American Journal of Computational Mathematics, 2013, 3, 185-194
http://dx.doi.org/10.4236/ajcm.2013.33027 Published Online September 2013 (http://www.scirp.org/journal/ajcm)
Stochastic Oscillators with Quadratic Nonlinearity Using
WHEP and HPM Methods
Amnah S. Al-Johani1,2
1Department of Applied Mathematics, College of Science, Northern Borders University, Arar, Saudi Arabia
2College of Home Economics, Northern Borders University, Arar, Saudi Arabia
Email: xxwhitelinnetxx@hotmail.com
Received May 2, 2013; revised June 8, 2013; accepted July 2, 2013
Copyright © 2013 Amnah S. Al-Johani. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, quadratic nonlinear oscillators under stochastic excitation are consid ered. The Wiener-Hermite expansion
with perturbation (WHEP) method and the homotopy perturbation method (HPM) are used and compared. Different
approximation orders are considered and statistical moments are computed in the two methods. The two methods show
efficiency in estimating the sto chastic response of the nonlinear differential equation s.
Keywords: Nonlinear Stochastic Differential Equations; W iener-Hermite Expansion; W HEP Tech ni q u e; Homotopy
Perturbation Method
1. Introduction
Quadrate oscillation arises through many applied models
in applied sciences and engineering when studying oscil-
latory systems [1]. These systems can be exposed to a lot
of uncertainties through the external forces, the damping
coefficient, the frequency and/or the initial or boundary
conditions. These input uncertainties cause the output
solution process to be also uncertain. For most of the
cases, getting the probability density function (p.d.f.) of
the solution process may be impossible. So, developing
approximate techniques through which approximate sta-
tistical moments can be obtained, is an important and
necessary work.
Since Meecham and his co-workers [2] developed a
theory of turbulence involving a truncated Wiener-Her-
mite expansion (WHE) of the velocity field, many au-
thors studied problems concerning turbulence [3-8]. A lot
of general applications in fluid mechanics were also stud-
ied in [9-11]. Scattering problems attracted the WHE
applications through many authors [12-16]. The nonlin-
ear oscillators were considered as an opened area for the
applications of WHE as can be found in [17-23]. There
are a lot of applications in boundary value problems [24 ,
25] and generally in different mathematical studies [26-
29]. The WHE properties and description of its usage are
given in [30].
In HPM technique [31-34], the response of nonlinear
differential equations can be obtained analytically as a
series solution. The basic idea of the homotopy method is
to deform continuously a simple problem (and easy to
solve) into the difficult problem under study [35]. The
HPM method is a special case of homotopy analysis
method (HAM) propounded by Liao in 1992 [36]. The
HAM was systematically described in Liao’s book in
2003 [37] and was applied by many authors in [38-41].
The HAM method possesses auxiliary parameters and
functions which can control the convergence of the ob-
tained series solution .
The stochastic oscillator with cubic nonlinearity (Duffing
oscillator) was considered in [17,42]. The nonlinear term
is due to the restoring nonlinear force. In some applica-
tions, the restoring force is quadratic and it is required to
estimate the response in this case. The main goal of this
paper is to consider the quadratic nonlinear oscillator
under stochastic excitation. The WHEP and HPM meth-
ods are used and compared.
This paper is organized as follows. The problem for-
mulation is outlined in Section 2. The WHEP technique
is described and applied to the stochastic quadratic oscil-
lator in Section 3. The HPM is outlined in Section 4 and
applied also to the quadratic oscillator. A comparison
between the two methods is shown in Section 5.
2. Problem Formulation
In this section, the following quadratic nonlinear oscilla-
tory equation is cons idered:
C
opyright © 2013 SciRes. AJCM
A. S. AL-JOHANI
186
 
222
;2 ;,0,
x
twxwxwxFtt
 
 
 T (1)
under stochastic excitation
;
F
t
with deterministic
initial condition s

00
0,0
x
xx x

,
where
w: frequency of oscillation,
: damping coefficient,
: deterministic nonlinearity scale,
,,p


