Research Journal of Mathematics and Statistics 3(1): 12-19, 2011
ISSN: 2040-7505
© Maxwell Scientific Organization, 2011
Received: July 12, 2010
Accepted: September 28, 2010
Published: February 15, 2011
An Analytical Approximate Solution of Fourth Order Damped-Oscillatory
Nonlinear Systems
1
Habibur Rahman, 1B.M. Ikramul Haque and 2M. Ali Akbar
Department of Mathematics, Khulna University of Engineering and Technology (KUET),
Khulna-9203, Bangladesh
2
Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
1
Abstract: In this study, the Krylov-Bogoliubov-Mitroplskii (KBM) method has been extended for obtaining
the solution of fourth order damped-oscillatory nonlinear systems. The method is illustrated by an example. The
results obtained by the presented technique agree nicely with the results (considered as exact solution) obtained
by the numerical method.
Key words: Eigen-values, perturbation method, weakly nonlinear systems
In this study, we have investigated solutions of the
fourth order damped oscillatory nonlinear systems when
two of the eigen-values are complex conjugates and the
other two are real and negative. The results obtained by
the presented method agree nicely with those obtained by
the numerical method.
INTRODUCTION
The Krylov-Bogoliubov-Mitropolskii (KBM) method
(1947, 1961) is one of the widely used techniques to
obtain analytical approximate solution of weakly
nonlinear systems. The method was originally developed
for system with periodic solution was later extended by
Popov (1956) for nonlinear damped oscillatory systems.
Owing to physical importance, Mendelson (1970)
rediscovered Popov’s results. Murty and Deekshatulu
(1969) expanded the method to solve over-damped
nonlinear systems. Murty (1971) presented a unified
KBM method for solving second order nonlinear systems
which cover the un-damped, damped and over-damped
cases. Bojadziev and Hung (1984) developed a technique
based on the KBM method to solve damped oscillations
modeled by a 3-dimensional time dependent system.
Alam (2001) developed a new perturbation technique to
find the analytical approximate solution of nonlinear
systems with large damping. Later, Alam (2002a)
extended the method for n-th order nonlinear systems.
Alam and Sattar (2001) examined third order timedependent oscillating systems with large damping. Alam
and Sattar (1997) also presented a unified method for
obtaining solution of third order damped oscillatory and
over-damped nonlinear systems. Akbar et al. (2002)
investigated a technique for solving fourth order overdamped nonlinear systems. Later, Akbar et al. (2003)
extended the technique for damped oscillatory nonlinear
systems in the case when the four eigen-values are
complex conjugates. But, none of the above authors
investigated solution of fourth order nonlinear systems
when two of the eigen-values are real and negative and
the rest of the two are complex conjugates.
METHODOLOGY
Consider a weakly nonlinear damped oscillatory
system governed by the differential equation:
d4x
d 3x
d2x
dx
+
+
+ c3
+ c4 x
c
c
1
2
4
3
2
dt
dt
dt
dt
= −ε f ( x , x& , &&
x , &&&
x)
(1)
where g is a small positive quantity, f is the nonlinear
function and c1, c2, c3, c4 are the characteristic parameters
defined by:
4
4
c1 =
∑λ , c = ∑λ λ
2
i
i =1
i
j
i , j =1
i≠ j
4
c3 =
∑λ λ λ
i
i , j , k =1
i≠ j≠k
j k
4
and c4 =
∏λ
i
i =1
where –81, –82, –83, –84 are four eigen-values of the
unperturbed Eq. (1). We consider, two of the eigen-values
say –81, –82 are real and negative and the other two say
–83, –84 are complex conjugates.
Corresponding Author: M. Ali Akbar, Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
12
Res. J. Math. Stat., 3(1): 12-19, 2011
The unperturbed solution (when g = 0) of the Eq. (1) is:
x( t ,0) =
4
∑a
i ,0 e
− λi t
(2)
i =1
where, ai,0 (i = 1,2,3,4) are constants of integration. If g … 0, following Alam (2002b), we seek the solution of the Eq.
(1) in of the form:
x( t , ε ) =
4
∑ae
i =1
i
− λi t
(
)
(
)
+ ε u1 a1 , a 2 , a 3 , a 4 , t + ε 2 u2 a1 , a 2 , a 3 , a 4 , t + ε 3 ...
