Research ArticleApplication of Hybrid Functions for SolvingDuffing-Harmonic Oscillator
Mohammad Heydari1 Ghasem Brid Loghmani1
Seyed Mohammad Hosseini2 and Seyed Mehdi Karbassi3
1 Department of Mathematics Yazd University Yazd Iran2Department of Mathematics Islamic Azad University Shahrekord Branch Shahrekord Iran3Department of Mathematics Islamic Azad University Yazd Branch Yazd Iran
Correspondence should be addressed to Mohammad Heydari mheydari85gmailcom
Received 9 April 2014 Revised 26 July 2014 Accepted 28 July 2014 Published 14 August 2014
Academic Editor Athanassios G Bratsos
Copyright copy 2014 Mohammad Heydari et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A numerical method for finding the solution of Duffing-harmonic oscillator is proposedThe approach is based on hybrid functionsapproximationThe properties of hybrid functions that consist of block-pulse and Chebyshev cardinal functions are discussedTheassociated operational matrices of integration and product are then utilized to reduce the solution of a strongly nonlinear oscillatorto the solution of a system of algebraic equationsThemethod is easy to implement and computationally very attractiveThe resultsare compared with the exact solution and results from several recently published methods and the comparisons showed properaccuracy of this method
1 Introduction
Most phenomena in our world are essentially nonlinear andare described by nonlinear ordinary differential equationsNonlinear oscillation in mechanics physics and appliedmathematics has been a topic of intensive research for manyyears Difficulty of solving the nonlinear problems or gettingan analytic solution leads one to use numerical methodsSeveral methods have been used to find approximate solu-tions to these nonlinear problems Some of these well-knownmethods are harmonic balance method [1] multiple scalesmethod [2] Krylov-Bogoliubov-Mitropolsky method [3 4]modified Lindstedt-Poincare method [5] linearized pertur-bation method [6] energy balance method [7] iterationperturbation method [8] bookkeeping parameter pertur-bation method [9] amplitude frequency formulation [10]maximum approach [11] Mickens iteration procedure [12]rational harmonic balance method [13] Adomian decom-position method [14] variational iteration method [15]modified variational iteration method [16 17] homotopy
Recently hybrid functions have been applied extensivelyfor solving differential equations or systems and proved to bea useful mathematical tool The pioneering work in the solu-tion of linear systemswith inequality constraints via hybrid ofblock-pulse functions and Legendre polynomials was led in[21] that first derived an operational matrix for the integralsof the hybrid function vector Razzaghi and Marzban in [22]the variational problems are solved using hybrid of block-pulse and Chebyshev functions Razzaghi and Marzban [23]applied the hybrid of block-pulse andChebyshev functions tofind approximate solution of systems with delays in state andcontrol Solution of time-varying delay systems is approxi-mated using hybrid of block-pulse functions and Legendrepolynomials in [24] Maleknejad and Tavassoli Kajani in [25]introduced aGalerkinmethod based on hybrid Legendre andblock-pulse functions on interval [0 1) to solve the linearintegrodifferential equation system Razzaghi and Marzban
Hindawi Publishing CorporationJournal of Difference EquationsVolume 2014 Article ID 210754 9 pageshttpdxdoiorg1011552014210754
2 Journal of Difference Equations
