small

parameter may lead to ideal results, however, an unsuitable choice of small

parameter results in badly eﬀects, sometimes serious ones.

(3) Even if a suitable small parameter exists, the approximate solutions solved by

the perturbation methods are valid, in most cases, only for the small values

of the parameter. For example, the approximations solved by the method of

multiple scales are uniformly valid as long as a speciﬁc system parameter is

small. However, we cannot rely fully on the approximations because there is no

criterion on how small the parameters should be. Thus checking numerically

and/or experimentally the validity of the approximations is essential.

For

example, we consider the following two equations:

113

+ u − ε = 0 , u(0) = 1 (1.1)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1143

and

− u − ε = 0 , u(0) = 1 . (1.2)

The solution of Eq. (1.1) is u(t) = ε+(1 −ε)e

−t

, while the unperturbed solution

of Eq. (1.1) is u

(t) = e

−t

. In case ε  1, the perturbation solution leads to high

accuracy, for |u(t) − u

(t)| ≤ ε.

By similar analysis for Eq. (1.2), we have

|u(t) − u

(t)| = ε|1 − e

|. (1.3)

It is obvious that u

can never be considered as an approximate solution of

Eq. (1.2) in case t  1, so the traditional perturbation method is not valid for this

case even when ε  1.

In what follows, we should introduce some new developed methods for problem

solving in areas where traditional techniques have been unsuccessful.

There exist some alternative analytical asymptotic approaches, such as the non-

perturbative method,

weighted linearization method,

10,105

Adomian decompo-

sition method,

1,9,35,118,119

modiﬁed Lindstedt–Poincare method,

28,65

–

67,91

varia-

tional iteration method,

49,70,74

–

energy balance methods,

tanh-method,

5,13,36

F -expansion method

106,115,116

and so on. Just recently, some new perturbation

methods such as artiﬁcial parameter method,

69,88

δ-method,

11,16

perturbation in-

cremental method,

homotopy perturbation method,

–

48,50

–

52,62,72

parameter-

ized perturbation method,

and bookkeeping artiﬁcial parameter perturbation

method

which does not depend on on the small parameter assumption are pro-

posed. A recent study

also reveals that the numerical technique can also be pow-

erfully applied to the perturbation method. A review of perturbation techniques in

phase change heat transfer can be found in Ref. 12, and a review of recently de-

veloped analytical techniques can be found in Ref. 71. Recent advance in nonlinear

dynamics, vibrations and control in China can be found in Ref. 93.

A wide body of literature dealing with the problem of approximate solutions

to nonlinear equations with various diﬀerent methodologies also exists. Although

a comprehensive review of asymptotic methods for nonlinear problems would be

in order at this time, this paper does not undertake such a task. Our emphasis is

mainly put on the recent developments of this ﬁeld in China, hence, our references

to literature will not be exhaustive. Rather, our purpose in this paper is to present a

comprehensive review of the past and current work on some new developed asymp-

totic techniques for solving nonlinear problems, with the goal of setting some ideas

and improvements which point toward new and interesting applications with these

methods. The content of this paper is partially speculative and heuristic, but we

feel that there is enough theoretical and computational evidence to warrant further

investigation and most certainly, reﬁnements and improvements, should the reader

choose to pursue the ideas.

In this review article, we limit ourselves to nonlinear order diﬀerential equations,

most nonlinear partial diﬀerential equations can be converted into order diﬀerential

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1144 J.-H. He

ones by some linear translation, for example, the KdV equation

+ uu

+ εu

xxx

= 0 . (1.4)

If the wave solution is solved, we can assume that u(x, t) = u(ξ) = u(x −ct), where

c is the wave speed. Substituting the above traveling wave solution into Eq. (1.4),

we have

−cu

+ uu

+ εu

ξξξ

= 0 . (1.5)

Integrating Eq. (1.5), we have an ordinary diﬀerential equation:

−cu +

+ εu

ξξ

= D , (1.6)

where D is an integral constant.

In Sec. 2, variational approaches to soliton solution, bifurcation, limit cycle, and

period solutions of nonlinear equations are illustrated, including the Ritz method,

energy method, variational iteration method; In Sec. 3, we ﬁrst introduce some

modiﬁcations of Lindstedt–Poincare method, then parameter-expanding methods

are illustrated where a parameter (or a constant) is expanded into a series in ε which

can be a small parameter in the equation, or an artiﬁcial parameter, or a book-

keeping parameter; Sec. 4 introduces parametrized perturbation method where the

expanding parameter is introduced by a linear transformation. Homotopy perturba-

tion method is systemically illustrated in Sec. 5. In Sec. 6, the iteration technology

is introduced to the perturbation method; Sec. 7 brieﬂy introduces some ancient

Chinese methods and their modern applications are illustrated. Finally, conclusions

are made in Sec. 8.

2. Variational Approaches

2.1. Ritz method

Variational methods

38,55

–

57,89,90,121

such as Raleigh–Ritz and Bubnov–Galerkin

techniques have been, and continue to be, popular tools for nonlinear analysis.

When contrasted with other approximate analytical methods, variational methods

combine the following two advantages:

(1) They provide physical insight into the nature of the solution of the problem.

For example, we consider the statistically unstable Duﬃng oscillator, deﬁned

− u + εu

= 0 . (2.1)

Its variational formulations can be obtained with ease

J(u) =



−

εu



dt . (2.2)

If we want to search for a period solution for Eq. (2.1), then from the energy integral

(2.2) it requires that the potential −

εu

must be positive for all t > 0, so

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1145

an oscillation about the origin will occur only if εA

> 2, where A is the amplitude

of the oscillation.

(2) The obtained solutions are the best among all the possible trial-functions. As

an illustration, consider the following chemical reaction

nA → C + D

which obeys the equation

= k(a − x)

, (2.3)

where a is the number of molecules A at t = 0, and x is the number of molecules C

(or D) after time t, and k is a reaction constant. At the start of reaction (t = 0),

there are no molecules C (or D) yet formed so that the initial condition is x(0) = 0.

In order to obtain a variational model, we diﬀerentiate both sides of Eq. (2.3)

with respect to time, resulting in

= −kn(a − x)

n−1

. (2.4)

Substituting Eq. (2.3) into Eq. (2.4), we obtain the following 2-order diﬀerential

equation

= −k

n(a − x)

2n−1

. (2.5)

The variational model for the diﬀerential equation (2.5) can be obtained with ease

by the semi-inverse method,

which reads

J(x) =

∞

(





(a − x)

)

dt . (2.6)

It is easy to prove that the stationary condition of the above functional, Eq. (2.4),

is equivalent to Eq. (2.3).

We apply the Ritz method to obtain an analytical solution of the discussed

problem. When all molecules A are used up, no further increase in the number of

molecules C can occur, so dx/dt = 0. Putting this into Eq. (2.3), the ﬁnal number

of molecules C is x

∞

= a. By the above analysis, we choose a trial-function in the

form

x = a(1 − e

−ηt

) (2.7)

where η is an unknown constant to be further determined.

It is obvious that the funtion (2.7) satisﬁes the initial condition. Substituting

the function (2.7) into Eq. (2.6), we obtain

J(η) =

∞



−2ηt

−2nηt



dt =

η +

4nη

. (2.8)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1146 J.-H. He

Minimizing the functional, Eq. (2.6), with respect to x, is approximately equivalent

to minimizing the above function, Eq. (2.8), with respect to η. We set

∂J

∂η

−

4nη

= 0 (2.9)

which leads to the result

η =

√

n−1

. (2.10)

So we obtain an explicit analytical solution for the discussed problem

x = a[1 − exp(−n

−1/2

n−1

t)] . (2.11)

The unknown constant, η, in the function (2.7) can be identiﬁed by various

other methods, for example various residual methods,

but the result (2.10) is the

most optimal one among all other possible choices.

The solution (2.11) states that x increases approximately exponentially from 0

to a ﬁnal value x = a, with a time constant given by the result (2.10). Chemists

and technologists always want to know its halfway through the change, i.e. x = a/2

when t = t

1/2

. An approximate halfway time obtained from the problem (2.11)

reads

1/2

= −

ln 0.5 = −

−1/2

n−1

ln 0.5 . (2.12)

To compare with the exact solution, we consider the case when n = 2. Under such

a case, we can easily obtain the following exact solution:

exact

= a



1 −

1 − akt



. (2.13)

Its halfway time reads

1/2

= −

. (2.14)

From the half-way time (2.12) we have

1/2

= −

−1/2

ln 0.5 = −

0.98

. (2.15)

The 2% accuracy is remarkable good. Further improvements in accuracy can be

achieved by suitable choice of trial-functions, and are outside the purpose of the

paper.

The above technology can be successfully applied to many ﬁrst order nonlinear

diﬀerential equations arising in physics. Now we consider the Riccati equation

117

in the form

+ p(x)y + q(x)y

+ r(x) = 0 . (2.16)

Diﬀerentiating Eq. (2.16) with respect to x, and eliminating y

in the resulted

equation, Eq. (2.16) becomes

+ (p

+ p

+ 2qr)y + (q

+ pq + 2pq)y

+ 2q

+ r

+ pr = 0 . (2.17)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1147

Its variational formulation can be expressed as

J(y) =



−

+ p

+ 2qr)y

+ pq + 2pq)y

+ (r

+ pr)y



dx . (2.18)

Variational approach can also be applied to simplify nonlinear equation, consider

the Lambert equation:

y = (1 − n)

. (2.19)

By the semi-inverse method,

its variational formulation reads

J(y) =



−

2n−2



dx . (2.20)

Hinted by the above energy form, Eq. (2.20), we introduce a transformation:

z = y

. (2.21)

Functional (2.20) becomes

J(z) =



−



dx . (2.22)

Its Euler–Lagrange equation is

+ k

z = 0 , (2.23)

which is a linear equation. The solution of Eq. (2.19), therefore, has the form:

y = (C cos kx + D sin kx)

1/n

. (2.24)

2.2. Soliton solution

There exist various approaches to the search for soliton solution for nonlinear wave

equations.

–

21,39

Here, we suggest a novel and eﬀective method by means of vari-

ational technology. As an illustrative example, we consider the KdV equation

∂u

∂t

− 6u

∂u

∂x

∂

∂x

= 0 . (2.25)

We seek its traveling wave solutions in the following frame

u(x, t) = u(ξ) , ξ = x − ct , (2.26)

Substituting the solution (2.26) into Eq. (2.25) yields

−cu

− 6uu

+ u

000

= 0 , (2.27)

where prime denotes the diﬀerential with respect to ξ.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1148 J.-H. He

Integrating Eq. (2.27) yields the result

−cu − 3u

+ u

= 0 . (2.28)

By the semi-inverse method,

the following variational formulation is established

J =

∞

+ u



dξ



dξ . (2.29)

By Ritz method, we search for a solitary wave solution in the form

u = p sech

(qξ) , (2.30)

where p and q are constants to be further determined.

Substituting Eq. (2.30) into Eq. (2.29) results in

J =

∞



sech

(qξ) + p

sech

(qξ) +

sech

(qξ) tanh

(qξ)



dξ

∞

sech

(z)dz +

∞

sech

(z)dz + 2p

∞

{sech

(z) tanh

(z)}dz

15q

. (2.31)

Making J stationary with respect to p and q results in

∂J

∂p

2cp

24p

15q

8pq

= 0 , (2.32)

∂J

∂q

= −

−

15q

= 0 , (2.33)

or simplifying

5c + 12p + 4q

= 0 , (2.34)

−5c − 8p + 4q

= 0 . (2.35)

From Eqs. (2.34) and (2.35), we can easily obtain the following relations:

p = −

c , q =

. (2.36)

So the solitary wave solution can be approximated as

u = −

sech

(x − ct − ξ

) , (2.37)

which is the exact solitary wave solution of KdV equation (2.25).