: a triple probability space with
as
the sample space,
is a
-algebra on events in
and P is a probability measure.
3. WHEP Technique
The application of the WHE aims at finding a truncated
series solution to the solution process of differential eq ua-
tions. The truncated series composes of two major parts;
the first is the Gaussian part which consists of the first
two terms, while the rest of the series constitute the non-
Gaussian part. In nonlinear cases, there exists always dif-
ficulties of solving the resultant set of deterministic inte-
gro-differential equations got from the applications of a
set of comprehensive averages on the stochastic integro-
differential equation obtained after the direct application
of WHE. Many authors introduced different methods to
face these obstacles. Among them, the WHEP technique
was introduced in [22] using the perturbation technique
to solve perturbed nonlinear problems.
The WHE method utilizes the Wiener-Hermite poly-
nomials which are the elements of a complete set of sta-
tistically orthog onal random functions [30]. The Wiener-
Hermite polynomial

12
,, ,
i
i
H
tt t satisfies the fol-
lowing recurrence relation:









12 2
11
1212 1
-1 2
1
,, ,,, ,
,,,, 2
i
ii
ii
ii
iiiim i
m
i
H
tttHtttH t
Htt ttti



(2)
where
 
 





 
 

 




 












01211
12121 2
32111
12312312321 3
431222
12341234123413 242314
1,, ,,
(, , ),()(),
(, , ,), ,,,,,
HHtntH ttHtHttt
HtttH ttHtHtttHttt
H
ttttHtttHtH ttttH ttttH tttt


 


(3)
in which n(t) is the white noise with the following statis-
tical properties
 
 
12 12
0, ,EntEnt ntt t
 (4)
where

.
is the Dirac delta function and E denotes
the ensemble average operator.
The Wiener-Hermite set is a statistically or thogonal set,
i.e.
 
0
ij
EHHi j.
 (5)
The average of almost all H functions vanishes, par-
ticularly,

0 for 1.
i
EH i
(6)
Due to the completeness of the Wiener-Hermite set,
any random function
;Gt
can be expanded as















0112 2
11112 121
33
123123123
;;d +;,
;,,,, ddd
GtGtGttHttGtttHt ttt
GttttHtttttt

 

  




2
,dd
(7)
where the first two terms are the Gaussian part of G(t; ω).
The rest of the terms in the expansion represent the
non-Gau ssi an part of G(t; ω). The average of G(t; ω) is



0
;
GEG tGt

 (8)
The cov ariance of
;Gt
is
 

















112 2
11112 1212
33 3 3
123123132231123
Cov;,;;;
;,d +2;,,,dd
2;,,,,,,,,,,,ddd
GG
GtGEGttG
GttGttG tttGtt tt
GttttG tttGtttG tttttt
 



 

  
 





(9)
Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI 187

;Gt
The variance of is
dd
(10)
The WHE method can be elementary used in solving
stochastic differential equations by expanding the solu-
tion process as well as the stochastic input processes via
the WHE. The resultant equation is more complex than
the original one due to being a stochastic integro-differ-
ential equation. Taking a set of ensemble averages to-
gether with using the statistical properties of the WH
po
 















22
212
11 1212
2
333
12312 312313212 3
33
123231123
Var;;;d +2;,dd
2;,,ddd2,,,,,,d
2,,,,,, ddd
G
GtEGttGtttGtt ttt
GtttttttGttttGtttt ttt
GttttGtttt ttt


 
 
    

  

  
 

 



 
.
 


lynomials, a set of deterministic integro-differential
equations are obtained in the deterministic kernels


;,0,1,2,
i
Gt i
. To obtain an approximate solu-
tions for these deterministic kernels, one can use pertur-
bation theory in the case of having a perturbed system
depending on, say,
. Expanding the kernels as a power
series of
, another set of simpler iterative equations in
the kernel series components are obtained. This is the
main algorithm of the WHEP technique. The technique
ied to several nonlinear stochastic
equations; see [20,22,23,25].
The WHEP technique can be applied on linear or
nonlinear perturbedsystems described by ordinary or
partial difrential equations. The solution can be modi-
fied in the sense that additional parts of the Wiener-
was successfully appl
fe
Hermite expansion can always be taken into considera-
tions and the required order of approximatio
ways be made. It can be even run through a package if it
is coded in some sort of symbolic languages.
Case-Study
ns can al-
The quadr atic n onlinear oscillatory problem, Equation (1 )
under stochastic excitation