in [26] a direct method for solving multidelay systemsusing hybrid of block-pulse functions and Taylor series ispresented Marzban et al [27] implemented hybrid of block-pulse functions and Lagrange-interpolating polynomials tofind approximate solution of Volterrarsquos population modelThe Lane-Emden type equations are solved in [28] usinghybrid functions of block-pulse and Lagrange-interpolatingpolynomials The hybrid of block-pulse functions and Taylorseries is employed in [29] to solve the linear quadratic optimalcontrol with delay systems Application of hybrid of block-pulse functions and Lagrange polynomials for solving thenonlinear mixed Volterra-Fredholm-Hammerstein integralequations is investigated in [30]
In this study we consider the following nonlinearDuffing-harmonic oscillation [31ndash35]
11990610158401015840+
1199063
1 + 1199062= 0 119906 (0) = 119860 119906
1015840
(0) = 0 (1)
where is an example of conservative nonlinear oscillatorysystems having a rational form for the restoring force Notethat for small values of 119906 (1) is that of a Duffing-typenonlinear oscillator that is
11990610158401015840+ 1199063≃ 0 (2)
while for large values of 119906 the equation approximates that ofa linear harmonic oscillator that is
11990610158401015840+ 119906 ≃ 0 (3)
Hence (1) is called the Duffing-harmonic oscillator [31] Thesystemwill oscillate between symmetric bounds [minus119860 119860] andthe frequency and corresponding periodic solution of thenonlinear oscillator are dependent on the amplitude 119860 [33]
In this paper we introduce an alternative numericalmethod to solve Duffing-harmonic oscillator The methodconsists of reducing this equation to a set of algebraic equa-tions by first expanding the candidate function as a hybridfunction with unknown coefficients These hybrid functionswhich consist of block-pulse functions plus Chebyshev cardi-nal functions are first introduced The operational matricesof integration and product are givenThese matrices are thenused to evaluate the coefficients of the hybrid function for thesolution of strongly nonlinear oscillators
The outline of this paper is as follows In Section 2 thebasic properties of hybrid block-pulse functions and Cheby-shev cardinal functions required for subsequent developmentare described In Section 3 we apply the proposed numericalmethod to the Duffing-harmonic oscillation Results andcomparisons with existing methods in the literature arepresented in Section 4 and finally conclusions are drawn inSection 5
2 Properties of Hybrid Functions
Marzban et al in [27 29 30] used the hybrid of block-pulsefunctions and Lagrange-interpolating polynomials based onzeros of the Legendre polynomials But no explicit formulasare known for the zeros of the Legendre polynomials In
this study we used Chebyshev cardinal functions which arespecial cases of Lagrange-interpolating polynomials basedon zeros of the Chebyshev polynomials of the first kindto overcome this problem In this paper we present theproperties of hybrid functions which consist of block-pulsefunctions plus Chebyshev cardinal functions similar to [2729 30] The hybrid functions are first introduced and theoperational matrices of integration and product are thenderived
21 Hybrid Functions of Block-Pulse and Chebyshev CardinalFunctions Hybrid functions 120579
119899119898(119905) 119899 = 1 2 119873 119898 =
0 1 119872 minus 1 are defined on the interval [0 119905119891) as
120579119899119898
(119905)
=
119862119898(2119873
119905119891
119905 minus 2119899 + 1) 119905 isin [(119899 minus 1
119873) 119905119891119899
119873119905119891)
0 otherwise(4)
where 119899 and 119898 are the order of block-pulse functions andChebyshev cardinal functions respectively Here 119862
119898(119905) are
defined as [36ndash38]
119862119898(119905) =
119879119872
(119905)
119879119872119905
(119905119898) (119905 minus 119905
119898) 119898 = 0 1 119872 minus 1 (5)
where 119879119872(119905) is the first kind Chebyshev polynomial of order
119872 in [minus1 1] defined by
119879119872
(119905) = cos (119872arccos (119905)) (6)
subscript 119905 denotes 119905-differentiation and 119905119898 119898 = 0 1
