- Email: [email protected]
Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parametrized perturbation technique
No.1
. . . (II) Parametrized
HE: Some New Approaches
Perturbation
...
81
Some New Approaches to Duffing Equation with Strongly and High Order Nonlinearity (II) Parametrized Perturbation Technique 2o Jihuan HE ( Shanghai Institute of Applied Mathematics Shanghai 200072, China) Email: [email protected]
and Mechanics, Shanghai University,
Abstract: For the strongly nonlinear equations without small parameter, a transformation u = ,kIv can be introduced for the nonlinear equation L(U) + N(U) = 0, where L and N are general linear and nonlinear differential operators respectively, @ is sufficiently small. Therefore, a strongly non-linear system is transformed into a small parameter system with respect to the new introduced parameter 8, thus various traditional perturbation techniques can be applied. By Lindstedt-Poincare method, a perturbation solution for Duffing equation with 5th order nonlinearity is obtained, which is valid not only for the small parameter E in the equation, but also for very large values of E. Keywords: perturbation technique, nonlinear equation
Introduction In this paper, we will study the strongly nonlinear equation without small parameter. Consider the well-known Duffing equation with 5th order nonlinearity[1-3] u(0) = A,
u” + 21+ &U5 = 0,
u’(0) = 0
(1)
where E needs not be small in the present study, i.e. 0 5 E < 00. In order to use the traditional perturbation methods, it is necessary to introduce an artificial small parameter /3. We let ‘11= pv (2) in Eq.(l)
and obtain
A 40) = 3’
v” + v + &P4V5 = 0,
1 Method Applying
v’(0) = 0
(3)
& It’s Solution the Lindstedt-Poincare
method, i.e. letting 7- =
(4
wt
in Eq.(3), we have the following equation w2v”
+ v + &P4V5 = 0,
where the primes denote differentiation 20The paper was received on Feb.6, 1999
v(0)
= $,
with respect to r.
v’(0)
= 0
(5)
82
Communications
in Nonlinear
Science & Numerical
Vo1.4, No.1 (Apr. 1999)
Simulation
Suppose that the solution of Eq.(5) and w2 can be expressed in the forms w = wo + p4w1 + /I%2 + . *.
(6)
w2 = 1+ @WI + pswz +. . .
(7)
Substituting Eqs.(G) and (7) into (5) and equating coefficients of like powers of /3 yield the following equations
w; + wo = 0,
A vo(O) = -7 P 5-O - )
w:‘+wl+wlw::+EwO
w(!)(O)= 0
w(O) = 0,
w:(o) = 0
(8) (9)
Solving Eq. (8) results in
A 210= -cosr P Eq.(9), therefore, can be rewritten
vy+q+
(10)
as
(~-wl)~cosT+~cos37+~cos5gi=0
(11)
Avoiding the presence of a secular term needs
5&A4 wl = sg4 Solving Eq.( ll),
(12)
we obtain
EA5 u1 = - -(COST 128jj5 If, for example, its first-order
&A5 - COS3T)- -(COST 384P5
approximation
- cos57)
(13)
is sufficient, then we have
21= pw = P(wo + P4Wl)
&A5 &A5 = Acoswt - 128 (coswt - cos3wt) coswt - cos5wt) 384( where the angular frequency can be written
(14
in the form (15)
w=&j%&/l+$A4
(15)
Observe that for small E, i.e. 0 < E << 1, it follows that
w = 1 + $A2
(16)
Consequently, in this limit,, the present method gives exactly the same results as the standard Lindstedt-Poincare method[‘l. To illustrate the remarkable accuracy of the obtained result,, we compare the approximate period
(17)
No.1
ZENG: Entropy
Function for
83
.
with the exact oneL2] with Ic = tsA4/(1
What is rather surprising about the remarkable range of validity actual asymptotic period as E + 00 is also of high accuracy. lim cA4+m
dz
T
1 + co&
2m = ___ 7r + COSQZ
+ $A4)
(18)
of (17) is that the
x 1.14811=
1.0008
Therefore, for any value of E, it can be easily proved that the maximal relative error is less than 0.08%.
2 Conclusion Alternatively we introduce a small parameter solutions do not depend upon the parameter valid regardless the values of the parameter E transformation (2) can be written in the form details will be discussed in another paper. Acknowledgment The work is supported (98QN47).
B instead of E, so the obtained perturbation E in the original equations, and naturally in the equations. For more generality, the u = BV + b, where b is a constant. More
by the Shanghai Education
Foundation
for Young Scientists
References [l] Nayfeh, A. H., Problems in Perturbation, New York: John Wiley & Sons, 1985 [2] He, J. H., Variational iteration method: a kind of nonlinear analytical technique, some examples, Int. J. Non-linear Mech., 1999, 34(4): 699-708 [3] He, J. H., Some new approaches to Duffing equation with strongly and high order nonlinearity (I) linearized perturbation technique, Communications in Nonlinear Science and Numerical Simulation, 1999, 4(1):79
Entropy Function Thermodynamics
for Multifractal 21
Qiuhua ZENG (Department of Physics, Sichuan University, Email: [email protected]
Chengdu
610064, China)
Abstract: The theory on multifractal thermodynamics has been studied by the method of series expansion. The method is able to overcome the shortages of Kohmoto’s steepest desent method and the results have general meanings. Keywords: multifractal thermodynamics, classical thermodynamics, entropy function *‘The paper was received on Apr.9, 1999
. . . (II) Parametrized
HE: Some New Approaches
Perturbation
...