(3)
where, each ai (i = 1,2,3,4) satisfies the first order differential equation:
da i ( t )
= εAi a1 , a 2 , a 3 , a 4 , t + ε 2 Bi a1 , a 2 , a 3 , a 4 , t + ε 3 ...
dt
(
)
(
)
(4)
Differentiating (3) four times with respect to t, substituting x and the derivatives, in the original Eq. (1), using the
relation given in (4) and finally extracting the coefficients of ,, we obtain:
4
∏
i =1
⎛ d
⎞
⎜ + λi ⎟ u1 +
⎝ dt
⎠
4
∑e
− λi t
i =1
⎛ 4 ⎛ d
⎞ ⎞⎟
⎜
⎜ − λi + λ k ⎟ ⎟ Ai = f
⎜
⎠⎠
⎝ k = 1,i ≠ k ⎝ dt
∏
( 0) = f x , x& , &&
where, f
( 0 0 x0 , &&&x0 ) and x0 =
4
( 0)
(a1 , a 2 , a 3 , a 4 , t )
(5)
∑ a ( t )e λ
i
− it
i =1
In general, the functional f (0) can be expended in the Tailor series as (Murty and Deekshatulu, 1969):
f
( 0)
∞ ,...,∞
∑
=
Fm1 ,m2 ,m3 ,m4 a1 1 , a 2 2 , a 3 3 , a 4 4 e (
m
m
m
m
)
− m1λ1 , − m2 λ 2 , − m3λ 3 , − m4 λ 4 t
m1 = −∞ ,...,m4 = −∞
According to the KBM method, u1 does not contain the fundamental terms (Alam, 2001, 2002b; Murty and
Deekshatulu, 1969). Therefore, Eq. (5) can be separated into five equations for the unknown functions A1, A2, A3, A4
and u1. Substituting the value of f (0) into the Eq. (5) and equating the coefficients of e
(
)(
)(
− λi t
(i = 1,2,3,4), we obtain:
)
e − λ1t D − λ1 + λ 2 D − λ1 + λ 3 D − λ1 + λ 4 A1
=
∑F
m1
m2
m1 ,m2 ,m3 ,m4 a1 , a 2
m3 = m4 , m1 = m2 + 1,
, a3 3 , a4 4 e(
D=
m
m
)
− m1λ1 , − m2 λ 2 , − m3λ 3 , − m4 λ 4 t
d
dt
13
(6)
Res. J. Math. Stat., 3(1): 12-19, 2011
(
)(
)(
)
e − λ2 t D − λ 2 + λ1 D − λ 2 + λ 3 D − λ 2 + λ 4 A1
=
∑F
m1
m2
m1 ,m2 ,m3 ,m4 a1 , a 2
, a3 3 , a4 4 e(
m
m
)
(7)
)
(8)
)
(9)
− m1λ1 , − m2 λ 2 , − m3λ 3 , − m4 λ 4 t
m3 = m4 , m1 = m2 − 1
(
)(
)(
)
e − λ3t D − λ3 + λ1 D − λ 3 + λ 2 D − λ 3 + λ 4 A3
=
∑F
m1
m2
m1 ,m2 ,m3 ,m4 a1 , a 2
, a3 3 , a4 4 e(
m
m
− m1λ1 , − m2 λ 2 , − m3λ 3 , − m4 λ 4 t
m1 = m2 , m3 = m4 + 1
(
)(
)(
)
e − λ4 t D − λ 4 + λ1 D − λ 4 + λ 3 D − λ 4 + λ 3 A4
=
∑F
m1
m2
m1 ,m2 ,m3 ,m4 a1 , a 2
, a3 3 , a4 4 e(
m
m
− m1λ1 , − m2 λ 2 , − m3λ 3 , − m4 λ 4 t
m1 = m2 , m3 = m4 − 1
and
( D + λ1 )( D + λ2 )( D + λ3 )( D + λ4 )u1
= ∑ ′ Fm1 ,m2 ,m3 ,m4 a1 1 , a 2 2 , a 3 3 , a 4 4 e (
m
m
m
m
(10)
)
− m1λ1 , − m2 λ 2 , − m3λ 3 , − m4 λ 4 t
where 3! excludes those terms for m1 = m2 ± 1, m3 = m4 ± 1.
The particular solutions of Eq. (6)-(10) give the unknown functions A1, A2, A3, A4 and u1. Therefore, the
determination of the first order approximate solution is completed.