119872minus1 are the zeros of 119879119872(119905) defined by cos((2119898+1)120587)2119872)
119898 = 0 1 119872 minus 1 with the Kronecker property
119862119898(119905119894) = 120575119898119894
= 1 if 119894 = 119898
0 if 119894 = 119898(7)
where 120575119898119894
is the Kronecker delta function
22 Function Approximation A function 119891(119905) defined overthe interval [0 119905
119891) may be expanded as
119891 (119905) =
infin
sum119899=1
infin
sum119898=0
119888119899119898
120579119899119898
(119905) (8)
If the infinite series in (8) is truncated then (8) can beexpressed as
119891 (119905) ≃
119873
sum119899=1
119872minus1
sum119898=0
119888119899119898
120579119899119898
(119905) = 119862119879Θ (119905) (9)
where
119862 = [11988810 119888
1119872minus1 11988820 119888
2119872minus1 119888
1198730 119888
119873119872minus1]119879
(10)
Θ (119905) = [12057910
(119905) 1205791119872minus1
(119905) 12057920
(119905)
1205792119872minus1
(119905) 1205791198730
(119905) 120579119873119872minus1
(119905)]119879
(11)
Journal of Difference Equations 3
In (10) and (11) 119888119899119898 119899 = 1 2 119873 119898 = 0 1 119872 minus 1
are the expansion coefficients of the function 119891(119905) in the119899th subinterval [((119899 minus 1)119873)119905
119891 (119899119873)119905
119891) and 120579
119899119898(119905) 119899 =
1 2 119873 119898 = 0 1 119872 minus 1 are defined as in (4) Withthe aid of (7) the coefficients 119888
119899119898can be obtained as
119888119899119898
= 119891(119905119891
2119873(119905119898
+ 2119899 minus 1)) (12)
23TheOperationalMatrix of Integration In this section theoperational matrix of integration is derived The integrationof the vector Θ(119905) defined in (11) can be approximated as
int119905
0
Θ (119904) 119889119904 ≃ 119875Θ (119905) (13)
where119875 is the119872119873times119872119873 operationalmatrix of integration forChebyshev cardinal functions The matrix 119875 can be obtainedby the following process Let
Case 3 If 119899 gt 119897 then (119897119873)119905119891lt (1199051198912119873)(119905
119898+ 2119899 minus 1) and we
obtain119888119899119898
= int(119897119873)119905119891
((119897minus1)119873)119905119891
119862119896(2119873
119905119891
119904 minus 2119897 + 1)119889119904 =119905119891
2119873int1
minus1
119862119896(V) 119889V
(18)
Comparing (13) (14) and (15) we obtain
119875 = (
119864 119867 119867 sdot sdot sdot 119867
0 119864 119867 sdot sdot sdot 119867
0 0 119864 sdot sdot sdot 119867
0 0 0 sdot sdot sdot 119864
) (19)
where 119864 and 119867 are 119872 times 119872 matrices that can be obtained asfollows
Let
119864 = (119890119894119895) 119867 = (ℎ
119894119895) (20)
then for 119894 119895 = 0 1 119872 minus 1 we have
119890119894119895=
119905119891
2119873int119905119895
minus1
119862119894(V) 119889V
ℎ119894119895=
119905119891
2119873int1
minus1
119862119894(V) 119889V
(21)
where 119905119895 119895 = 0 1 119872 minus 1 are the zeros of the first
kind Chebyshev polynomial of order 119872 It is noted that 119864 isthe operational matrix of integration for Chebyshev cardinalfunctions over interval [((119899 minus 1)119873)119905
119891 (119899119873)119905
119891)
Remark 1 To calculate the entries 119890119894119895and ℎ
119894119895 119894 119895 = 0 1
119872 minus 1 we have
119862119898(V) =
119879119872
(V)119879119872V (119905119898) (V minus 119905
119898)=
120573
119879119872V (119905119898)
times
119872minus1
prod119896=0119896 =119898
(V minus 119905119896)
(22)
where 120573 = 2119872minus1 is the coefficient of 119905
119872 in the Chebyshevpolynomial function 119879
119872(V) Using (22) we get
119890119894119895=
119905119891120573
2119873119879119872119905
(119905119894)int119905119895
minus1
119872minus1
prod119896=0119896 =119894
(V minus 119905119896) 119889V
ℎ119894119895=
119905119891120573
2119873119879119872119905
(119905119894)int1