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1149

2.3. Bifurcation

Bifurcation arises in various nonlinear problems.

45,47,48,79,122

–

125

Variational

method can also be applied to the search for dual solutions of a nonlinear equation,

and to approximate identiﬁcation of the bifurcation point. Consider the Bratu’s

equation in the form:

+ λe

= 0 , u(0) = u(1) = 0 , (2.38)

where λ is a positive constant called as the eigenvalue. Equation (2.38) comes orig-

inally from a simpliﬁcation of the solid fuel ignition model in thermal combustion

theory,

and it has an exact solution

u(x) = −2 log



cosh



0.5(x − 0.5)θ

cosh(θ/4)



where θ solves

θ =

√

2λ cosh(θ/4) .

Due to its importance, the Bratu problem has been studied extensively. First, it

arises in a wide variety of physical applications, ranging from chemical reaction the-

ory, radiative heat transfer and nanotechnology

114

to the expansion of universe.

Second, because of its simplicity, the equation is widely used as a benchmarking tool

for numerical methods. Khuri

applied Adomian decomposition method to study

the problem, the procedure is tedious though the expression is a lovely closed-form,

furthermore only one solution is found when 0 < λ < λ

, and error arises when

λ → λ

The present problem can be easily solved with high accuracy by the variational

method, furthermore, two branches of the solutions can be simultaneously deter-

mined, and the bifurcation point can be identiﬁed with ease. It is easy to establish

a variational formulation for Eq. (2.38), which reads

J(u) =



− λe



dx . (2.39)

In view of the Ritz method, we choose the following trial-function:

u = Ax(1 − x) , (2.40)

where A is an unknown constant to be further determined. Substituting the function

(2.40) into the formula (2.39) results in:

J(A) =



(1 − 2x)

− λe

Ax(1−x)



dx . (2.41)

Minimizing the formula (2.41) to determine the constant, this yields

∂J

∂A

{A(1 − 2x)

− λx(1 − x)e

Ax(1−x)

}dx = 0 . (2.42)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1150 J.-H. He

Equation (2.42) can be readily solved by some mathematical software, for example,

MATHEMATICA, Matlab. There are two solutions to Eq. (2.42) for values of 0 <

λ < λ

, and no solutions for λ > λ

. There is only one solution for a critical value

of λ = λ

, which solves

{(1 − 2x)

− λ

(1 − x)

x(1−x)

}dx = 0 . (2.43)

Dividing Eq. (2.43) by Eq. (2.42) produces

{x(1 − x)e

x(1−x)

}dx

(1 − x)

x(1−x)

}dx

. (2.44)

Solving Eq. (2.44), we have A

= 4.72772. Substituting the result into Eq. (2.43)

results in λ

= 3.56908. The exact critical value is

= 3.51383071. The 1.57%

accuracy is remarkable good in view of the crude trial-function, Eq. (2.40).

In order to compare with Khuri’s results, we consider the case λ = 1. In such

case, Eq. (2.43) becomes

{A(1 − 2x)

− x(1 − x)e

Ax(1−x)

}dx = 0 . (2.45)

Solving this equation by Mathematica:

Find Root [Integrate [A

∗

(1 − 2

∗

∧

2 − x

∗

(1 − x)Exp[A

∗

(1 − x)], {x, 0, 1}]

= 0, {A, 1}] ,

we have A

= 0.559441, and A

= 16.0395.

In the case λ = 1, the Bratu equation has two solutions u

and u

with

(0) = 0.549 and u

(0) = 10.909. Our solutions predict that u

(0) = 0.559441,

and u

(0) = 16.0395, with accuracies of 1.9% and 46.94%, respectively. Accu-

racy can be improved if we choose a more suitable trial-function, for example,

u = Ax(1 − x)(1 + Bx + Cx

2.4. Limit cycle

Energy approach (Variational approach) to limit cycle is suggested by He,

58,59

which

can be applied to not only weakly nonlinear equations, but also strongly ones, and

the obtained results are valid for whole solution domain. D’Acunto

called the

energy approach as He’s variational method.

To illustrate the basic idea of the method, we consider the following nonlinear

oscillation:

¨x + x + εf(x, ˙x) = 0 , (2.46)

In the present study the parameter ε need not be small.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1151

Generally speaking, limit cycles can be determined approximately in the form

x = b + a(t) cos ωt +

n=1

cos nωt + D

sin nωt) , (2.47)

where b, C

and D

are constant.

Substituting Eq. (2.47) into Eq. (2.46) results in the following residual

R(t) = ¨x + x + εf(x, ˙x) . (2.48)

In general, the residual might not be vanishingly small at all points, the error

depends upon the inﬁnite “work”, dw, done by the “force” R in the inﬁnite distance

dx:

dw = Rdx . (2.49)

We hope the totally work done in a period is zero, that requires

Rdx = 0 , (2.50)

where A

and A

are minimum and maximum amplitudes, respectively.

Equation (2.50) can be equivalently written in the form

R ˙xdt = 0 , (2.51)

where T is the period, this technique is similar to the method of weighted residuals.

We assume that the amplitude weakly varies with time, so we write the ampli-

tude in the form

a(t) = Ae

αt

≈ A , |α|  1 . (2.52)

Accordingly we approximately have the following expressions

˙x ≈ αx −ω

− x

, (2.53)

¨x ≈ (α

− ω

)x − 2αω

− x

. (2.54)

Now we consider the van der Pol equation:

¨x + x − ε(1 − x

) ˙x = 0 , (2.55)

We will not re-illustrate the solution procedure, but we identify α in Ref. 58 instead

with

α =

− 4)ωπ

4(2πω − εA

)

, (2.56)

and the frequency equation (33) in Ref. 58 should be replaced by

1 − ω

− 4)

16(2πω − εA

)

−

− 4)ωπ

4(2πω − εA

)

= 0 . (2.57)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1152 J.-H. He

Now consider the limit ε  1, by assumption ω = 1 + aε

+ O(ε

), Eq. (2.57)

becomes



−2a +

− 4)



+ O(ε

) = 0 . (2.58)

After identiﬁcation the constant, a, we have

ω = 1 + ε



− 4)

128

− 4



+ O(ε

) . (2.59)

In the case ε → ∞, the frequency can be approximated as

ω =

+ O





, (2.60)

From Eq. (2.57), we have

1 −





− 4)

16(2πb/ε − εA

)

−

εb(A

− 4)π

4(2πb/ε − εA

+ O





= 0 . (2.61)

If we keep the order of 1/ε, then Eq. (2.61) can be simpliﬁed as

1 +

b(A

− 4)π

= 0 , (2.62)

from which the constant, b, can be readily determined. As a result, we have:

ω = −

− 4)πε

+ O





. (2.63)

If we choose A = 2 which is identiﬁed by perturbation method,

104

then the last

two terms in Eq. (2.57) vanish completely, leading to the invalidity of the method

as pointed out by Rajendran et al.

104

The perturbation method provides us with

a choice of the approximate value of A, and many alternative approaches to the

identiﬁcation of A exist.

Assuming that period solution has the form: x = A cos ωt, we obtain the fol-

lowing residual:

R = (1 − ω

)A cos ωt + Aεω(1 − A

cos

ωt) sin ωt . (2.64)

By substituting Eq. (2.59) or Eq. (2.63) into Eq. (2.64) respectively, we have

R = Aε(1 − A

cos

ωt) sin ωt + O(ε

) , ε → 0 , (2.65)

and

R/ε = A(1 − A

cos

ωt) sin ωt + O(1/ε) , 1/ε → 0 . (2.66)

The value of A can be determined in the parlance of vanishing small residual by

the method of weighted residuals.

Applying the Galerkin method

T/4

R sin ωtdt =

T/4

A(1 − A

cos

ωt) sin

ωtdt = 0 , T = 2π/ω , (2.67)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1153

leading to the result: A = 2, which agrees with the standard result of the pertur-

bation method.

The simplest method among the method of weighted residuals is the method of

collocation. Collocating at ωt = π/3, ωt = π/4, and ωt = π/6 gives, respectively,

the approximate values of A: A = 2, A = 1.414, and A = 1.155. The average value

of A reads A = 1.522.

We can also apply the least squares method to ﬁnd the approximate value of A

by the formulation:

T/4

∂R

∂A

dt = 0 , (2.68)

i.e.

T/4

A(1 − A

cos

ωt)(1 −3A

cos

ωt) sin

ωtdt = 0 , (2.69)

which leads to the result A =

8/3 = 1.633.

Now we choose A = 1.633, then Eq. (2.59) becomes

ω = 1 −

+ O(ε

) = 1 − 0.0694ε

+ O(ε

) . (2.70)

We write down the perturbation solution for reference, which reads

ω = 1 −

+ O(ε

) = 1 − 0.0625ε

+ O(ε

) . (2.71)

Equation (2.63) becomes

ω =

πε

+ O





πε

+ O





2.566

+ O





. (2.72)

The exact frequency under the limit ε → ∞ is

ω =

3.8929

+ O





. (2.73)

So the obtained solution is valid for both ε → 0 and ε → ∞.

2.5. Variational iteration method

The variational iteration method

70,74

–

has been shown to solve eﬀectively, easily,

and accurately a large class of nonlinear problems with approximations converg-

ing rapidly to accurate solutions. Most authors found that the shortcomings aris-

ing in Adomian method can be completely eliminated by the variational iteration

method.

6,7,17,96,97,103,111,112

To illustrate the basic idea of the variational iteration method, we consider the

following general nonlinear system:

Lu + N u = g(x) , (2.74)

where L is a linear operator, and N is a nonlinear operator.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1154 J.-H. He

According to the variational iteration method, we can construct the following

iteration formulation:

n+1

(x) = u

(x) +

λ{Lu

(s) + N ˜u

(s) − g(s)}ds (2.75)

where λ is called a general Lagrange multiplier, which can be identiﬁed optimally

via the variational theory, ˜u

is considered as a restricted variation, i.e. δ˜u

= 0.

We consider Duﬃng equation as an example to illustrate its basic computational

procedure.

+ u + εu

= 0 , (2.76)

with initial conditions u(0) = A, u

(0) = 0.

Supposing that the angular frequency of the system (2.76) is ω, we have the

following linearized equation:

+ ω

u = 0 . (2.77)

So we can rewrite Eq. (2.76) in the form

+ ω

u + g(u) = 0 , (2.78)

where g(u) = (1 − ω

)u + εu

Applying the variational iteration method, we can construct the following

functional:

n+1

(t) = u

(t) +

λ{u

(τ) + ω

(τ) + ˜g(u

)}dτ , (2.79)

where ˜g is considered as a restricted variation, i.e. δ˜g = 0.

Calculating variation with respect to u

, and noting that δ˜g(u

) = 0, we have

the following stationary conditions:











(τ) + ω

λ(τ) = 0 ,

λ(τ)|

τ=t

= 0 ,

1 − λ

(τ)|

τ=t

= 0 .

(2.80)

The multiplier, therefore, can be identiﬁed as

λ =

sin ω(τ − t) . (2.81)

Substituting the identiﬁed multiplier into Eq. (2.79) results in the following iteration

formula:

n+1

(t) = u

(t) +

sin ω(τ − t){u

(τ) + u

(τ) + εu

(τ)}dτ . (2.82)

Assuming its initial approximate solution has the form:

(t) = A cos ωt , (2.83)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1155

and substituting Eq. (2.83) into Eq. (2.76) leads to the following residual

(t) =



1 − ω

εA



A cos ωt +

εA

cos 3ωt . (2.84)

By the formulation (2.82), we have

(t) = A cos ωt +

(τ) sin ω(τ − t)dτ . (2.85)

In order to ensure that no secular terms appear in u

, resonance must be avoided.