Example: As an example of the above method, we consider a weakly nonlinear damped oscillatory system governed
by the fourth order differential equation:
d4x
dt 4
+ c1
d 3x
dt 3
+ c2
d2x
dt 2
+ c3
dx
+ c4 x = −ε x& 3
dt
(11)
For example (11), we have, f = x3 and:
0
f ( ) = a13λ13e − 3λ1t + a23λ32 e − 3λ2 t + a33λ33e − 3λ3t + a43λ34 e − 3λ4 t
− 2λ + λ t
− 2λ + λ t
− 2λ + λ t
+ 3 a12 a2 λ12 λ2 e ( 1 2 ) + a12 a3λ12 λ3e ( 1 3 ) + a12 a4 λ12 λ4 e ( 1 4 )
{
− 2λ + λ t
− 2λ + λ t
− 2λ + λ t
+ a22 a1λ22 λ1e ( 2 1 ) + a22 a3λ22 λ3e ( 2 3 ) + a22 a4 λ22 λ4 e ( 2 4 )
− 2λ + λ t
− 2λ + λ t
− 2λ + λ t
+ a32 a1λ23 λ1e ( 3 1 ) + a32 a2 λ23 λ2 e ( 3 2 ) + a32 a4 λ32 λ4 e ( 3 4 )
− 2λ + λ t
− 2λ + λ t
− 2λ + λ t
+ a 2 a λ2 λ e ( 4 1 ) + a 2 a λ2 λ e ( 4 2 ) + a 2 a λ2 λ e ( 4 3 )
4 1 4 1
{
4 2 4 2
4 3 4 3
− λ +λ +λ t
− λ +λ +λ t
+ 6 a1a2 a3λ1λ2 λ3e ( 1 2 3 ) + a1a2 a4 λ1λ2 λ4 e ( 1 2 4 )
− λ +λ +λ t
− λ +λ +λ t
+ a1a3a4 λ1λ3λ4 e ( 1 3 4 ) + a2 a3a4 λ2 λ3λ4 e ( 2 3 4 )
14
}
}
(12)
Res. J. Math. Stat., 3(1): 12-19, 2011
Equating the like terms as have been considered in (6)-(10), yield:
(
)(
)(
)
e − λ1t D − λ1 + λ2 D − λ1 + λ 3 D − λ1 + λ 4 A1
{
− 2λ + λ t
− λ +λ +λ t
= − 3a12 a 2 λ12 λ2 e ( 1 2 ) + 6a1a 3 a 4 λ1 λ3 λ 4 e ( 1 3 4 )
(
)(
)(
)
e − λ2 t D − λ2 + λ1 D − λ2 + λ 3 D − λ2 + λ 4 A2
{
− 2λ + λ t
− λ +λ +λ t
= − 3a 22 a1 λ22 λ1e ( 2 1 ) + 6a 2 a 3 a 4 λ2 λ3 λ 4 e ( 2 3 4 )
(
)(
)(
)
e − λ3t D − λ3 + λ1 D − λ3 + λ2 D − λ3 + λ 4 A3
{
− 2λ + λ t
− λ +λ +λ t
= − 3a 32 a 4 λ23 λ4 e ( 3 4 ) + 6a1a 2 a 3 λ1 λ 2 λ 3 e ( 1 2 3 )
(
)(
)(
)
e − λ4 t D − λ4 + λ1 D − λ4 + λ 2 D − λ 4 + λ 3 A4
{
− 2λ + λ t
− λ +λ +λ t
= − 3a 42 a 3 λ24 λ 3 e ( 4 3 ) + 6a1a 2 a 4 λ1 λ2 λ 4 e ( 1 2 4 )
and
}
(13)
}
(14)
}
(15)
}
(16)
( D + λ1 )( D + λ2 )( D + λ3 )( D + λ4 )u1 = − {a13 λ13e − 3λ t + a 23 λ32 e − 3λ t + a 33 λ33e − 3λ t
1
2
3
− 2λ + λ t
− 2λ + λ t
+ a 43 λ34 e − 3λ4 t + 3a13 a 3 λ33 λ3 e ( 1 3 ) + 3a12 a 4 λ12 λ 4 e ( 1 4 )
− 2λ + λ t
− 2λ + λ t
− 2λ + λ t
+ 3a 22 a 3 λ32 λ3 e ( 2 3 ) + 3a 22 a 4 λ22 λ4 e ( 2 4 ) + 3a 32 a1 λ23 λ1e ( 3 1 )
− 2λ + λ t
− 2λ + λ t
− 2λ + λ t