minus1
119872minus1
prod119896=0119896 =119894
(V minus 119905119896) 119889V
(23)
for 119894 119895 = 0 1 119872 minus 1
24 The Operational Matrix of Product The following prop-erty of the product of two hybrid function vectors will also beused Let
Θ (119905)Θ119879
(119905) 119865 ≃ 119865Θ (119905) (24)
where
119865 = [11989110 119891
1119872minus1 11989120 119891
2119872minus1 119891
1198730 119891
119873119872minus1]119879
(25)
4 Journal of Difference Equations
and 119865 is an 119872119873 times 119872119873 product operational matrix To find119865 we apply the following procedure First by using (11) and(25) we obtain
Θ (119905)Θ119879
(119905) 119865 =
((((((((((((((((((((((((((
(
12057910
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
12057911
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198730
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198731
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
120579119873119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
))))))))))))))))))))))))))
)
(26)
Using (9) any function 120579119894119895(119905)120579119897119896(119905) 119894 119897 = 1 2 119873 119895 119896 =
therefore we find the 119872119873 times 119872119873 matrix 119865 as
119865 = diag [11989110 119891
1119872minus1 11989120
1198912119872minus1
1198911198730
119891119873119872minus1
] (30)
Lemma 2 The functions 119862119895(119905) 119895 = 0 1 119872 minus 1 are
orthogonal with respect to 119908(119905) = 1radic1 minus 1199052 on [minus1 1] andsatisfy the orthogonality condition
⟨119862119894(119905) 119862
119895(119905)⟩119908
= int1
minus1
119862119894(119905) 119862119895(119905)
radic1 minus 1199052119889119905 =
120587
119872 119894119891 119895 = 119894
0 119894119891 119895 = 119894
(31)
The proof of this lemma is presented in [39]
Remark 3 Since 120579119899119898
(119905) consists of block-pulse functions andChebyshev cardinal functions which are both complete andorthogonal the set of hybrid of block-pulse functions andChebyshev cardinal functions is a complete orthogonal set inthe Hilbert space 119871
2[0 119905119891)
Lemma 4 Let MN vectors 119862 and 119862119901be hybrid functions
coefficients of 119906(119905) and 119906119901(119905) respectively If
119862 = [11988810 119888
1119872minus1 11988820 119888
2119872minus1 119888
1198730 119888
119873119872minus1]119879
(32)
then
119862119901≃ [119888119901
10 119888
119901
1119872minus1 119888119901
20 119888
119901
2119872minus1 119888
119901
1198730 119888
119901
119873119872minus1]119879
(33)
where 119901 ge 1 is a positive integer
Proof When 119901 = 1 (33) follows at once from 119906119901(119905) = 119906(119905)
Suppose that (33) holds for119901 we will deduce it for119901+1 Since119906119901+1
(119905) = 119906(119905)119906119901(119905) from (24) and (30) we have
where 119862119901can be calculated in a similar way to matrix 119865 in
(24) Now using (33) we obtain
119862119901+1
= 119862119879119862119901= [119888119901+1
10 119888
119901+1
1119872minus1 119888119901+1
20
119888119901+1
2119872minus1 119888
119901+1
1198730 119888
119901+1
119873119872minus1]119879
(35)
Therefore (33) holds for 119901 + 1 and the lemma isestablished
3 Hybrid Functions Method to SolveDuffing-Harmonic Oscillator
In this section by using the results obtained in the previoussection about hybrid functions an effective and accuratemethod for solving Duffing-harmonic oscillator (1) is pre-sented
Consider the following nonlinear Duffing-harmonicoscillator
Table 1 Comparison of various approximate angular frequencies with exact angular frequency
119860 120596HBM 120596EBM 120596Tiw 120596exHFM
119873 119872 119905119891
120596HFM
001 000866 000866 000866 000847 10 5 750 000846
005 004326 004326 004327 004232 10 5 150 004236
01 008628 008627 008624 008439 10 5 75 008446