To do so, the coeﬃcient of cos ωt in Eq. (2.84) requires to be zero, i.e.

ω =

1 +

εA

. (2.86)

So, from Eq. (2.85), we have the following ﬁrst-order approximate solution:

(t) = A cos ωt +

εA

4ω

cos 3ωt sin ω(τ − t)dτ

= A cos ωt +

εA

32ω

(cos 3ωt − cos ωt) . (2.87)

It is obvious that for small ε, i.e. 0 < ε  1, it follows that

ω = 1 +

εA

. (2.88)

Consequently, in this limit, the present method gives exactly the same results

as the standard Lindstedt–Poincare method.

98,100

To illustrate the remarkable ac-

curacy of the obtained result, we compare the approximate period

T =

2π

1 + 3εA

(2.89)

with the exact one

√

1 + εA

π/2

1 − k sin

, (2.90)

where k = 0.5εA

/(1 + εA

What is rather surprising about the remarkable range of validity of Eq. (2.89)

is that the actual asymptotic period as ε → ∞ is also of high accuracy.

lim

ε→∞

3/4

π/2

1 − 0.5 sin

= 0.9294 .

Therefore, for any value of ε > 0, it can be easily proved that the maximal

relative error of the period Eq. (2.89) is less than 7.6%, i.e. |T − T

|/T

< 7.6%.

Due to the very high accuracy of the ﬁrst-order approximate solution, we always

stop before the second iteration step. That does not mean that we cannot obtain

higher order approximations.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1156 J.-H. He

3. Parameter-Expanding Methods

We consider the following nonlinear oscillator

+ ω

u + εf (u) = 0 . (3.1)

Various perturbation methods have been applied frequently to analyze Eq. (3.1).

The perturbation methods are, however, limited to the case of small ε and mω

> 0,

that is, the associated linear oscillator must be statically stable in order that linear

and nonlinear response be qualitatively similar.

Classical methods which are applied to weakly nonlinear systems include the

Lindstedt–Poincare, Krylov–Bogoliubov–Mitropolski averaging, and multiple time

scale methods. These are described by many textbooks by Bellman,

Nayfeh,

and Nayfeh and Mook,

100

among others. These methods are characterized by ex-

pansions of the dependent variables in power series in a small parameter, resulting

in a collection of linear diﬀerential equations which can be solved successively. The

solution to the associated linear problem thus provides a starting point for the gen-

eration of the “small perturbation” due to the nonlinearity, with various devices

employed to annul secularities.

Classical techniques which have been used for both weakly and strongly nonlin-

ear systems include the equivalent linearization method,

105

the harmonic balance

method.

A feature of these methods is that the form of solution is speciﬁed in

advance. The methods work well provided that the ﬁlter hypothesis is satisﬁed,

that is, higher harmonics output by the nonlinearities are substantially attenuated

by the linear part of the system.

3.1. Modiﬁcations of Lindstedt–Poincare method

A number of variants of the classical perturbation methods have been proposed to

analyze conservative oscillators similar to Eq. (3.1). The Shohat expansion

108

for the

van der Pol equation is of remarkable accuracy not only for small parameter ε, but

also for large ε. But recently Mickens

shows that it does not have a general validity

as a “perturbation” scheme that can be extended to large values of ε for Duﬃng

equation, as was originally thought to be. Burton

23,24

deﬁned a new parameter

α = α(ε) in such a way that asymptotic solutions in power series in α converge

more quickly than do the standard perturbation expansions in power series in ε.

α =

εA

4 + 3εA

. (3.2)

It is obvious that α < 1/3 for all εA

, so the expansion in power series in α is

quickly convergent regardless of the magnitude of εA

The Lindstedt–Poincare method

98,100

gives uniformly valid asymptotic expan-

sions for the periodic solutions of weakly nonlinear oscillations, in general, the

technique does not work in case of strongly nonlinear terms.

For the sake of comparison, the standard Lindstedt–Poincare method is brieﬂy

recapitulated ﬁrst. The basic idea of the standard Lindstedt–Poincare method is to

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1157

introduce a new variable

τ = ω(ε)t , (3.3)

where ω is the frequency of the system. Equation (3.1) then becomes

+ ω

u + εf (u) = 0 , (3.4)

where prime denotes diﬀerentiation with respect to τ. The frequency is also ex-

panded in powers of ε, that is

ω = ω

+ εω

+ ε

+ ···. (3.5)

Various modiﬁcations of Lindstedt–Poincare method appeared in open literature.

It is of interest to note that a much more accurate relation for frequency ω can be

found by expanding ω

, rather than ω, in a power series in ε.

= 1 + εω

+ ε

+ ···. (3.6)

Cheung et al. introduced a new parameter

α =

εω

+ εω

. (3.7)

and expand the solution and ω

in power series in α.

All those modiﬁcations are limited to conservation system, instead of linear

transformation (3.3), a nonlinear time transformation can be successfully annul

such shortcoming.

We can re-write the transformation of standard Lindstedt–Poincare method in

the form

τ = t + f(ε, t) , f(0, t) = 0 , (3.8)

where f(ε, t) is an unknown function, not a functional. We can also see from the

following examples that the identiﬁcation of the unknown function f is much easier

than that of unknown functional F in Dai’s method.

31,32

From the relation (3.8), we have

∂

∂t



1 +

∂f

∂t



∂

∂τ

= G

∂

∂τ

. (3.9)

where

G(ε, t) =



1 +

∂f

∂t



. (3.10)

Applying the Taylor series, we have

G(ε, t) = G

+ εG

+ ··· , (3.11)

where G

(i = 1, 2, 3, . . .) can be identiﬁed in view of no secular terms. After iden-

tiﬁcation of G

(i = 1, 2, 3, . . .), from the Eq. (3.10), the unknown function f can

be solved. The transformation Eq. (3.8) can be further simpliﬁed as

τ = f(t, ε) , f(t, 0) = t . (3.12)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1158 J.-H. He

So we have

∂

∂t



∂f

∂t



∂

∂τ

. (3.13)

We expand (∂f/∂t)

in a series of ε:



∂f

∂t



= 1 + εf

+ ε

+ ···. (3.14)

Now we consider the van der Pol equation.

+ u − ε(1 − u

= 0 . (3.15)

By the transformation τ = f (t, ε), Eq. (3.15) becomes



∂f

∂t



∂

∂τ

+ u = ε(1 − u

)

∂f

∂t

∂u

∂τ

. (3.16)

Introducing a new variable ω, which is deﬁned as



∂f

∂t



. (3.17)

So Eq. (3.16) reduces to

∂

∂τ

+ ω

u = εω(1 − u

)

∂u

∂τ

. (3.18)

By simple operation, we have



∂f

∂t



1 +

. (3.19)

Solving Eq. (3.19), subject to the condition f(t, 0) = t, gives

τ = f =

1 +

. (3.20)

Its period can be expressed as

T = 2π

1 +

. (3.21)

It is obvious that when ε  1, the results are same as those obtained by standard

perturbation methods.

However, for large values of ε (ε  1), we have the following approximation

for exact period

≈ 2ε

√



− 3vdv



= 1.614ε , (ε  1) . (3.22)

From Eq. (3.20), it follows

T ≈

√

ε = 2.22ε , (ε  1) . (3.23)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1159

It is obvious that for large ε, the approximate period has the same feature as

the exact one, and we have

lim

ε→∞

1.614

2.22

= 0.727 .

So the maximal relative error for all 0 < ε < ∞ is less than 37.5%. We can

readily obtain higher-order approximations by MATHEMATICA.

Now we return to Eq. (3.1). Most methods are limited to odd nonlinearities and

require that ω

> 0 and m > 0. We must devise a method to solve Eq. (3.1) for

the case when (1) m > 0, ω

≤ 0; (2) ω

≥ 0, m ≤ 0; (3) even nonlinearities ex-

ist. Parameter-expanding methods including modiﬁed Lindsted–Poincare method

and bookkeeping parameter method

can successfully deal with such special cases

where classical methods fail. The methods need not have a time transformation

like Lindstedt–Poincare method, the basic character of the method is to expand

the solution and some parameters in the equation. If we search for periodic solution

of Eq. (3.1), then we can assume that

u = u

+ εu

+ ε

+ ···, (3.24)

= ω

+ pω

+ p

+ ··· , (3.25)

m = 1 + pm

+ p

+ ··· . (3.26)

Hereby ε can be a small parameter in the equation, or an artiﬁcial parameter or a

bookkeeping parameter.

As an illustration, consider the motion of a ball-bearing oscillating in a glass

tube that is bent into a curve such that the restoring force depends upon the cube

of the displacement u. The governing equation, ignoring frictional losses, is

+ εu

= 0 , u(0) = A , u

(0) = 0 . (3.27)

The standard Lindstedt–Poincare method does not work for this example. Now

we re-write Eq. (3.27) in the form

+ 0 · u + εu

= 0 . (3.28)

Supposing the solution and the constant zero in Eq. (3.28) can be expressed as

u = u

+ εu

+ ε

+ ···, (3.29)

0 = ω

+ εc

+ ε

+ ···. (3.30)

Substituting Eqs. (3.29) and (3.30) into Eq. (3.28), and processing as the standard

perturbation method, we have

+ ω

= 0 , u

(0) = A , u

(0) = 0 , (3.31)

+ ω

+ c

+ u

= 0 , u

(0) = 0 , u

(0) = 0 . (3.32)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1160 J.-H. He

Solving Eq. (3.31), we have u

= A cos ωt. Substituting u

into Eq. (3.32) results

into

+ ω

+ A





cos ωt +

cos 3ωt = 0 . (3.33)

Eliminating the secular term needs

= −

. (3.34)

If only the ﬁrst-order approximate solution is searched for, then from Eq. (3.30),

we have

0 = ω

−

εA

, (3.35)

which leads to the result

ω =

√

1/2

A . (3.36)

Its period, therefore, can be written as

T =

4π

√

−1/2

−1

= 7.25ε

−1/2

−1

. (3.37)

Its exact period can be readily obtained, which reads

= 7.4164ε

−1/2

−1

. (3.38)

It is obvious that the maximal relative error is less than 2.2%, and the obtained

approximate period is valid for all ε > 0.

Now consider another example:

1 + εu

= 0 . (3.39)

with initial conditions u(0) = A and u

(0) = 0.

We rewrite Eq. (3.39) in the form

+ 1 · u + εu

= 0 . (3.40)

Proceeding with the same manipulation as the previous example, we obtain the

following angular frequency:

ω =

1 +

εA

. (3.41)

It is obvious that for small ε, it follows that

ω = 1 −

εA

. (3.42)

Consequently, in this limit, the present method gives exactly the same results as

the standard perturbations.

To illustrate the remarkable accuracy of the obtained

results, we compare the approximate period

T = 2π

1 +

εA

(3.43)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1161

with the exact one

T = 4

√

ln(1 + εA

) − ln(1 + εu

)

= 4

√

ln[(1 + εA

)/(1 + εu

)]

(3.44)

In the case εA

→ ∞, Eq. (3.44) reduces to

lim

εA

→∞

T = 4

√

2(ln A −ln u)

= 2

√

2εA

ln(1/s)

= 4

√

2εA

∞

exp(−x

)dx = 2

√

2πεA . (3.45)

It is obvious that the approximate period (3.43) has the same feature as the exact

one for ε  1. And in the case ε → ∞, we have

lim

A→∞

√

2πεA

√

3επA

= 0.9213 .