+ 3a 32 a 2 λ23 λ2 e ( 3 2 ) + 3a 42 a1 λ24 λ1e ( 4 1 ) + 3a 42 a 2 λ24 λ 2 e ( 4 2 )
(17)
}
Solving Eq. (13)-(16) and substituting, 81 = k1 – T1, 82 = k1 – T1, 83 = k2 – iT2 and 84 = k2 – iT2 we obtain:
(
)(
2
3a12 a 2 ( k 1 − ω 1 ) ( k 1 + ω 1 )e − 2 k t
)(
)
A1 =
+
2( k 1 − ω 1 )(3k 1 + k 2 − ω 1 + iω 2 )(3k 1 − k 2 − ω 1 − iω 2 ) ( k 2 − ω 1 )( k 1 + k 2 − ω 1 + iω 2 )( k 1 + k 2 − ω 1 − iω 2 )
3a1a 3 a 4 k 1 − ω 1 k 2 − iω 2 k 2 + iω 2 e − 2 k2 t
A2 =
(
1
)(
2
3a1a 22 ( k 1 − ω 1 )( k 1 + ω 1 ) e − 2 k t
)(
)
+
( k 2 + ω1 )( k1 + k 2 + ω1 + iω 2 )(k1 + k 2 + ω1 − iω 2 ) 2(k1 + ω1 )(3k1 − k 2 + ω1 + iω 2 )(3k1 − k 2 + ω1 − iω 2 )
3a 2 a 3 a 4 k 1 + ω 1 k 2 − iω 2 k 2 + iω 2 e − 2 k2 t
(
)(
(
)(
1
2
3a 32 a 4 ( k 2 − iω 2 ) ( k 2 + iω 2 )e − 2 k t
)(
)
A3 =
+
( k1 − iω 2 )( k1 + k 2 + ω1 − iω 2 )( k1 + k 2 − ω1 − iω 2 ) 2(k 2 − iω 2 )(3k 2 − k1 + ω1 + iω 2 )(3k 2 − k1 − ω1 − iω 2 )
3a1a 2 a 3 k 1 − ω 1 k 1 + ω 1 k 2 − iω 2 e − 2 k1t
2
and
A4 =
2
3a 3 a 42 ( k 2 − iω 2 )( k 2 + iω 2 ) e − 2 k t
)(
)
+
( k1 + iω 2 )(k1 + k 2 + ω1 + iω 2 )( k1 + k 2 − ω1 − iω 2 ) 2( k 2 + iω 2 )(3k 2 − k1 + ω1 + iω 2 )(3k 2 − k1 − ω1 + iω 2 )
3a1a 2 a 4 k 1 − ω 1 k 1 + ω 1 k 2 + iω 2 e − 2 k1t
2
(18)
Substituting the values of (18) into Eq. (4) and neglecting the second and higher powers of , (since , is very small), we
obtain:
15
Res. J. Math. Stat., 3(1): 12-19, 2011
2
⎧⎪ 3a a a ( k − ω )( k − iω )( k + iω ) e − 2 k2 t
⎫⎪
3a12 a2 ( k1 − ω1 ) ( k1 + ω1 ) e − 2 k1t
da1
1 2 3 1
1
2
2
2
2
= ε⎨
+
⎬
dt
⎪⎩ ( k1 − ω1 )( k1 + k 2 − ω1 + iω 2 )( k1 + k2 − ω1 − iω 2 ) 2( k1 − ω1 )( 3k1 + k 2 − ω1 + iω 2 )( 3k1 − k 2 − ω1 − iω 2 ) ⎪⎭
(
)(
)(
)(
)
(
)(
)(
)(
)
(
)(
)(
)(
)
(
)(
(
)(
(
)(
)
⎧⎪ 3a a a k + ω k − iω k + iω e − 2 k2 t
3a1a 22 k 1 − ω 1 k 1 + ω 1 e − 2 k1t
da 2
2 3 4 1
1
2
2
2
2
= ε⎨
+
dt
2 k 1 + ω 1 3k 1 − k 2 + ω 1 + iω 2 3k 1 − k 2 + ω 1 − iω 2
⎩⎪ k 2 + ω 1 k 1 + k 2 + ω 1 + iω 2 k 1 + k 2 + ω 1 − iω 2
(
)(
) (
)(
)(
)
)
⎫⎪
⎬
⎭⎪
2
⎧
3a1a 2 a 3 k 1 − ω 1 k 1 + ω 1 k 2 − iω 2 e − 2 k1t
3a 32 a 4 k 2 − iω 2 k 2 + iω 2 e − 2 k2 t
da 3
⎪
= ε⎨
+
dt
2 k 2 + iω 2 3k 2 − k 1 + ω 1 + iω 2 3k 2 − k 1 − ω 1 − iω 2