05 039736 039638 039423 038737 10 5 165 038757
10 065465 065164 064359 063678 15 5 10 063673
50 097435 097343 096731 096698 14 10 65 096699
100 099340 099314 099095 099092 14 10 65 099093
A
09
08
07
06
05
04
03
02
01
0
0 2 4 6 8 10
A
120596
120596
HFMHBMEBM
TiwExact
A
120596
HFMHBMEBM
TiwExact
0940
0935
0930
0925
0920
3 305 310 315 320 325
0976
0975
0974
0973
0972
0971
0970
0969
0968
0967500 505 510 515 520 525
A
120596
HFMHBMEBM
TiwExact
0990
0989
0988
0987
8 805 810 815 820 825
Figure 1 Comparison of the approximate frequencies with corresponding exact frequency
where Θ(119905) is defined in (11) and 119880 is a vector with 119872119873
unknowns as follows
119880 = [11990610 119906
1119872minus1 11990620 119906
2119872minus1 119906
1198730 119906
119873119872minus1]119879
(40)
By expanding 119906(0) = 119860 and 119891(119905) = 1 in terms of hybridfunctions we get
119906 (0) = 119860 = A119879Θ (119905) (41)
119891 (119905) = 1 = 119864119879Θ (119905) (42)
whereA = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119860 119860 119860]
119879 and 119864 = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 1 1]
119879 Integrating(39) from 0 to 119905 and using (41) we obtain
1199061015840
(119905) = int119905
0
119880119879Θ (119904) 119889119904 + 119906
1015840
(0) ≃ 119880119879119875Θ (119905)
119906 (119905) = int119905
0
1199061015840
(119904) 119889119904 + 119906 (0)
≃ int119905
0
119880119879119875Θ (119904) 119889119904 +A
119879Θ (119905) ≃ 119880
1198791198752Θ (119905) +A
119879Θ (119905)
= U119879Θ (119905)
(43)
6 Journal of Difference Equations
0010
0005
0
minus0005
minus0010
100 200 400300 600500 700
t
t t t
t t
u(t)
u(t)
u(t)
u(t)
u(t)
u(t)
A = 001 N = 10M = 5 A = 005 N = 10M = 5
004
002
0
minus002
minus004
50 100 150
010
005
0
minus005
minus010
10 20 30 40 50 60 70
A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
04
02
0
minus02
minus04
2 4 6 8 10 12 14 16
1
05
0
minus05
minus1
2 4 6 8 10
4
2
0
minus2
minus4
1 2 3 4 5 6
Figure 2 Plots of displacement 119906 versus time 119905 Solid line HFM solid circle RK4
whereU = (1198801198791198752+A119879)
119879 and 119875 is the operational matrix ofintegration given in (13) Using Lemma 4 the functions 1199062(119905)and 119906
3(119905) can be expanded as
1199062
(119905) ≃ U119879
2Θ (119905) (44)
1199063
(119905) ≃ U119879
3Θ (119905) (45)
Therefore by using (39) and (42)ndash(45) the right side of (38)can be approximated as
(1 + 1199062
(119905)) 11990610158401015840
(119905) + 1199063
(119905)
≃ (119864119879+U119879
2)Θ (119905) Θ
119879
(119905) 119880⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Θ(119905)
+U119879
3Θ (119905)
≃ [(119864119879+U119879
2) +U
119879
3]Θ (119905) = 0
(46)
where can be calculated in a similar way tomatrix119865 in (24)Since the above equation is satisfied for every 119905 isin [0 119905
119891) we
can get
(119864119879+U119879
2) +U
119879
3= 0 (47)
This is a system of algebraic equations with 119872119873 equationsand 119872119873 unknowns which can be solved by Newtonrsquositeration method to obtain the unknown vector 119880
Table 2 Error percentage comparison between previous results andHFM with various 119860
119860Error percentage
HBM EBM Tiw HFM001 2242 2242 2242 0118
005 2223 2219 2212 0095
01 2239 2225 2197 0083
05 2579 2326 1771 0052
10 2807 2334 1070 0008
50 0763 0667 0034 0001
100 0250 0224 0003 0001
Remark 5 The approximate period and frequency of thehybrid functions method (HFM) can be obtained as follows
119879HFM = 120572 120596HFM =2120587
120572 (48)
where120572 is the first positive root of equation119906(119905)minus119860 = 0 Herewe use the famous Newtonrsquos iteration method for finding aproper approximation 120572 of nonlinear equation 119906(119905) minus 119860 = 0in the following form
120572119899+1
= 120572119899minus
119865 (120572119899)