Therefore, for any values of ε, it can be easily proven that the maximal relative

error is less than 8.54% on the whole solution domain (0 < ε < ∞).

3.2. Bookkeeping parameter

In most cases the parameter, ε, might not be a small parameter, or there is no

parameter in the equation, under such conditions, we can use a bookkeeping pa-

rameter. Consider the mathematical pendulum:

+ sin u = 0 . (3.46)

We re-write Eq. (3.46) in the following form

+ 0 · u + sin u = 0 . (3.47)

Supposing the solution and the constant, zero, can be expressed in the forms:

u = u

+ pu

+ p

+ ···, (3.48)

0 = ω

+ pc

+ p

+ ···. (3.49)

Here, p is a bookkeeping parameter, c

can be identiﬁed in view of no secular

terms in u

(i = 1, 2, 3, . . .). For example, the constant c

in Eq. (3.49) can also be

identiﬁed in view of no secular terms in u

, that is

A cos ωt{c

A cos ωt + sin(A cos ωt)}dt = 0 , (3.50)

where T = 2π/ω.

From Eq. (3.50), c

can be determined explicitly by Bessel function. The angular

frequency can be determined as

ω =

(A)

, (3.51)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1162 J.-H. He

where J

is the ﬁrst-order Bessel function of the ﬁrst kind.

The parameter-expanding methods can also be applied to non-periodic solu-

tions. Consider another simple example.

+ u

= 1 , u(0) = 0 , (3.52)

which has an exact solution

1 − e

−2t

1 + e

−2t

. (3.53)

No small parameter exits in the equation, so in order to carry out a straightforward

expansion, Liu

introduced an artiﬁcial parameter ε embedding in Eq. (3.52) so

that he obtained the following equation

+ (u + 1)(εu − 1) = 0 , ε = 1 . (3.54)

Supposing that the solution can be expanded in powers of ε, and processing in

a traditional fashion of perturbation technique, Liu obtained the following result

u = (1 − e

−t

) + εe

−t

+ t − 1) . (3.55)

Let ε = 1 in Eq. (3.55). We obtain the approximate solution of the original

Eq. (3.52):

u(t) = (1 − e

−t

) + e

−t

+ t − 1) . (3.56)

The obtained approximation (3.56) is of relative high accuracy.

In order to illustrate the basic idea of the proposed perturbation method, we

rewrite Eq. (3.52) in the form

+ 0 · u + 1 · u

= 1 , u(0) = 0 . (3.57)

Supposing that the solution and the coeﬃcients (0 and 1) can be expressed in the

forms:

u = u

+ pu

+ p

+ ···, (3.58)

0 = a + pa

+ p

+ ···, (3.59)

1 = pb

+ p

+ ···. (3.60)

Substituting Eqs. (3.58)–(3.60) into Eq. (3.57), collecting the same power of p, and

equating each coeﬃcient of p to zero, we obtain

+ au

= 1 , u

(0) = 0 , (3.61)

+ au

+ a

+ b

= 0 , (3.62)

+ au

+ a

+ b

+ 2b

= 0 . (3.63)

We can easily solve the above equations sequentially for u

(i = 1, 2, 3, . . .), the

initial conditions for u

(i = 1, 2, 3, . . .) should satisfy the condition

i=1

(0) = 0.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1163

Solving Eq. (3.61), we have

(1 − e

−at

) . (3.64)

Substituting u

into Eq. (3.62) results in

+ au

(1 − e

−at

) +

(1 − e

−at

)

= 0 . (3.65)

+ au





−





−at

−2at

= 0 . (3.66)

The requirement of no term te

−at

in u

needs

= 0 . (3.67)

If we keep terms to O(p

) only, from Eqs. (3.59) and (3.60), we have

a + a

= 0 and b

= 1 . (3.68)

From the Eqs. (3.67) and (3.68), we obtain

a =

√

2 . (3.69)

Solving Eq. (3.66) subject to the initial condition u

(0) = 0 gives

= −



−



(1 − e

−at

) +

−2at

− e

−at

) . (3.70)

Setting p = 1, we have the ﬁrst-order approximation:

u(t) = u

(t) + u

(t) =



−



(1 − e

−at

) +

−2at

− e

−at

) , (3.71)

where a =

√

2. It is easy to prove that the obtained result (3.71) is uniformly valid

on the whole solution domain.

Consider the Thomas–Fermi equation

41,56,82

(x) = x

−1/2

3/2

, u(0) = 1 , u(∞) = 0 . (3.72)

The energy (in atomic units) for a neutral atom of atomic number Z is

E =



3π



2/3

7/3

(0) .

So it is important to determine a highly accurate value for the initial slope of the

potential u

(0).

The equation is studied by δ-method.

The basic idea of the δ-method is to

replace the right hand side of the Thomas–Fermi equation by one which contains

the parameter δ, i.e.

(x) = u

1+δ

−δ

. (3.73)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1164 J.-H. He

The solution is assumed to be expanded in a power series in δ

u = u

+ δu

+ δ

+ ··· . (3.74)

This, in turn, produces a set of linear equations for u

− u

= 0 ,

− u

= u

ln(u

/x)

− u

= u

+ u

ln(u

/x) +

/x)

with associated boundary conditions u

(0) = 1, u

(∞) = 0 and u

(0) = u

(∞) = 0

for n > 1.

To solve u

for n > 1, we need some unfamiliar functions, so it might meet some

diﬃculties in promoting this method.

We rewrite the equation in the form

(x) − 0 · u = 1 · x

−1/2

3/2

. (3.75)

Supposing that the solution and the coeﬃcients (0 and 1) can be expressed in the

forms

u = u

+ pu

+ p

+ ···, (3.76)

0 = β

+ pa

+ p

+ ···, (3.77)

1 = pb

+ p

+ ··· . (3.78)

By simple operation, we have the following equations

(x) − β

= 0 , (3.79)

(x) − β

= b

−1/2

3/2

+ a

, (3.80)

with associated boundary conditions u

(0) = 1, u

(∞) = 0 and u

(0) = u

(∞) = 0

for n > 1. The ﬁrst-order approximate solution is

= e

−1.19078255x

. (3.81)

The second-order approximate solution is approximately solved as

u(x) =



1 −



−1.19078255x

. (3.82)

In 1981, Nayfeh

studied the free oscillation of a nonlinear oscillation with

quadratic and cubic nonlinearities, and the governing equation reads

+ u + au

+ bu

= 0 , (3.83)

where a and b are constants.

We can also assume that u

(x) + 0 · u = 1 · x

−1/2

3/2

, then the constant β

will be proven to

be negative.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1165

In order to carry out a straightforward expansion for small but ﬁnite ampli-

tudes for Eq. (3.83), we need to introduce a small parameter because none appear

explicitly in this equation. To this end, we seek an expansion in the form

u = qu

+ q

+ ···, (3.84)

where q is a small dimensionless parameter that is a measure of the amplitude of

oscillation. It can be used as a bookkeeping device and set equal to unity if the am-

plitude is taken to be small. As illustrated in Ref. 98, the straightforward expansion

breaks down due to secular terms. And Nayfeh used the method of renormaliza-

tion to render the straightforward expansion uniform. Unfortunately, the obtained

results are valid only for small amplitudes.

Hagedorn

also used an artiﬁcial parameter to deal with a mathematical pen-

dulum. For Eq. (3.83), we can assume that

u = u

+ pu

+ p

+ ···, (3.85)

= ω

+ pc

+ p

+ ···, (3.86)

a = pa

+ p

+ ···, (3.87)

b = p

+ p

+ ···. (3.88)

We only write down the ﬁrst-order angular frequency, which reads

ω =

1 +

(1 +

)

−

. (3.89)

We write down Nayfeh’s result

for comparison:

ω = 1 +



b −



, (3.90)

which is obtained under the assumption of small amplitude. Consequently, in this

limit, the present method gives exactly the same results as Nayfeh’s.

4. Parametrized Perturbation Method

In Refs. 73 and 71, an expanding parameter is introduced by a linear transforma-

tion:

u = εv + b , (4.1)

where ε is the introduced perturbation parameter, b is a constant.

We reconsider Eq. (3.52). Substituting Eq. (4.1) into Eq. (3.52) results in

(

+ εv

+ 2bv + (b

− 1)/ε = 0 ,

v(0) = −b/ε .

(4.2a)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1166 J.-H. He

Setting b = 1 for simplicity, so we have the following equation with an artiﬁcial

parameter ε:

(

+ εv

+ 2v = 0 ,

v(0) = −1/ε .

(4.2b)

Assume that the solution of Eq. (4.2b) can be written in the form

v = v

+ εv

+ ε

+ ···. (4.3)

Unlike the traditional perturbation methods, we keep

(0) = v(0) , and

i=1

(0) = 0 . (4.4)

We write the ﬁrst-order approximate solution of Eq. (4.2b)

v = v

+ εv

= −

−2t

2ε

−4t

− e

−2t

) . (4.5)

In view of the transformation (4.1), we obtain the approximate solution of the

original Eq. (3.52):

u = εv + 1 = 1 − e

−2t

−4t

− e

−2t

) . (4.6)

It is interesting to note that in Eq. (4.6) the artiﬁcial parameter ε is eliminated

completely. It is obvious that even its ﬁrst-order approximate solution (4.6) is of

remarkable accuracy.

We now consider the following simple example to take the notation

+ u − u

= 0 , u(0) =

(4.7)

where there exists no small parameter. In order to apply the perturbation tech-

niques, we introduce a parameter ε by the transformation

u = εv (4.8)

so the original Eq. (4.7) becomes

+ v − εv

= 0 , v(0) = A (4.9)

where εA = 1/2.

Supposing that the solution of Eq. (4.9) can be expressed in the form

v = v

+ εv

+ ε

+ ··· (4.10)

and processing in a traditional fashion of perturbation technique, we obtain the

following equations

+ v

= 0 , v

(0) = A (4.11)

+ v

− v

= 0 , v

(0) = 0 . (4.12)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1167

Solving Eqs. (4.11) and (4.12), we obtain

= Ae

−t

(4.13)

= A

−t

− e

−2t

) . (4.14)

Using the fact A = 1/(2ε), we obtain the ﬁrst-order approximate solution of

Eq. (4.9)

v = v

+ εv

+ O(ε

) = Ae

−t

+ εA

−t

− e

−2t

) + O(ε

)

2ε

−t

4ε

−t

− e

−2t

) + O(ε

) . (4.15)

In view of the transformation (4.8), we have the approximate solution of the

original Eq. (4.7)

u = εv =

−t

−

−2t

(4.16)

which is independent upon the artiﬁcial parameter ε, as it ought to be. The exact

solution of Eq. (4.7) reads

−t

1 + e

−t

= e

−t

(1 − e

−t

+ e

−2t

+ ···) . (4.17)

It is obvious that the obtained approximate solution (3.16) is of high accuracy.

Example 1. Now we consider the Duﬃng equation with 5th order nonlinearity,

which reads

+ u + αu

= 0 u(0) = A , u

(0) = 0 (4.18)

where α needs not be small in the present study, i.e. 0 ≤ α < ∞.