⎪⎩ k 1 − iω 2 k 1 + k 2 + ω 1 − iω 2 k 1 + k 2 − ω 1 − iω 2
(
)(
) (
)(
)(
)
⎫
⎪
⎬
⎪⎭
and
)
2
⎧
3a1a 2 a 4 k 1 − ω 1 k 1 + ω 1 k 2 + iω 2 e − 2 k1t
3a 3 a 42 k 2 − iω 2 k 2 + iω 2 e − 2 k2 t
da 4
⎪
+
= ε⎨
dt
2 k 2 + iω 2 3k 2 − k 1 + ω 1 + iω 2 3k 2 − k 1 − ω 1 + iω 2
⎪⎩ k 1 + iω 2 k 1 + k 2 + ω 1 + iω 2 k 1 + k 2 − ω 1 + iω 2
(
)(
ϕ1
) (
)(
)(
)
⎫
⎪
⎬ (19)
⎪⎭
2 , a1 = ae −ϕ1 2 , a3 = beiϕ 2 2 , a4 = beiϕ 2 2 into Eq. (19) and simplifying, we obtain:
Now replacing a1 = ae
(
)
(
da
db
= ε l1a 3e − 2 k1t + l2 ab 2 e − 2 k2 t ,
= ε r1a 2be − 2 k1t + r2b 3e − 2 k2 t
dt
dt
)
(
dϕ1
dϕ 2
= ε m1a 2 e − 2 k1t + m2b 2 e − 2 k2 t and
= ε s1a 2 e − 2 k1t + s2b 2 e − 2 k2 t
dt
dt
(
)
)
(20)
where,
⎡
⎤
k 12 − ω 12 9 k 12 + k 22 + ω 12 + ω 22 − 6k 1 k 2
3⎢
⎥
l1 = − ⎢
⎥,
2
2
8⎢
3k 1 − k 2 − ω 1 + ω 22 3k 1 − k 2 + ω 1 + ω 22 ⎥
⎣
⎦
⎡
2
2
2
2
2
2
2
2
3
⎢ 2 k 1 + k 2 + ω 1 + ω 2 + 2 k 1 k 2 k 1 k 2 − ω 1 + 4ω 1 k 1 − k 2
l 2 = − k 22 + ω 22 ⎢
2
2
8
⎢ k 22 − ω 12 k 1 + k 2 − ω 1 + ω 22 k 1 + k 2 + ω 1 + ω 22
⎣
{(
(
)(
}{(
)
(
)
(
(
)
)
)(
){(
}
)
}{(
)
(
)
) ⎤⎥
} ⎥⎦
⎥
r1 = −
⎡ 2
⎤
2
2
2
2
2
2
2
3 ⎢ k 1 − ω 1 k 1 + k 2 − ω 1 − ω 2 + 2 k 1 k 2 k 1 k 2 + ω 2 − 2ω 2 k 1 + k 2 k 2 − k 1 ⎥
⎥,
2
2
4 ⎢⎢
⎥
k 12 + ω 22 k 1 + k 2 + ω 1 + ω 22 k 1 + k 2 + ω 1 + ω 22
⎣
⎦
r2 = −
⎡
k 22 + ω 22 k 12 + 9 k 22 − ω 12 − ω 22 − 6k 1 k 2
3⎢
2
2
8 ⎢⎢
k − 3k 2 − ω 1 + ω 22 k 1 − 3k 2 + ω 1 + ω 22
⎣ 1
(
)(
(
{(
)(
){(
(
)(
)
)
⎡
k 12 − ω 12 ω 1 3k 1 − k 2
3⎢
m1 = − ⎢
2
4⎢
3k 1 − k 2 − ω 1 + ω 22 3k 1 − k 2 + ω 1
⎣
(
{(
) (
}{(
)
}{(
)
}{(
)
)
)
}
(
)
}
)(
)
⎤
⎥
⎥
⎥
⎦
) 2 + ω 22 }
⎤
⎥
⎥,
⎥
⎦
⎡ 2
k + k 22 + ω 12 + ω 22 + 2 k 1 k 2 k 1 − k 2 ω 1 + 2 k 1 − k 2 ω 1 k 1 k 2 − ω 12
3 2
2 ⎢ 1
m2 = −
k2 + ω2 ⎢
2
2
4
⎢
k 22 − ω 12 k 1 + k 2 − ω 1 + ω 22 k 1 + k 2 + ω 1 + ω 22
⎣
(
)
(
(
)(
){(
)
)
16
}{(
(
) (
)
}
) ⎤⎥
⎥
⎥
⎦
Res. J. Math. Stat., 3(1): 12-19, 2011
⎡
2ω 2 k 1 + k 2 k 1 k 2 + ω 22 + ω 2 k 2 − k 1 k 12 + k 22 − ω 12 − ω 22 + 2 k 1 k 2
3 2
2 ⎢
s1 = −
k1 − ω1 ⎢
2
2
4
⎢
k 12 − ω 22 k 1 + k 2 − ω 1 + ω 22 k 1 + k 2 + ω 1 + ω 22
⎣
(
(
)
)(
)
){(
(
(
)
⎡
ω 2 3k 2 − k 1 k 22 + ω 22
3⎢
s2 = − ⎢
2
4⎢
k − 3k 2 − ω 1 + ω 22 k 1 − 3k 2 + ω 1