81
Some New Approaches to Duffing Equation with Strongly and High Order Nonlinearity (II) Parametrized Perturbation Technique 2o Jihuan HE ( Shanghai Institute of Applied Mathematics Shanghai 200072, China) Email: [email protected]
and Mechanics, Shanghai University,
Abstract: For the strongly nonlinear equations without small parameter, a transformation u = ,kIv can be introduced for the nonlinear equation L(U) + N(U) = 0, where L and N are general linear and nonlinear differential operators respectively, @ is sufficiently small. Therefore, a strongly non-linear system is transformed into a small parameter system with respect to the new introduced parameter 8, thus various traditional perturbation techniques can be applied. By Lindstedt-Poincare method, a perturbation solution for Duffing equation with 5th order nonlinearity is obtained, which is valid not only for the small parameter E in the equation, but also for very large values of E. Keywords: perturbation technique, nonlinear equation
Introduction In this paper, we will study the strongly nonlinear equation without small parameter. Consider the well-known Duffing equation with 5th order nonlinearity[1-3] u(0) = A,
u” + 21+ &U5 = 0,
u’(0) = 0
(1)
where E needs not be small in the present study, i.e. 0 5 E < 00. In order to use the traditional perturbation methods, it is necessary to introduce an artificial small parameter /3. We let ‘11= pv (2) in Eq.(l)
and obtain
A 40) = 3’
v” + v + &P4V5 = 0,
1 Method Applying
v’(0) = 0
(3)
& It’s Solution the Lindstedt-Poincare
method, i.e. letting 7- =
(4
wt
in Eq.(3), we have the following equation w2v”
+ v + &P4V5 = 0,
where the primes denote differentiation 20The paper was received on Feb.6, 1999
v(0)
= $,
with respect to r.
v’(0)
= 0
(5)
82
Communications
in Nonlinear
Science & Numerical
Vo1.4, No.1 (Apr. 1999)
Simulation
Suppose that the solution of Eq.(5) and w2 can be expressed in the forms w = wo + p4w1 + /I%2 + . *.
(6)
w2 = 1+ @WI + pswz +. . .
(7)
Substituting Eqs.(G) and (7) into (5) and equating coefficients of like powers of /3 yield the following equations
w; + wo = 0,
A vo(O) = -7 P 5-O - )
w:‘+wl+wlw::+EwO
w(!)(O)= 0
w(O) = 0,
w:(o) = 0
(8) (9)
Solving Eq. (8) results in
A 210= -cosr P Eq.(9), therefore, can be rewritten
vy+q+
(10)
as
(~-wl)~cosT+~cos37+~cos5gi=0
(11)
Avoiding the presence of a secular term needs
5&A4 wl = sg4 Solving Eq.( ll),
(12)
we obtain
EA5 u1 = - -(COST 128jj5 If, for example, its first-order
&A5 - COS3T)- -(COST 384P5
approximation
- cos57)
(13)
is sufficient, then we have
21= pw = P(wo + P4Wl)
&A5 &A5 = Acoswt - 128 (coswt - cos3wt) coswt - cos5wt) 384( where the angular frequency can be written
(14
in the form (15)
w=&j%&/l+$A4
(15)
Observe that for small E, i.e. 0 < E << 1, it follows that
w = 1 + $A2
(16)
Consequently, in this limit,, the present method gives exactly the same results as the standard Lindstedt-Poincare method[‘l. To illustrate the remarkable accuracy of the obtained result,, we compare the approximate period
(17)
No.1
ZENG: Entropy
Function for
83
.
with the exact oneL2] with Ic = tsA4/(1
What is rather surprising about the remarkable range of validity actual asymptotic period as E + 00 is also of high accuracy. lim cA4+m
dz
T
1 + co&
2m = ___ 7r + COSQZ
+ $A4)
(18)
of (17) is that the
x 1.14811=
1.0008
Therefore, for any value of E, it can be easily proved that the maximal relative error is less than 0.08%.
2 Conclusion Alternatively we introduce a small parameter solutions do not depend upon the parameter valid regardless the values of the parameter E transformation (2) can be written in the form details will be discussed in another paper. Acknowledgment The work is supported (98QN47).
B instead of E, so the obtained perturbation E in the original equations, and naturally in the equations. For more generality, the u = BV + b, where b is a constant. More
by the Shanghai Education
Foundation
for Young Scientists
References [l] Nayfeh, A. H., Problems in Perturbation, New York: John Wiley & Sons, 1985 [2] He, J. H., Variational iteration method: a kind of nonlinear analytical technique, some examples, Int. J. Non-linear Mech., 1999, 34(4): 699-708 [3] He, J. H., Some new approaches to Duffing equation with strongly and high order nonlinearity (I) linearized perturbation technique, Communications in Nonlinear Science and Numerical Simulation, 1999, 4(1):79
Entropy Function Thermodynamics
for Multifractal 21
Qiuhua ZENG (Department of Physics, Sichuan University, Email: [email protected]
Chengdu
610064, China)
Abstract: The theory on multifractal thermodynamics has been studied by the method of series expansion. The method is able to overcome the shortages of Kohmoto’s steepest desent method and the results have general meanings. Keywords: multifractal thermodynamics, classical thermodynamics, entropy function *‘The paper was received on Apr.9, 1999