We let

u = εv (4.19)

in Eq. (4.18) and obtain

+ 1 · v + αε

= 0 , v(0) = A/ε , v

(0) = 0 . (4.20)

Applying the parameter-expanding method (modiﬁed Lindstedt–Poincare

method

) as illustrated in Sec. 3.1, we suppose that the solution of Eq. (4.20)

and the constant 1, can be expressed in the forms

v = v

+ ε

+ ··· (4.21)

1 = ω

+ ε

+ ··· . (4.22)

Note: If we assume that

v = v

+ εv

+ ε

· · · and 1 = ω

+ εω

+ ε

· · ·

then it is easy to ﬁnd that v

= v

= 0 and ω

= ω

= 0, so that the secular terms will

not occur.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1168 J.-H. He

Substituting Eqs. (4.21) and (4.22) into Eq. (4.20) and equating coeﬃcients of

like powers of ε yields the following equations

+ ω

= 0 , v

(0) = A/ε , v

(0) = 0 (4.23)

+ ω

+ αv

= 0 , v

(0) = 0 , v

(0) = 0 . (4.24)

Solving Eq. (4.23) results in

cos ωt . (4.25)

Equation (4.24), therefore, can be re-written down as

+ ω



5αA

8ε

+ ω



cos ωt +

5αA

16ε

cos 3ωt +

αA

16ε

cos 5ωt = 0 . (4.26)

Avoiding the presence of a secular term needs

= −

5αA

8ε

(4.27)

Solving Eq. (4.26), we obtain

= −

αA

128ε

(cos ωt − cos 3ωt) −

αA

384ε

(cos ωt − cos 5ωt) . (4.28)

If, for example, its ﬁrst-order approximation is suﬃcient, then we have

u = εv = ε(v

+ ε

) = A cos ωt −

αA

128ω

(cos ωt − cos 3ωt)

−

αA

384ω

(cos ωt − cos 5ωt) (4.29)

where the angular frequency can be written in the form

ω =

1 − ε

1 +

αA

, (4.30)

which is valid for all α > 0.

Example 2. Consider the motion of a particle on a rotating parabola, which is

governed by the equation (Exercise 4.8 in Ref. 99)

(1 + 4q

)

+ α

u + 4q





= 0 , t ∈ Ω (4.31)

with initial conditions u(0) = A, and u

(0) = 0.

Herein α and q are known constants, and need not to be small. We let u = εv

in Eq. (4.31) and obtain

(1 + 4ε

)

+ α

v + 4q





= 0 , v(0) = A/ε , v

(0) = 0 . (4.32)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1169

By parameter-expanding method(the modiﬁed Lindsted–Poincare method

we assuming that α

and solution of the Eq. (4.32) can be written in the form

= Ω

+ ε

+ ··· (4.33)

v = v

+ ε

+ ···. (4.34)

Substituting Eqs. (4.33) and (4.34) into Eq. (4.32), and equating the coeﬃcients

of like powers of ε results in the following equations

+ Ω

= 0 , v

(0) = A/ε , v

(0) = 0 (4.35)

+ Ω

+ ω

+ 4q

+ v

) = 0 , v

(0) = 0 , v

(0) = 0 . (4.36)

Solving Eq. (4.35) yields

(τ) =

cos Ωt . (4.37)

Equation (4.36), therefore, can be re-written down as

+ Ω

Aω

cos Ωt +

Ω

(−cos

Ωt + cos Ωt sin

Ωt) = 0 . (4.38)

Using trigonometric identities, Eq. (4.38) becomes

+ Ω



Aω

−

Ω



cos Ωt −

Ω

cos 3Ωt = 0 . (4.39)

Eliminating the secular term demands that

Ω

. (4.40)

Solving Eq. (4.39), we obtain

Ω

4ε

(cos Ωt − cos 3Ωt) . (4.41)

If, for example, its ﬁrst-order approximation is suﬃcient, then we have the ﬁrst-

order approximate solution of Eq. (4.32)

u = ε(v

+ ε

) = A cos ωt +

Ω

(cos ωt − cos 3ωt) (4.42)

where the angular frequency ω can be solved from Eqs. (4.33) and (4.40)

= Ω

+ ε

= Ω

+ 2q

Ω

(4.43)

which leads to

Ω =

1 + 2(qA)

. (4.44)

Its period can be approximately expressed as follows

app

2π

1 + 2q

. (4.45)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1170 J.-H. He

In case qA is suﬃciently small, i.e. 0 < qA  1, it follows that

app

2π

(1 + q

) .

Consequently, in this limit, the present method gives exactly the same results

as the standard Lindstedt–Poincare method.

However, in our present study, qA

need not be small, even in case qA → ∞, the present results also have high accuracy

lim

|qA|→∞

app

= lim

|qA|→∞

π/2

1 + 4q

cos

t dt

1 + 2q

√

= 0.900 (4.46)

where T

is the exact period, which reads

απ

π/2

1 + 4q

cos

t dt . (4.47)

Therefore, for any values of qA, it can be easily proven that

0 ≤

− T

app

≤ 10% .

Example 3. Consider the equation (Exercise 4.4 in Ref. 99)

(1 + u

+ u = 0 , u(0) = A , ¬u

(0) = 0 . (4.48)

We let u = εv in Eq. (4.48) and obtain

+ 1 · v + ε

= 0 , v(0) = A/ε , v

(0) = 0 . (4.49)

Supposing that the solution of Eq. (4.49) and ω

can be expressed in the form

v = v

+ ε

+ ··· (4.50)

1 = ω

+ ε

+ ··· . (4.51)

Substituting Eqs. (4.50) and (4.51) into Eq. (4.49) and equating coeﬃcients of

like powers of ε yields the following equations

+ ω

= 0 , v

(0) = A/ε , v

(0) = 0 (4.52)

+ ω

+ v

= 0 , v

(0) = 0 , v

(0) = 0 . (4.53)

Solving Eq. (4.52) results in

cos ωt . (4.54)

Equation (4.53), therefore, can be re-written down as

+ ω

cos ωt −

cos

ωt = 0 (4.55)

+ ω



−

3ω

4ε



cos ωt −

4ε

cos 3ωt = 0 . (4.56)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1171

We let

3ω

4ε

(4.57)

in Eq. (4.56) so that the secular term can be eliminated. Solving Eq. (4.56) yields

32ε

(cos ωt − cos 3ωt) (4.58)

Thus we obtain the ﬁrst-order approximate solution of the original Eq. (4.48),

which reads

u = ε(v

+ ε

) = A cos ωt +

(cos ωt − cos 3ωt) . (4.59)

Substituting Eq. (4.57) into Eq. (4.51) results in

1 = ω

+ ε

= ω

3ω

(4.60)

which leads to

ω =

1 +

. (4.61)

To compare, we write Nayfeh’s result,

which can be written in the form

u = ε

1/2

a cos



1 −

εa



t + θ



(4.62)

where ε is a small dimensionless parameter that is a measure of the amplitude

of oscillation, the constants a and θ can be determined by the initial conditions.

Imposing the initial conditions, Nayfeh’s result can be re-written as

u = A cos



1 −



t . (4.63)

It is also interesting to point out that Nayfeh’s result is only valid only for small

amplitude A, while the present results (4.59) and (4.61) are valid not only for small

amplitude A, but also for very large values of amplitude.

5. Homotopy Perturbation Method

The homotopy perturbation method

–

48,50

–

52,62,72

provides an alternative ap-

proach to introducing an expanding parameter.

To illustrate its basic ideas of the method, we consider the following nonlinear

diﬀerential equation

A(u) − f (r) = 0 , r ∈ Ω (5.1)

with boundary conditions

B(u, ∂u/∂n) = 0 , r ∈ Γ (5.2)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1172 J.-H. He

where A is a general diﬀerential operator, B is a boundary operator, f(r) is a known

analytic function, Γ is the boundary of the domain Ω.

The operator A can, generally speaking, be divided into two parts L and N,

where L is linear, while N is nonlinear. Equation (5.1), therefore, can be rewritten

as follows

L(u) + N (u) −f(r) = 0 . (5.3)

By the homotopy technique, we construct a homotopy v(r, p) : Ω × [0, 1] → R

which satisﬁes

H(v, p) = (1 − p)[L(v) −L(u

)] + p[A(v) −f(r)] = 0 , (5.4a)

H(v, p) = L(v) − L(u

) + pL(u

) + p[N(v) − f(r)] = 0 , (5.4b)

where p ∈ [0, 1] is an embedding parameter, u

is an initial approximation of

Eq. (5.1), which satisﬁes generally the boundary conditions.

Obviously, from Eq. (5.4a) or Eq. (5.4b) we have

H(v, 0) = L(v) −L(u

) = 0 , (5.5)

and

H(v, 1) = A(v) − f (r) = 0 . (5.6)

It is obvious that when p = 0, Eq. (5.4a) or Eq. (5.4b) becomes a linear equation;

when p = 1 it becomes the original nonlinear one. So the changing process of p

from zero to unity is just that of L(v) − L(u

) = 0 to A(v) − f(r) = 0. The

embedding parameter p monotonically increases from zero to unit as the trivial

problem L(v)−L(u

) = 0 is continuously deformed to the problem A(v)−f(r) = 0.

This is a basic idea of homotopy method which is to continuously deform a simple

problem easy to solve into the diﬃcult problem under study.

The basic assumption is that the solution of Eq. (5.4a) or Eq. (5.4b) can be

written as a power series in p:

v = v

+ pv

+ p

+ ···. (5.7)

Setting p = 1 results in the approximate solution of Eq. (5.1):

u = lim

p→1

v = v

+ v

+ ···. (5.8)

Recently, some rather extraordinary virtues of the homotopy perturbation

method have been exploited. The method has eliminated limitations of the tra-

ditional perturbation methods. On the other hand it can take full advantage of

the traditional perturbation techniques so there has been a considerable deal of

research in applying homotopy technique for solving various strongly nonlinear

equations,

–

4,25,29,37,109,110,120

furthermore the diﬀerential operator L does not

need to be linear, see Ref. 29.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1173

Liao

–

proposes a kind of approximate solution technique based on the ho-

motopy technique. In Liao’s approach, the solution is expanded into a more gener-

alized Taylor series with a free parameter ~, a careful choice of the parameter ~, as

illustrated in Liao’s publications, leads to fast convergence.

Diﬀerent from Liao’s method, the homotopy perturbation method uses the

imbedding parameter p as a small parameter, only few iterations (2 or 3) are needed

to search for the needed asymptotic solution, unlike Liao’s method where inﬁnite

series is needed, the suitable choice of the free parameter ~ can guarantee the

convergence of the obtained series. Comparison between homotopy perturbation

method and Liao’s method is systematically discussed in Ref. 51.

5.1. Periodic solution

As an illustrative example, we consider a problem of some importance in plasma

physics concerning an electron beam injected into a plasma tube where the magnetic

ﬁeld is cylindrical and increases toward the axis in inverse proportion to the radius.

The beam is injected parallel to the axis, but the magnetic ﬁeld bends the path

toward the axis. The governing equation for the path u(x) of the electrons is

+ u

−1

= 0 , (5.9)

with initial conditions u(0) = A, u

(0) = 0.

Physical considerations show that this equation has a periodic solution. In order

to look for the periodic solution, we suppose that the frequency of Eq. (5.9) is ω,

and assume that the nonlinear term in Eq. (2.9) can be approximated by a linear

term:

−1

∼ ω

u . (5.10)

So we obtain a linearized equation for Eq. (5.9)

+ ω

u = 0 . (5.11)

We re-write Eq. (5.9) in the form

+ ω

u = ω

u − u

−1

. (5.12)

We construct the following homotopy

+ ω

u = p(ω

u − u

−1

) . (5.13)

The embedding parameter p monotonically increases from zero to unit as the trivial

problem, so Eq. (5.11) is continuously deformed to the original problem, Eq. (5.9).