⎣ 1
(
)
{(
)(
}{(
)
ϕ
Solving Eq. (17) and by replacing a1 = ae 1 2 , a2 = ae
82 = k1 – T1, 83 = k2 – iT2 and 84 = k2 – iT2, we obtain:
[ (
[ (
)
)(
}{(
) 2 + ω 22 }
−ϕ1
}
)
) ⎤⎥
⎥,
⎥
⎦
⎤
⎥
⎥
⎥
⎦
2 , a3 = beiϕ 2 2 , a4 = be −iϕ 2 2 and 81 = k1 – T1,
) ]
(
1 3 − 3k1t
a e
cosh 3 ω 1t + ϕ g1 + sinh 3 ω 1t + ϕ g 2
16
1 3 − 3k 2 t
−
b e
cos 3 ω 2 t + ψ g 3 − sin 3 ω 2 t + ψ g 4
16
3 2
− 2 k − 2ω + k t + 2ϕ
−
a b k1 − ω1 e ( 1 1 2 )
.cos ω 2 t + ψ k 2 k 1 − ω 1 k 1 + k 2
16
u1 = −
)
) ]
(
)[ {(
)(
)2
− 4ω 1 ( k 1 − ω 1 )( k 1 + k 2 ) + ( k 1 − ω 1 )(3ω 12 − ω 22 ) − 2ω 22 ( k 1 + k 2 ) + 4ω 1ω 22 }
+ ω 22 {(3k 1 + 3k 2 ) − 2ω 1 (7 k 1 + 3k 2 ) − ( k 22 − ω 22 ) + 11ω 12 }] h1
(
)
(
)[{k 2 ( k 2 − ω1 ) + ω 22 }
2
× {( k 1 + k 2 − ω 1 ) − 3ω 22 } − 4 k 2 ω 22 ( k 1 + k 2 − ω 1 )+ 4ω 22 ( k 2 − ω 1 )( k 1 + k 2 − ω 1 )]
− sin 2(ω 2 t + ψ )[4ω 2 ( k 1 + k 2 − ω 1 ){k 2 ( k 2 − ω 1 ) + ω 22 } + k 2 ω 2
{(k1 + k 2 − ω1 ) 2 − 3ω 22 } − ω 2 (k 2 − ω1 ) {(k1 + k 2 − ω1 ) 2 − 3ω 22 }⎤⎦⎥ ⎤⎥⎦ h2
−
3
− k −ω + 2 k t +ϕ
ab 2 k 1 − ω 1 e ( 1 1 2 ) ⎡⎢ cos 2 ω 2 t + ψ
⎣
16
(
)
(
)
(
)[ {(
) }{(k1 + k 2 ) 2
+ 4ω 1 ( k 1 + k 2 ) + (3ω 12 − ω 22 ) − 2 k 2 ω 22 ( k 1 + k 2 ) − 4 k 2 ω 1ω 22 + 2ω 22 ( k 1 + ω 1 )
+ 4ω 1ω 22 ( k 1 + ω 1 ) − sin(ω 2 t + ψ )[2 k 2 ω 2 ( k 1 + ω 1 )( k 1 + k 2 ) + 4 k 2 ω 1ω 2 ( k 1 + ω 1 )
2
+ 2ω 23 ( k 1 + k 2 ) + 4ω 1ω 23 + k 2 ω 2 ( k 1 + k 2 ) + 4 k 2 ω 1ω 2 ( k 1 + k 2 ) + k 2 ω 2 (3ω 12 − ω 22 )
2
− ω 2 ( k 1 + ω 1 )( k 1 + k 2 ) − 4ω 1ω 2 ( k 1 + ω 1 )( k 1 + k 2 ) − ω 2 ( k 1 + ω 1 )(3ω 12 − ω 22 )] h3
−
−
3 2
− 2 k + 2ω + k t + 2ϕ
a b k1 + ω1 e ( 1 1 2 )
.cos ω 2 t + ψ k 2 k 1 + ω 1 + ω 22
16
(
{
3 2
− k +ω + 2 k t −ϕ
ab ( k1 + ω1 ) e ( 1 1 2 ) .cos 2(ω 2 t + ψ ) ⎡ k 2 ( k 2 + ω1 ) + ω 22
⎣⎢
16
]
}
}{( k + k
1
2
+ ω1 )
2
− 3ω 22 − 4 k 2ω 22 ( k1 + k2 + ω1 ) + 4ω 22 ( k 2 + ω1 )( k1 + k 2 + ω1 ) − sin 2(ω 2 t + ψ )
[
× 4 k2ω 2 ( k 2 + ω1 ) ( k1 +
( k1 + k2 + ω1 ) + k2ω2 ( k1 + k2 + ω1 ) 2
2
− 3k 2ω 23 − ω 2 ( k 2 + ω1 )( k1 + k2 + ω1 ) − 3ω 23 ( k 2 + ω1 ) ] h4
where,
{
(
g1 = k 13 3k 1 − k 2
(
) 2 + k13 (9ω12 + ω 22 ) − 3k1ω12 (3k1 − k 2 ) 2 − 3k1ω12 (9ω12 + ω 22 )