;
F
t
with deterministic
initial conditions is solved using WHEP technique. The
solution process takes the following form:


;xt x









011
111
12312312 3
;d
dd
;,,,, ddd
txtt Htt
tt
xttttHtt tttt

 



22
12 1212
33
+;
,,xtttHtt

 


 (11)
Applying the WHEP technique, the following equa-
tions in the deterministic kernels are obtained:



2
00 1
Lx tw














2 2
2
22 2
11 1212
33 0
123132123
;d+2 ;,dd
;,,,,, ddd
xtwxt ttwxt tttt
xttttxttttttt Ft
 

 


  






2
3
212312 3
+
2;,, dddwxtt ttttt


 




3
(13)

3
3
1232311 23
;,,,,, dddx ttttx ttttttt
 
 (12)

























101 12
22
112122
122 3
22
2122 231232
23 1
223231231
,2,4; ;,d
+4,;,d +8;,;,,dd
4;,;,,dd,
Lx ttwxtxttwx ttxtttt
w xttxttttwxtttxtttttt
wxtt txtt ttttFtt



 
  

 


 

Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI
188































2021122
2
1212121323 3
13 1313
31323 32313 31233
33 3
256165 562
,,2,,,,4;,;, d
2; ;,,d2; ;,,d2; ;,,d
2;,,;,,dd4;,
Lxtttwxtxtttxttxttxtttxtttt
xtt xtttttxttxtttttxtt xttttt
xttttxtttttt xtt


  

 

 























3
56156 56
33 33
2565615615626556
33 33
156562 56165562 56
35
,;,,dd
3;,,;,,dd2;,,;,,dd
3;,, ;,,dd;,, ;,,dd
2;,
tt xtttttt
xttttxtttttt xttttxtttttt
xttttxtttttt xttttxtttttt
xttt

 
 
  
 
  



 
 









333
6256156265561 5612
,;,, dd;,,;,,dd,,txtt ttttxtt t txttt tttFttt
 
 

 
2
(14)
(15)
Let us take the simple case of evaluating the only
Gaussian part (first order approximation) of the solution
process of the previous case study, mainly
1
d





























333 0333
2
123132231123 132
23 2323
41 432441 4234424314
23
42 4
,,,,,,,,, 2,,,,,,,
4 ;,;,,d4;,;,,d4;,;,,d
4;, ;
Lxtt ttLxtt ttLxttttwxtxtt ttxtt ttxtt
xtttxttttt xtttxttttt xtttxttttt
xtttxtt

  

 

 



























23 23
134434124434214
232 323
413244 423144 431244
333
123 1322
,,d4 ;,;,,d4 ;,;,,d
4;, ;,,d4;,;,,d4;,;,,d
,,, ,,,,,
tttxtt txtt tttxtt txtt t tt
xtttxttttt xtttxttttt xtttxttttt
Gtttt GttttGtt

  

  

 




31
,tt
231
,,t t







1
01
11
;;
x
txtxttHt


t. (16)
In this case, the governing equations are
0








22
001
211
;dLx twx txtttG t

 
 

 


(17)






1
,
(18)
The ensemble average is
(19)
an
t (20
It has to be noticed that all the previous equations are
deterministic linear ones in the general form

1011
2
11
,2 ,Lx ttwxtxttG


tt



0
xtxt
d the variance is



2
1
211
;d
xtxtt



)

22
x
wx wxFt

wit
ns
 
0,0

tio h deterministic initial condi-
00
x
xx x

.