So if we can construct an iteration formula for Eq. (5.13), the series of approxi-

mations comes along the solution path, by incrementing the imbedding parameter

from zero to one; this continuously maps the initial solution into the solution of the

original Eq. (5.9).

Supposing that the solution of Eq. (5.13) can be expressed in a series in p:

u = u

+ pu

+ p

+ ···. (5.14)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1174 J.-H. He

We, therefore, obtain the following linear equations for u

and u

+ ω

= 0 , u

(0) = A , u

(0) = 0 , (5.15)

+ ω

= ω

− u

−1

, u

(0) = 0 , u

(0) = 0 . (5.16)

The solution of Eq. (5.15) is u

(t) = A cos ωt. Substituting u

into Eq. (5.16)

results in

+ ω

= A ω

cos ωt −

A cos ωt

. (5.17)

A rule of avoiding secular term in u

leads to the condition

π/(2ω)

cos ωt



A ω

cos ωt −

A cos ωt



dt = 0 , (5.18)

which leads to the result

ω =

√

2/A . (5.19)

Applying the variational iteration method as illustrated in Sec. 2.5, we can solve

from Eq. (5.17) under the initial conditions u

(0) = 0 and u

(0) = 0,

(t) =

sin ω(s − t)



Aω

cos ωs −

A cos ωs



ds . (5.20)

So we obtain the ﬁrst-order approximate solution by setting p = 1, which reads

u = u

+ u

= A cos ωt +

sin ω(s − t)



Aω

cos ωs −

A cos ωs



ds . (5.21)

The period of the system can be expressed in the form

T =

√

2πA = 4.442A . (5.22)

The exact period can be easily computed,

= 2

√

1/sds

= 2

√

ln A −ln u

. (5.23)

By transformation u = As, Eq. (5.23) leads to

= 2

√

ln(1/s)

= 2

√

2πA = 5.01325A . (5.24)

There exist alternative approaches to the construction of the needed homotopy. We

rewrite Eq. (5.9) in the form

+ u = 0 . (5.25)

We construct the following homotopy:

0 · u

+ 1 · u + pu

= 0 . (5.26)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1175

Please note that we recover the main term u

by multiplying zero. Assuming the

solution, the coeﬃcients of u and u

, 1 and 0, respectively, can be expanded, in the

following form

u = u

+ pu

+ p

+ ··· (5.27)

1 = ω

+ pa

+ p

+ ··· (5.28)

0 = 1 + pb

+ p

+ ··· . (5.29)

Substituting Eqs. (5.27)–(5.29) into Eq. (5.26) results in

(1 + pb

+ ···)(u

+ pu

+ ····) + (ω

+ pa

+ ···)(u

+ pu

+ ····)

+p(u

+ pu

+ ····)(u

+ pu

+ ····)

= 0 . (5.30)

Collecting terms of the same power of p, gives

+ ω

= 0 , (5.31)

+ ω

+ b

+ a

+ u

= 0 . (5.32)

The solution of Eq. (5.31) is u

(x) = A cos ωx. Substituting u

into Eq. (5.32), by

simple manipulation, yields

+ ω

−



− a



A cos ωx −

cos 3ωx = 0 . (5.33)

No secular term requires that

− a

= 0 . (5.34)

If the ﬁrst-order approximate solution is enough, then from Eqs. (5.28) and (5.29),

we have

1 = ω

+ a

(5.35)

0 = 1 + b

. (5.36)

Solving Eqs. (5.34)–(5.36) simultaneously, we obtain

ω =

√

−1

= 1.1547A

−1

. (5.37)

The accuracy 7.8% is remarkable good.

We can also rewrite Eq. (5.9) in the form

1 + u

u = 0 . (5.38)

Multiplying both sides of Eq. (5.38) by u

, we have

+ u

002

u = 0 . (5.39)

We construct a homotopy in the form

+ 0 · u + pu

002

u = 0 . (5.40)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1176 J.-H. He

Similarly, we expand the solution and the coeﬃcient of the middle term, 0, into the

forms

u = u

+ pu

+ p

+ ··· (5.41)

0 = ω

+ pa

+ p

+ ··· . (5.42)

As a result, we obtain

+ ω

= 0 , (5.43)

+ ω

+ a

+ u

002

= 0 . (5.44)

The solution of Eq. (5.43) is u

(x) = A cos ωx. Substituting u

into Eq. (5.44), by

simple manipulation, yields

+ ω





A cos ωx +

cos 3ωx = 0 . (5.45)

No secular terms in u

requires that

= −

. (5.46)

If we stop at the ﬁrst-order approximate solution, then Eq. (5.42) becomes

0 = ω

−

, (5.47)

from which we can obtain the same result as Eq. (5.37).

Now we consider Duﬃng equation

+ u + εu

= 0 , ε > 0 (5.48)

with initial conditions u(0) = A, and u

(0) = 0.

We construct the following homotopy:

+ ω

u + p(εu

+ (1 − ω

)u) = 0 , p ∈ [0, 1] . (5.49)

When p = 0, Eq. (5.49) becomes the linearized equation, u

+ ω

u = 0, when

p = 1, it turns out to be the original one. We assume that the periodic solution to

Eq. (5.49) may be written as a power series in p:

u = u

+ pu

+ p

+ ···. (5.50)

Substituting Eq. (5.50) into Eq. (5.49), collecting terms of the same power of p,

gives

+ ω

= 0 , u

(0) = A , u

(0) = 0 (5.51)

+ ω

+ εu

+ (1 − ω

= 0 , u

(0) = 0 , u

(0) = 0 . (5.52)

Equation (5.51) can be solved easily, giving u

= A cos ωt. If u

is substituted into

Eq. (5.52), and the resulting equation is simpliﬁed, we obtain

+ ω



1 +

εA

− ω



A cos ωt +

εA

cos 3ω = 0 . (5.53)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1177

No secular terms in u

requires that

ω =

1 +

εA

. (5.54)

The accuracy of the expression (5.54), is 7.6% when ε → ∞.

Secular terms arise for higher-order approximate solutions so we always stop

before the second iteration. If higher-order approximate solution is required, the

parameter-expanding method (the modiﬁed Lindstedt–Poincare method

) can be

applied. For example, we can construct the following homotopy

+ 1 · u + pεu

= 0 , (5.55)

and we expand the coeﬃcient of the linear term into a series of p:

1 = ω

+ pω

+ p

+ ··· (5.56)

where ω

can be identiﬁed in view of absence of secular terms in u

= −

εA

, ω

= −

128ω

. (5.57)

Substituting in Eq. (5.56) and setting p = 1, we have

1 = ω

−

εA

−

128ω

, (5.58)

ω =

1 +

εA

1 +

εA

. (5.59)

The accuracy of Eq. (5.59) reaches 5.9% when ε → ∞.

We can also construct a homotopy in the form

(1 − p)(v

+ ω

v − u

− ω

) + p(v

+ v + εv

) = 0 . (5.60)

By simple operation, we have

+ ω

− u

− ω

= 0 , (5.61)

+ ω

− (v

+ ω

− u

− ω

)(v

+ v

+ εv

) = 0 . (5.62)

We begin with

= u

= A cos ωt . (5.63)

Substituting Eq. (5.63) into Eq. (5.62), we have

+ ω

+ a



−ω

+ 1 +

εA



cos ωt +

εA

cos 3ωt = 0 , (5.64)

which is identical to Eq. (5.53).

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1178 J.-H. He

5.2. Bifurcation

In this section, we consider the following Duﬃng oscillator in space

+ ε(u − A

) = 0 , u(0) = u(π) = 0 , u(π/2) = 1 , ε ≥ 0 . (5.65)

For any ε ≥ 0, the above equation has the trivial solution u(t) = 0. The so-called

bifurcation occurs when a nontrivial solution exists for some of values of ε.

The traditional perturbation method predicts that

A = ±2

ε − 1

, ε ≥ 1 , (5.66)

which is only valid for small values of ε.

According to the homotopy perturbation, we construct the following simple

homotopy:

+ εu − εpA

= 0 . (5.67)

Assume that the solution can be expressed in the form

u = u

+ pu

+ p

+ ···. (5.68)

In the parlance of the parameter-expanding method (the modiﬁed Lindstedt–

Poincare method

), we expand the coeﬃcient of u in the middle term of Eq. (5.67)

in the form

ε = ω

+ pω

+ p

+ ···. (5.69)

Substituting Eqs. (5.68) and (5.69) into Eq. (5.67), and equating the terms with

the identical powers of p, we can obtain a series of linear equations and write only

the ﬁrst two linear equations:

+ ω

= 0 , u

(0) = u

(π) = 0 , u

(π/2) = 1 , (5.70)

+ ω

− εA

= 0 , u

(0) = u

(π) = 0 , u

(π/2) = 0 . (5.71)

The general solution of Eq. (5.70) is

(t) = a cos ωt + b sin ωt , (5.72)

where a and b are constant. Adjoining to the boundary conditions: u

(0) = u

(π) =

0, and the additional condition u

(π/2) = 1, we have a = 1, b = 0, and ω = 1. So

the solution of u

reads

= sin t . (5.73)

Substituting u

into Eq. (5.71), we obtain a diﬀerential equation for u

+ u

+ ω

sin t − εA

sin

t = 0 . (5.74)

+ u



−

εA



sin t −

εA

sin 3t = 0 . (5.75)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1179

No secular terms in u

requires that

εA

. (5.76)

The solution of Eq. (5.75), considering the boundary conditions u

(0) = u

(π) = 0,

and the additional condition u

(π/2) = 0, is

= −

εA

(sin 3t + sin t) . (5.77)

We, therefore, obtain the ﬁrst-order approximate solution which reads

u = u

+ u

= sin t −

εA

(sin 3t + sin t) , (5.78)

and

ε = 1 +

εA

. (5.79)

Since ε ≥ 0, the above equation has no solution when ε < 1. However, when ε > 1,

Eq. (5.79) has the solution

A = ±

√

1 −

. (5.80)

Therefore, the so-called simple bifurcation occurs at ε = 1. Our result is the same

with that obtained by Liao.

6. Iteration Perturbation Method

In this section, we will illustrate a new perturbation technique coupling with the

iteration method.

Consider the following nonlinear oscillation:

+ u + εf(u, u

) = 0 , u(0) = A , u

(0) = 0 . (6.1)

We rewrite Eq. (6.1) in the following form:

+ u + εu · g(u, u

) = 0 , (6.2)

where g(u, u

) = f/u.

We construct an iteration formula for the above equation:

n+1

+ u

n+1

+ εu

n+1

g(u

, u

) = 0 , (6.3)

where we denote by u

the nth approximate solution. For nonlinear oscillation,

Eq. (6.3) is of Mathieu type. We will use the perturbation method to ﬁnd approxi-

mately u

n+1

. The technique is called iteration perturbation method.