+ 12ω 14 3k 1 − k 2
{
)
k 2 + ω1 + 4ω 23
)} ⎡⎢⎣ ( k12 − 4ω12 ){(3k1 − 3ω1 − k 2 ) 2 + ω 22 }{(3k1 − 3ω1 − k 2 ) 2 + ω 22 }⎤⎥⎦
(
)
(
)
(
g 2 = 6k 13ω 1 3k 1 − k 2 − 18k 1ω 13 3k 1 − k 2 + 2ω 13 3k 1 + k 2
(
+ 2ω 13 9ω 12 + ω 22
)2
)} ⎡⎢⎣ ( k12 − 4ω12 ){(3k1 − 3ω1 − k 2 ) 2 + ω 22 }{(3k1 − 3ω1 − k 2 ) 2 + ω 22 }⎤⎥⎦
17
(21)
Res. J. Math. Stat., 3(1): 12-19, 2011
{
(
g 3 = k 23 3k 2 − k 1
) 2 + k 23 (9ω12 + ω12 ) + 3k 2ω 22 (3k 2 − k1 ) 2 − 3k 2ω 22 (9ω 22 + ω12 )
)} ⎡⎣⎢ ( k 22 − 4ω 22 ){(3k 2 − k1 + ω1 ) 2 + 9ω 22 }{(3k 2 − k1 − ω1 ) 2 + 9ω 22 }⎤⎦⎥
(
+ 12ω 24 3k 2 − k 1
{
(
)
(
)
(
g 4 = 6k 23ω 2 3k 2 − k 1 + 18k 2 ω 23 3k 2 − k 1 − 2ω 23 3k 2 − k 1
(
+ 2ω 23 9ω 22 + ω 12
)2
)} ⎡⎢⎣ ( k 22 − 4ω 22 ){(3k 2 − k1 + ω1 ) 2 + 9ω 22 }{(3k 2 − k1 − ω1 ) 2 + 9ω 22 }⎤⎥⎦
{( k + k − ω ) + ω }{( k + k − 3ω ) + ω }{( k − ω ) + ω }
h = {( k + k − ω ) + ω }{( k − ω ) + ω }{( k + k − ω ) + ω }
h = {( k + k − ω ) + ω }{( k + k − 3ω ) + ω }{( k + ω ) + ω }
h = {( k + k − ω ) + ω }{( k + k + ω ) + 9ω }{( k + ω ) + ω }
2
h1 =
1
2
1
2
1
2
1
3
1
2
1
4
1
2
1
2
2
2
2
2
2
1
2
2
2
2
2
1
2
2
2
1
2
2
1
2
2
2
2
2
1
2
1
2
2
1
1
1
2
1
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
1
2
2
2
1
The Eq. (20) has no exact solution. Since, da/dt, db/dt, dN1/dt, dN2/dt are proportional to the small parameter ,, therefore
they are slowly varying functions of time t. Therefore, we may assume that a and b are constants in the right hand side.
This assumption was first made by Murty et al. (1969). Thus integrating Eq. (20), we obtain:
)
(
) }
{ (
b = b + ε { r b (1 − e
) k + l a b (1 − e ) k } 2 ,
ϕ = ϕ (0) + ε { m a (1 − e
) k + m b (1 − e ) k } 2 ,
a = a 0 + ε l1a 03 1 − e − 2 k1t k 1 + l 2 a 0 b02 1 − e − 2 k2t k 2 2 ,
1
and
− 2 k2t
3
2 0
0
2
1 0
1
2
− 2 k1t
1
1
− 2 k1t
2
0 0
1
− 2 k2t
2
2 0
2
{
}
ϕ 2 = ϕ 2 (0) + ε s2 b02 (1 − e − 2 k2 t ) k 2 + s1a 02 (1 − e − 2 k1t ) k1 2
(22)
Therefore, the first order approximate solution of the Eq. (11) is:
(
)
(
)
x( t , ε ) = ae − k1t cosh ω 1 t + ϕ 1 + be − k 2 t cos ω 2 t + ϕ 2 + ε u1
(23)
where a, b, N1, N2 are given by (22) and u1 is given by (21).