0102 0
d
t
x
txtxhtsFs


s (21)
In which we have



2
2
22
122
22
1esin1,
1
11
ee
21 21
1ee
,
1
wt
mt qt
mt qt
htw t
w
t
t
 


 



where
,
2w
22
1, 1mwwqww
 
 .
When adding the first term in the non-Gaussian part
(the second approx imation) of the solutio n pro
inly cess of the
previous case study, ma
d











011
111
22
1212 12
;;
;,,dd
xtxtxttHtt
It has the general solution
x
tt tHtttt


 

, (22)

Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI 189
the governing equations become
0
(23)
1
,
(24)
 







22
001
211
2
2
21212
(); d
2;,dd
Lxtwxtxttt
wxtttttGt


 





















101
2
11
121
22122
,2 ,
4;;,d
Lx ttwxtx tt
wxttxtt ttGtt





,

212
02
2
,,
2,,
Lxt tt
wxt xttt









1 1
1212
22 2
132331
,
4;,;,d,,
x ttxtt
2
x
tt txttttGtt

t
(25)
The ensemble average is still got by Equation (19)
2
while the variance is got as





22
12
211 121
;d2;,dd
xtxtttxtt ttt


 

 

(26)
The WHEP technique uses the following expansion for
its deterministic kernels as corrections made under each
approximation order.



23
012 3,0,1,2,3,.
iiiii
xt xxxxi
 
 .(27)
Example:
Let us take

;e;, 0.
t
Ftq t
 
 3 (28)
in the previous case-study and then solving using the
WHEP technique. The following results are obtained, see
Figures 1-3.
4. The Homotopy Perturbation Method
(HPM)
In this technique, a parameter
0,1p is embedded in
a homotopy function
,: 0,1
vrp 
which sat-
isfies
(a) (b)
(c)
(d)
(e)
(f)
Figure 1. (a) The first order aximation of the mean at ε correction for different correcti; (b) The first order ap-
proximation of the mean at ε2 correction for different correction levels; (c) The first order approximation of the mean at ε3
correction; (d) The first order approximation of the mean at ε, ε2, ε3 correction; (e) The first order approximation of the
mean at ε, ε2, ε3 correction; (f) The first order approximation of the mean at ε, ε2, ε3 correction.
ppro on levels
Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI
190
(a)
(b)
(c)
(d)
Figure 2. (a) The first order approximation of the variance at ε correction for different correction levels; (b) The first order
approximation of the variance at ε2 Correction for different correction levels; (c) The first order approximation of the vari-
ance at ε3 correction for different correction levels; (d) The first order approximation of the variance at. ε, ε2, ε3 correction.
(a)
(b)
Figure 3. (a) The first order approximation of the variance at ε, ε2, ε3 correction; (b) The first order approximation of the
variance at ε, ε2, ε3 correction.
 


0
,1HvppLvLupAvf r

 


 0
(29)
where is an initial approximation to the solution of
the equ
0
u
ation
 
0,Aufrr
 (30)
with boundary conditions
,0,
u

Bu r
n


 (31)
in which A is a nonlinear differential operator which can
be decompose into a linear operator L and a nonlinear
operator N, B is a boundary operator, f(r) is a known
analytic function and is the boundary of
. The
homotopy introducesontinuously deformution
for the case of p = 0, , to the case of p
= 1, inalation
(30)motopyethod
which is to deform continuously a simple problem (and
easy to solve) into the difficult problem under study [35].
The basic assumption of the HPM method is that the
solution of the original Equation (29) can be expanded as
a power series in p as:
a c
Lv
, wh
sic id
ed sol
Equ
m


00Lu
ich is the orig
ea the ho

0Avf r
. This is the ba of
2
pv3
01 2 3
vv pvpv
  (32)
Now, setting p = 1, the approximate solution of Equa-
tion (23) is obt ai ned a s:
0123
1
lim
p
uvvvvv
 (33)
The rate of convergence of the method depends greatly
on the initial approximation
The idea of the imbeddedrameter can be utilized to
solve nonlinear problems by imbedding th is parameter to
the problem and then forcing it to be unity in the ob-
tained approximate solution if converge can be assured.
A simple technique enables the extension of th applica-
bility of the perturbation ods from small valued ap-
0
u.
pa
e
meth
Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI 191
plications to ge n er a l ones.
Example
Considering the same previous example of Sub-Sect
3.1.1, one can get the following results w.r.t. homo
ati
ion
topy
perturbon:
 
22
A
xLx wx
 ,

2
2Lxxwxwx
 
 ,

2
Nx x
,

;fr Ft
.
The homotopy function takes the following form:
 