68,71

Consider Eq. (3.27), and assume that initial approximate solution can be ex-

pressed as u

= A cos ωt, where ω is the angular frequency of the oscillation, we

rewrite Eq. (3.27) approximately as follows

+ εA

u cos

ωt = 0 , (6.4a)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1180 J.-H. He

εA

u +

εA

u cos 2ωt = 0 , (6.4b)

which is Mathieu type. Suppose that

u = u

+ εu

+ ε

+ ··· , (6.5)

εA

= ω

+ εc

+ ε

+ ··· (6.6)

and substituting Eqs. (6.5) and (6.6) into Eq. (6.4b), and equating the coeﬃcients

of the same power of ε, we have the following two diﬀerential equations for u

and

+ ω

+ c

cos 2ωtu

= 0 , u

(0) = 0 , u

(0) = 0 , (6.7)

+ ω

+ c

cos 2ωtu

= 0 , u

(0) = 0 , u

(0) = 0 , (6.8)

where u

= A cos ωt. Substituting u

into Eq. (6.7), the diﬀerential equation for

becomes

+ ω

+ A





cos ωt +

cos 3ωt = 0 . (6.9)

The requirement of no secular term requires that

= −

. (6.10)

Solving Eq. (6.9), subject to the initial conditions u

(0) = 0 and u

(0) = 0, yields

the result:

32ω

(cos 3ωt − cos ωt) . (6.11)

We, therefore, obtain its ﬁrst-order approximate solution, which reads

u = A cos ωt +

εA

32ω

(cos 3ωt − cos ωt) = A cos ωt +

(cos 3ωt − cos ωt) ,(6.12)

where the angular frequency is determined from the relations (6.6) and (6.10), which

reads

ω =

√

1/2

A . (6.13)

To obtain the second-order approximate solution, we substitute u

and u

into

Eq. (6.8), in the parlance of no secular, we ﬁnd

32ω

= −

128ω

. (6.14)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1181

Substituting the determined c

and c

into Eq. (6.6), we have

εA

= ω

−

εA

−

128ω

, (6.15)

which leads to the result:

ω = 0.8174ε

1/2

A . (6.16)

However, the second-order period T = 7.6867ε

−1/2

−1

is not more accurate than

the ﬁrst-order one, because Eq. (6.4b) is an approximate one, the obtained result

(6.16) is the approximate angular frequency of Eq. (6.4b), not that of the original

one. So we need not search for higher-order approximations. To obtain approximate

solutions with higher accuracy, we replace zeroth-order approximate solution by the

following

= A cos ωt +

(cos 3ωt − cos ωt) . (6.17)

So the original Eq. (3.27) can be approximated by the linear equation

+ ε



A cos ωt +

(cos 3ωt − cos ωt)



u = 0 . (6.18)

By the same manipulation, we can identify the angular frequency ω = 0.8475ε

1/2

and obtain the approximate period T = 7.413ε

−1/2

−1

. The relative error is about

0.05%.

Now we re-consider Eq. (5.9) in a more general form:

= 0 . (6.19)

We approximate the above equation by

cos

ωt

u = 0 , (6.20)

u + u

cos 2ωt = 0 . (6.21)

We introduce an artiﬁcial parameter in Eq. (6.21)

u + εu

cos 2ωt = 0 . (6.22)

And suppose that

= ω

+ εc

+ ε

+ ···, (6.23)

u = u

+ εu

+ ε

+ ···, (6.24)

we obtain the diﬀerential equation for u

+ ω

+ c

+ u

cos 2ωt = 0 , (6.25)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1182 J.-H. He

+ ω

= −Ac

cos ωt + Aω

cos ωt cos 2ωt



−Ac

Aω



cos ωt +

Aω

cos 3ωt . (6.26)

The requirement of no secular term needs

. (6.27)

So we identify the angular frequency as follows

ω =

. (6.28)

The approximate period can be written as

T =

√

3πA

1/2

5.44A

1/2

. (6.29)

Acton and Squire,

using the method of weighted residuals, obtained the following

result:

T =

16A

1/2

5.33A

1/2

. (6.30)

Now we consider two-dimensional viscous laminar ﬂow over an ﬁnite ﬂat-plain gov-

erned by a nonlinear ordinary diﬀerential equation:

61,86,87

000

= 0 , x ∈ [0, +∞] , (6.31)

with boundary conditions f(0) = f

(0) = 0, and f

(+∞) = 1.

The prime in Eq. (6.31) denotes the derivatives with respect to x which is deﬁned

x = Y

νX

and f(x) is relative to the stream function Ψ by

f(x) =

√

νUX

Here, U is the velocity at inﬁnity, ν is the kinematic viscosity coeﬃcient, X and

Y are the two independent coordinates.

In order to obtain a perturbation solution of Blasius equation, we introduce an

artiﬁcial parameter ε in Eq. (6.31)

000

εff

= 0 . (6.32)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1183

Processing in a traditional way of perturbation technique, and supposing that

(0) = σ, we obtain a solution of Eq. (6.32) in the form of a power series

f(x) =

+∞

k=0



−



k+1

(3k + 2)!

3k+2

where











1 (k = 0 and k = 1)

k−1

r=0

3k − 1

k−r−1

(k ≥ 2)

which is valid only for small x.

Now we begin with the initial approximate solution

(x) = x −

(1 − e

−bx

) , (6.33)

where b is an unknown constant.

Equation (6.31) can be approximated by the following equation:

000



x −

(1 − e

−bx

)



= 0 , (6.34)

We rewrite Eq. (6.34) in the form

000

+ bf



x −

(1 − e

−bx

) − 2b



= 0 , (6.35)

and embed an artiﬁcial parameter ε in Eq. (6.35):

000

+ bf



x −

(1 − e

−bx

) − 2b



= 0 . (6.36)

Suppose that the solution of Eq. (6.36) can be expressed as

f = f

+ εf

+ ···, (6.37)

we have the following linear equations:

000

+ bf

= 0 , f

(0) = f

(0) = 0 , and f

(+∞) = 1 , (6.38)

000

+ bf



x +

−bx

− 2b −



= 0 ,

(0) = f

(0) = 0 , and f

(+∞) = 0 .

(6.39)

The solution of Eq. (6.38) is f

(x) = x − (1 − e

−bx

)/b, substituting it into

Eq. (6.39) results in

000

+ bf

= −

(bx + e

−bx

− 2b

− 1)e

−bx

. (6.40)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1184 J.-H. He

The constant b can be identiﬁed by the following expression:

+∞

−bx

(bx + e

−bx

− 2b

− 1)e

−bx

dx = 0 . (6.41)

So we have

Γ(2) +

−

1 + 2b

= 0 , (6.42)

which leads to the result:

b = 1/

√

12 = 0.28867 . (6.43)

The exact solution of Eq. (6.40) is not required, the expression (6.41) requires no

terms of xe

−βx

(n = 1, 2, 3, . . .) in f

. So we can assume that the approximate

solution of Eq. (6.40) can be expressed as

(t) = Ae

−bx

−2bx

+ B , (6.44)

where the constants A and B can be identiﬁed from the initial conditions f

(0) =

(0) = 0, i.e. A = −1/4 and B = 1/8.

By setting p = 1, we obtain the ﬁrst-order approximate solution, i.e.

(x) = f

(x) + f

(x) , (6.45)

where f

and f

are deﬁned by Eqs. (6.33) and (6.44) respectively.

A highly accurate numerical solution of Blasius equation has been provided by

Howarth,

who gives the initial slop f

(0) = 0.332057. Comparing the approxi-

mate initial slop:

(0) = f

(0) + f

(0) = 0.3095

we ﬁnd that the relative error is 6.8%.

We can improve the accuracy of approximate solution to a high-order by the

iteration technique. Now we use Eq. (6.45) as the initial approximate solution of

Eq. (6.31):

(x) = x −

(1 − e

−bx

) −

−bx

−2bx

, (6.46)

where b is undermined constant.

By parallel operation, the constant b can be identiﬁed by the following relation:

∞

−bx



bx + e

−bx

− 2b

− 1 −

−bx

−2bx



−bx

dx = 0 (6.47)

which leads to the result

b = 0.3062 . (6.48)

It is obvious that

(0) = b + 0.25b

= 0.3296 reach a very high accuracy, and the

0.73% accuracy is remarkable good.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1185

7. Ancient Chinese Methods

China is one of the four countries with an ancient civilization. The ancient Chinese

mathematicians had made greatest contributions to the development of human cul-

ture, however, essentially nothing of a primary nature has come down to West con-

cerning ancient Chinese mathematics, little has been discussed on ancient Chinese

mathematics in some of the most famous monographs on history of mathematics.

So the present author feels strongly necessary to give a basic introduction to the

great classics of ancient Chinese mathematics for our Western colleagues, who are

unfamiliar with the Chinese language. This section concerns brieﬂy some famous

ancient Chinese algorithms, which might ﬁnd wide applications in modern science.

Jiu Zhang Suan Shu, Nine Chapters on the Art of Mathematics, comprises nine

chapters and hence its title is the oldest and most inﬂuential work in the history

of Chinese mathematics. It is a collection of 246 problems on agriculture, business

procedure, engineering, surveying, solution of equations, and properties of right tri-

angles. Rules of solution are given systematically, but there exists no proofs in Greek

sense. As it was pointed by Dauben

that the Nine Chapters can be regarded as a

Chinese counterpart to Euclid’s Elements, which dominated Western mathematics

in the same the Nine Chapters came to be regarded as the seminal work of ancient

Chinese mathematics for nearly two millennia. Great classics, when revisited in the

light of new developments, may reveal hidden pearls, as is the case with ancient

Chinese method and He Chengtian’s inequality, which will be discussed below.

7.1. Chinese method

The Chinese algorithm is called the Ying Buzu Shu in Chinese, which was proposed

in about 2nd century BC, known as the rule of double false position in West after

1202 AD. Chapter 7 of the Nine Chapters is the Ying Buzu Shu (lit. Method of

Surplus and Deﬁciency; too much, too less), an ancient Chinese algorithm, which is

the oldest method for approximating real roots of a nonlinear equation. To illustrate

the basic idea of the method, we consider the 15th example in this chapter, which

reads:

The weight of a jade with a cubic chun is 7 liang, while the weight of a stone

with a cubic chun is 6 liang. There is a cubic bowlder with a length of 3 chun, and

a mass of 11 jin (1 jin = 16 liang). What is mass of the jade and the stone in the

bowlder?

Answer: the volume of the jade in the bowlder is 14 cubic chun, the mass is

6 jin and 2 liang; the volume of the stone is 13 cubic chun, the mass is 4 jin and

14 liang.

The solution procedure: Assume that the bowlder is made of jade only, then

the mass is overestimated by 13 liang. If we assume that bowlder is made of stone

only, the mass is underestimated by 14 liang. According to the Ying Buzu Shu, the

mass of jade can be determined.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1186 J.-H. He

In the language of modern mathematics, let x and y be the volumes of the jade

and stone, respectively, then we have

x + y = 3

= 27

and

7x + 6y = 11 ×16 = 176 .

We write R(x, y) = 7x+6y −198, and let x

= 27, y

= 0 (assume that the bowlder

is made of jade only), we have R

(27, 0) = 13; if let x

= 0, y

= 27 (assume that

bowlder is made of stone only), then we obtain R

(0, 27) = −14. According the

ancient Chinese method, we can calculate the value of x in the form

x =

− x

− R

27 × 14

13 + 14

= 14 .

The ancient Chinese method can also be powerfully applied to nonlinear prob-

lems. We consider the last example of Chapter 7. It reads:

There is a wall with a thickness of 5 chi. Two mice excavate a hole in the wall

from both sides. The big mouse burrows 1 chi in ﬁrst day, and doubles its headway

everyday afterwards; while the small one burrows 1 chi in ﬁrst day also, but halves

its headway everyday afterwards. When the hole can be completed? What is the

length for each mouse?

Answer: 2 and 2/17 days. The big mouse penetrates totally 3 and 12/17 chi,

while the small one 1 and 5/17 chi.

The solution procedure: Assume that it needs two days, then 5 chun (1 chun

= 0.1 chi) is left. If we assume that it needs three days, then 3 chi and 0.75 chun

is redundant. According to the Ying Buzu Shu, the days needed for excavation can

be determined.