0.6
a0 = 0.25
b0 = 0.25
Φ 1 (0) = π/6
Φ 2 (0) = π/6
0.5
0.8
k 1 = 1/3
k 2 = 0.25
ω1 = 0.15
0.3
k 1 = 1/3
k 2 = 0.25
ω1 = 0.15
0.6
0.4
2
0.2
a0 = 0.5
b0 = 0.5
Φ 1 (0) = π/6
Φ 2(0) = π/6
1.0
X
X
0.4
1.2
, = 0.1
2
0.2
0.1
0
0
0
2
4
6
8
t
10
12
14
16
Fig. 1: Solution of Eq. (11): (i) Perturbation solution is denoted
by solid line, ( — ) and (ii) Numerical solution in broken
line, (---)
0
2
4
6
t
8
10
12
14
16
Fig. 2: Solution of Eq. (11): (i) Perturbation solution denoted
by solid line, ( — ) and (ii) Numerical solution in broken
line, (---)
18
Res. J. Math. Stat., 3(1): 12-19, 2011
Alam, M.S. and M.A. Sattar, 1997. A unified KrylovBogoliubov-Mitropolskii method for solving third
order nonlinear systems. Indian J. Pure Appl. Math.,
28: 151-167.
Alam, M.S. and M.A. Sattar, 2001. Time dependent thirdorder oscillating systems with damping. J. Acta
Ciencia Indica, 27: 463-466.
Alam, M.S., 2001. Perturbation theory for nonlinear
systems with large damping. Indian J. Pure Appl.
Math., 32: 1453-1461.
Alam, M.S., 2002a. Perturbation theory for n-th order
nonlinear systems with large damping. Indian J. Pure
Appl. Math., 33: 1677-1684.
Alam, M.S., 2002b. A unified Krylov-BogoliubovMitropolskii method for solving n-th order nonlinear
systems. J. Frank. Inst., 339: 239-248.
Bogoliubov, N.N. and Y. Mitropolskii, 1961. Asymptotic
Methods in the Theory of Nonlinear Oscillations.
Gordan and Breach, New York.
Bojadziev, G.N. and C.K. Hung, 1984. Damped
oscillations modeled by a 3-dimensional time
dependent differential systems. Acta Mechanica, 53:
101-114.
Krylov, N.N. and N.N. Bogoliubov, 1947. Introduction to
Nonlinear Mechanics. Princeton University Press,
New Jersey.
Mendelson, K.S., 1970. Perturbation theory for damped
nonlinear oscillations. J. Math. Phys., 11: 3413-3415.
Murty, I.S.N. and B.L. Deekshatulu, 1969. Method of
variation of parameters for over-damped nonlinear
systems. J. Control, 9(3): 259-266.
Murty, I.S.N., B.L. Deekshatulu and G. Krishna, 1969.
On an asymptotic method of Krylov-Bogoliubov for
over-damped nonlinear systems. J. Frank. Inst., 288:
49-65.
Murty, I.S.N., 1971. A unified Krylov-Bogoliubov
method for solving second order nonlinear systems.
Int. J. Nonlinear Mech., 6: 45-53.
Popov, I.P., 1956. A generalization of the bogoliubov
asymptotic method in the theory of nonlinear
oscillations (in Russian). Dokl. Akad. USSR, 3:
308-310.
RESULTS
It is customary to compare the perturbation results
obtained by a certain perturbation method to the
numerical results (considered being exact) to test the
accuracy of the method. To this end, we have calculated
x(t, ,) by (23), in which a, b, N1 and N2 are calculated
from (22) and u1 is calculated by (21) for different sets of
initial conditions. The corresponding numerical solution
has also been computed by Runge-Kutta method with a
small time increment )t = 0.05 and the results are plotted
in Fig. 1 and 2. From the figures it is clear that the
perturbation solutions (23) together with (22) agree with
the numerical solutions.
CONCLUSION
The KBM method has been extended in this article
for solving fourth order damped oscillatory nonlinear
systems when two of the eigen-values of the
corresponding linear equation are real and negative
numbers and other two are complex conjugates. The
results obtained by this method match accurately with
those obtained by the numerical method. The solution can
also be used for over-damped systems replacing T by -iT.
This is the importance of this technique.
ACKNOWLEDGMENT
Authors thanks to the management of Maxwell
Scientific Organization for financing the manuscript for
publication.
REFERENCES
Akbar, M.A., A.C. Paul and M.A. Sattar, 2002. An
asymptotic method of Krylov-Bogoliubov for fourth
order over-damped nonlinear systems, Ganit. J.
Bangladesh Math. Soc., 22: 83-96.
Akbar, M.A., M.S. Alam and M.A. Sattar, 2003.
Asymptotic method for fourth order damped
nonlinear systems, Ganit. J. Bangladesh Math. Soc.,
23: 41-49.
19