0
,1 0HvppLvLupAvfr

 



or equivalently,



22
0;0LvLup Lt


 

. (34)
Lett 23
01 2 3
pvpv , substituting in
Equation (34) and equating the equal powers of p in both
sides of the equation, one can get the following results:
1)
 
Lv Ly, in which one may consider the fol-
0
uwv F

ing
lo
2)
vv pv
00
wing simple solution:
 
00
,0,0.vyyxyx

0000
22
1001
;,0LvFtLvwvv


 
2
2,00,00Lvwvv vv
 
. 1
0,00v
.
3) 20
122
4)

22
310233
2,00,00Lvw vvvvv
 
.

4103124 4
2,00,Lvvvvvvv
00
 
. 5)
The approximate solution is
0123
;limxtv vvvv
1
p

which can be considered to any approximation order. On
can notice that the algorithm of the solution is straigh
t a lot of flexibilities can be made. For
any choices in guessing the initial
approximation together with its initial condition
zero initial conditions, we can choose
e
t
forward and tha
example, we have ms. For
0
v0 which leads
to:
5 012345
;xtxv vv vvv


;dht sF s
 

22
1
00
213
0
;d
2;;d
tt
t
swht sv ss
whtsvs vss


 

(35)
Figures 4-7 are obtained for 0.5
: [42].
5. Comparisons between WHEP and HPM
Methods
Figure [8] shows comparisons between the WHEP and
HPM methods for different values of the nonlinearity
strength,
. As the nonlinearity strength increases, the
deviation between the two methods is also increasing.
(a)
(b)
e mean for different correction levels; (b) The first and second
levels.
Figure 4. (a) The first and second order approximation of th
order approximation of the variance at for different correction
(a)
(b)
Figure 5. (a) The third order approximation of the mean for different correction levels; (b) The third order approximation of
the variance for different correction levels.
Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI
192
(a) (b)
Figure 6. (a) A comparison between first, second order and th
first, second order and the, third or de r oariance at ε = 0e n
f the v.1.
third order of the mean at ε = 0.1; (b) Comparison betwee
(a)
(b)
(c)
(d)
Figure 7. (a) A comparison between first, second order and the third order of the mean at ε = 0.3; (b) A comparison between
first, second order and the, third order of the variance at ε = 0.3; (c) A comparison between first, second order and the o third
order of the mean at ε = 0.7; (d) A comparison between first, second order and the third order of the variance at ε = 0.7.
(a)
(b)
(c)
(d)
Copyright © 2013 SciRes. AJCM
A. S. AL-JOHANI
Copyright © 2013 SciRes. AJCM
193
(f)
(e)
Figure 8. (a) A comparison between homotopy perturbation and Wiener-Hermite of the mean at ε = 0.1; (b) A comparison
between homotopy perturbation and Wiener-Hermite of the variance at ε = 0.1; (c) A comparison between homotopy per-
turbation and Wiener-Hermite of the mean at ε = 0.3; (d) A comparison between homotopy perturbation and Wie-
ner-Hermite of the variance at ε = 0.3; (e) A comparison between homotopy pe rturbation and Wiener-Hermite of the mean at
ε = 0.7; (f) A comparison betw ee n homotopy perturbation and Wiener-Hermite of the variance at ε = 0.7.
This is due to the convergence condition of the WHEP
technique which depends on
. For small values of
,
e the WHEP technique converbut after a certain val
of ges u
it will diverge. The M is more accurate for HP
higher values of
. The HPM has advantages when used
in solving differential equations with large nonlinearities.
6. Conclusion
The quadratic nonlinear oscillator with stochastic excita-
tion is considered. The solution was obtained using the
WHEP technique with different orders and different num-
ber of corrections. The HPM is used also with different
approximations. The WHEP technique is more efficient
but it converges only for certain limit of the nonlinearity
strength. The HPM is more difficult in the stochastic
differential equations but it is more preferable for high
values of the nonlinearity stth. The two methods are
shown to be efficient in estimating the stochastic re-
sponse of the quadratic nonlinear oscillators.
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