30.75 ×2 + 5 × 3

5 + 30.75

≈ 2

Consider an algebraic equation,

f(x) = 0 . (7.1)

Let x

and x

be the approximate solutions of the equation, which lead to the

remainders f (x

) and f(x

) respectively, the ancient Chinese algorithm leads to

the result

x =

f(x

) − x

f(x

)

f(x

) − f(x

)

. (7.2)

Now we rewrite Eq. (7.2) in the form

f(x

) − x

f(x

)

f(x

) − f(x

)

= x

−

f(x

)(x

− x

)

f(x

) − f (x

)

. (7.3)

If we introduce the derivative f

) deﬁned as

) =

f(x

) − f (x

)

− x

, (7.4)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1187

then we have

x = x

−

f(x

)

, (7.5)

which is the well-known Newton iteration formulation proposed by Newton (1642–

1727). Chinese mathematicians had used the theory for more than one millennium.

The Chinese algorithm seems to have some advantages over the well-known Newton

iteration formula if the two points (x

and x

) locate two sides of the exact root,

i.e. f(x

) · f(x

) < 0. Consider an example

f(x) = sin x . (7.6)

We begin with x

= 0.1 and x

= 4.7. It is obvious that f

= 0.1 > 0 and

= −1.0 < 0, so there is a root locating x ∈ (x

, x

). By Eq. (7.2), we have

f(x

) − x

f(x

)

f(x

) − f (x

)

= 1.43 .

Calculating f (x

) results in f

= 0.99 > 0, so the root locates at x ∈ (x

, x

). Using

Eq. (7.2), we have

f(x

) − x

f(x

)

f(x

) − f (x

)

= 3.06 .

We see the procedure converges very fast to the exact root, the accuracy reaches

2.5% by only two iterations even for very poor initial prediction.

Consider the Duﬃng equation

+ u + εu

= 0 , u(0) = A , u

(0) = 0 . (7.7)

We use the trial functions u

(t) = A cos t and u

= A cos ωt, which are, respec-

tively, the solutions of the following linear equations

+ ω

u = 0 , ω

= 1 , (7.8)

and

+ ω

u = 0 , ω

= ω

. (7.9)

The residuals are

(t) = εA

cos

t , (7.10)

(t) = A(1 − ω

) cos ωt + εA

cos

ωt . (7.11)

By the Chinese method, we have

(0) − ω

(0)

(0) − R

(0)

1 − ω

+ εA

− ω

εA

1 − ω

= 1 + εA

, (7.12)

i.e.,

ω =

1 + εA

. (7.13)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1188 J.-H. He

The approximate period reads

T =

2π

√

1 + εA

. (7.14)

Table 1 illustrates the comparison between the approximate period with the exact

one when εA

< 1.

The obtained solution obtained by the Chinese method is valid for the whole

solution domain, even in the case εA

→ ∞, the accuracy reaches 7.6%.

7.2. He Chengtian’s method

In a history book, it writes

He Chengtian uses 26/49 as the strong, and 9/17 as the weak. Among the strong

and the weak, Chengtian tries to ﬁnd a more accurate denominator of the fractional

day of the Moon. Chengtian obtains 752 as the denominator by using 15 and 1 as

weighting factors, respectively, for the strong and the weak. No other calendar can

reach such a high accuracy after Chengtian, who uses heuristically the strong and

weak weighting factors.

The above statement is generally called Tiao Ri Fa (Lit. Method for Modiﬁca-

tion of Denominator). Here, we call it as He Chengtian’s interpolation or He Cheng-

tian’s method. There exists many other famous interpolations in ancient Chinese

mathematics.

In modern mathematical view, we illustrate the statement as follows:

According to the observation data, He Chengtian ﬁnds that

days > 1 Moon > 29

days .

Using the weighting factors (15 and 1), He Chengtian obtains

the fractional day =

26 × 15 + 9 × 1

49 × 15 + 17 × 1

399

752

1 Moon = 29

399

752

days .

The error is about 0.1 second, it is the most accurate in He Chengtian’s time.

He Chengtian establishes Yuan-Jia Calendar, which is the most famous lunar cal-

endar in ancient China. The calendar was lunar with intercalary months to keep in

approximate synchronization with the seasons.

Table 1. Comparison of approximate period with the exact one.

εA

0 0.042 0.087 0.136 0.190 0.25

6.283 6.187 6.088 5.986 5.879 5.767

T (Eq. 7.14) 6.283 6.155 6.026 5.895 5.760 5.620

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1189

He Chengtian uses 26/49 as the strong weighing factor because it is much more

accurate than that of the weak weighing factor, 9/17. The observation shows that

the accuracy of the former is about 0.005%, while the accuracy for the latter is

about 0.22%. So the weighting factor for 26/49 is much larger than that for 9/17.

He Chengtian actually uses the following inequality:

< x <

, (7.15)

where a, b, c and d are real numbers, then

ma + nd

mb + nc

, (7.16)

and x is approximated by

x(m, n) =

ma + nd

mb + nc

, (7.17)

where m and n are weighting factors. We call x(1, 1) = (a + d)/(b + c) the He

Chengtian’s average of a/b and d/c.

Generally speaking m and n can be freely chosen, and the accuracy of the

obtained result is better than a/b or d/c. For example, if we choose m = 100 and

n = 1, then the improved fractional day can be calculated as

the fractional day =

26 × 100 + 9 × 1

49 × 100 + 17 × 1

2609

4917

The accuracy is about 0.0043%, and is improved compared with that of 26/49.

Equation (7.16) can be rewritten equivalently in the form

x =

ma + nd

mb + nc

ka + d

kb + c

, (7.18)

where k = m/n.

It is obvious that

lim

k→0

x =

, (7.19)

and

lim

k→∞

x =

. (7.20)

The changing process of k from zero to inﬁnite is just that of x from d/c to a/b.

There must exist a certain value of k, while the corresponding value of x locates at

its exact solution.

There exists no general rule of choosing the value of k. It should be larger than 1

when a/b is more accurate than a/b. The more accurates a/b is, the larger value of

k. If the accuracy of a/b is lower than that of d/c, the value of k should be smaller

than 1. The less accurate a/b is, the smaller value of k.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1190 J.-H. He

In He Chengtian’s time, ancient Chinese mathematicians knew that 157/50 <

π < 22/7 (π = 3.14 was suggested by Liu Hui (263), and π = 22/7 was obtained

by Zu Chongzhi and/or his son Zu Geng.)

Using the weighting factors 1 and 9

, we have

π =

157 + 9 × 22

50 + 9 × 7

355

113

= 3.1415929 .

So we obtain

355

113

< π <

which was obtained by Zu Chongzhi (430–501).

Example 1. Consider the complete elliptic integral of the ﬁrst kind

π/2

1 − k sin

x dx .

It is obvious that

π/2

1 − k sin

x dx <

√

1 − k . (7.21)

By He Chengtian’s method, we have

π/2

1 − k sin

x dx =

2(p + q)

(p + q

√

1 − k)

2(1 + ξ)

(1 + ξ

√

1 − k) , (7.22)

where p and q are weighting factors, ξ = q/p.

In the case k = 0, we have ξ = 0, and when k = 1, it follows ξ =

π−2

, so we

can approximately determine ξ in the form:

ξ =

π − 2

k . (7.23)

We, therefore, have

π/2

1 − k sin

x dx =

2 + (π − 2)k



1 +

π − 2

√

1 − k



. (7.24)

Example 2. Consider the equation

(1 + u

+ u = 0 , u(0) = A , u

(0) = 0 . (7.25)

We rewrite Eq. (7.25) in the form

= −

1 + u

u . (7.26)

22/7 is more accurate than 157/50, so the value of weight factor for the former is bigger than

that of the latter.

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1191

If we choose the trial-function in the form u = A cos ωt, where ω is the frequency,

then the maximal and minimal values of 1/(1 + u

) are, respectively, 1 and 1/(1 +

). So we immediately obtain

< ω

1 + A

. (7.27)

According to He Chengtian’s interpolation, we set

m + n

m + n(1 + A

)

1 + kA

, (7.28)

where m and n are weighting factors, k = n/(m + n).

So the frequency can be approximated as

ω =

√

1 + kA

. (7.29)

To compare, we write Nayfeh’s result,

which can be written in the form

u = A cos



1 −



t , (7.30)

which is valid only for small amplitude A. To match Nayfeh’s result, we set k = 3/4

in Eq. (7.29), yielding the result

ω =

1 +

. (7.31)

The accuracy reaches 8.5% even when A → ∞, which is remarkably good.

Example 3. Consider Duﬃng equation, which reads

+ u + εu

= 0 , u(0) = A , u

(0) = 0 (7.32)

where ε needs not be small in the present study, i.e. 0 ≤ ε < ∞.

We rewrite Eq. (7.32) in the form

= −(1 + εu

)u , (7.33)

which has a similar form of u

= −ω

u. Assume that period solution of Eq. (7.32)

can be written in the form

u = A cos ωt , (7.34)

where ω is the frequency.

Observe that the square of frequency, ω

, is never less than that in the solution

(t) = A cos t (7.35)

of the following oscillation

= −(1 + εu

min

)u = −u . (7.36)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

1192 J.-H. He

In addition, ω

never exceed the square of frequency of the solution

(t) = A cos

1 + εA

t (7.37)

of the following oscillation

= −(1 + εu

max

)u = −(1 + εA

)u . (7.38)

Hence, it follows that

< ω <

1 + εA

. (7.39)

According to He Chengtian’s interpolation, we have

ω =

m + n(1 + εA

)

m + n

1 + kεA

(7.40)

where k = n/(m + n).

In the case ε  1, the perturbation technique gives

pert

= 1 +

εA

. (7.41)

Matching Eq. (7.40) with Eq. (7.41) when ε  1, we obtain k = 3/4. So the

frequency can is obtained as follows

ω =

1 +

εA

. (7.42)

Example 4. Consider the motion of a particle on a rotating parabola, which is

governed by the equation

(1 + 4q

+ ε

u + 4q

= 0 , (7.43)

with initial conditions u(0) = A, and u

(0) = 0.

We rewrite Eq. (7.43) in the form

= −f(u)u , (7.44)

where

f(u) =

+ 4q

1 + 4q

. (7.45)

We begin with the trial function u(t) = A cos ωt, then we have

f(u) =

+ 4q

sin

ωt

1 + 4q

cos

ωt

. (7.46)

The maximum and minimum of f(u) are

max

(u) =

1 + 4q

, (7.47)

min

(u) = ε

, (7.48)

April 13, 2006 10:4 WSPC/140-IJMPB 03379

Some Asymptotic Methods for Strongly Nonlinear Equations 1193

respectively. So we have the following inequality

< ω <

1 + 4q

. (7.49)

By He Chengtian’s method, we have

ω =

(m + n)ε

m + n(1 + 4q

)

1 + 4kq

. (7.50)

The perturbation solution of Eq. (7.43) when ε  1 reads

pert

1 + 2q

. (7.51)

Comparison of Eq. (7.50) and (7.51) leads to the result: k = 1/2, so we obtain

ω =

1 + 2(qA)

. (7.52)

The accuracy reaches 10% even when qA → ∞.

8. Conclusions

We have reviewed a few new asymptotic techniques with numerous examples. All

reviewed methods can be applied to various kinds of nonlinear problems, and the

examples studied in the present review article can be used as paradigms for real-

life physics problems. For the nonlinear oscillators, all the reviewed methods yield

high accurate approximate periods, but the accuracy of the amplitudes cannot be

ameliorated by iteration, the discussion of the problem is available in Journal of

Sound and Vibration, Vol. 282, pages 1317–1320 (2005).

Acknowledgment

This work is supported by the Program for New Century Excellent Talents in

University of P.R. China and the William M.W. Wong Engineering Research Fund.

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