1198651015840 (120572119899) 119899 = 0 1 (49)
where 119865(119905) = 119906(119905) minus 119860 = (U119879 minus A119879)Θ(119905) and 1205720is initial
approximation
Journal of Difference Equations 7
25
20
15
10
05
0
25
20
15
10
05
0
times10minus7
times10minus7
times10minus6
times10minus6
times10minus6
times10minus7
0 0100 50 100 150200 300 400 500 600 700
120591i
120591i 120591i 120591i
120591i 120591i
14
12
10
80
60
40
20
00 10 20 30 40 50 60 70
90
80
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
20
18
16
14
12
10
08
06
04
02
0 02 214 436 658 10
70
60
50
40
30
20
10
0
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
in [26] a direct method for solving multidelay systemsusing hybrid of block-pulse functions and Taylor series ispresented Marzban et al [27] implemented hybrid of block-pulse functions and Lagrange-interpolating polynomials tofind approximate solution of Volterrarsquos population modelThe Lane-Emden type equations are solved in [28] usinghybrid functions of block-pulse and Lagrange-interpolatingpolynomials The hybrid of block-pulse functions and Taylorseries is employed in [29] to solve the linear quadratic optimalcontrol with delay systems Application of hybrid of block-pulse functions and Lagrange polynomials for solving thenonlinear mixed Volterra-Fredholm-Hammerstein integralequations is investigated in [30]
In this study we consider the following nonlinearDuffing-harmonic oscillation [31ndash35]
11990610158401015840+
1199063
1 + 1199062= 0 119906 (0) = 119860 119906
1015840
(0) = 0 (1)
where is an example of conservative nonlinear oscillatorysystems having a rational form for the restoring force Notethat for small values of 119906 (1) is that of a Duffing-typenonlinear oscillator that is
11990610158401015840+ 1199063≃ 0 (2)
while for large values of 119906 the equation approximates that ofa linear harmonic oscillator that is
11990610158401015840+ 119906 ≃ 0 (3)
Hence (1) is called the Duffing-harmonic oscillator [31] Thesystemwill oscillate between symmetric bounds [minus119860 119860] andthe frequency and corresponding periodic solution of thenonlinear oscillator are dependent on the amplitude 119860 [33]
In this paper we introduce an alternative numericalmethod to solve Duffing-harmonic oscillator The methodconsists of reducing this equation to a set of algebraic equa-tions by first expanding the candidate function as a hybridfunction with unknown coefficients These hybrid functionswhich consist of block-pulse functions plus Chebyshev cardi-nal functions are first introduced The operational matricesof integration and product are givenThese matrices are thenused to evaluate the coefficients of the hybrid function for thesolution of strongly nonlinear oscillators
The outline of this paper is as follows In Section 2 thebasic properties of hybrid block-pulse functions and Cheby-shev cardinal functions required for subsequent developmentare described In Section 3 we apply the proposed numericalmethod to the Duffing-harmonic oscillation Results andcomparisons with existing methods in the literature arepresented in Section 4 and finally conclusions are drawn inSection 5
2 Properties of Hybrid Functions
Marzban et al in [27 29 30] used the hybrid of block-pulsefunctions and Lagrange-interpolating polynomials based onzeros of the Legendre polynomials But no explicit formulasare known for the zeros of the Legendre polynomials In
this study we used Chebyshev cardinal functions which arespecial cases of Lagrange-interpolating polynomials basedon zeros of the Chebyshev polynomials of the first kindto overcome this problem In this paper we present theproperties of hybrid functions which consist of block-pulsefunctions plus Chebyshev cardinal functions similar to [2729 30] The hybrid functions are first introduced and theoperational matrices of integration and product are thenderived
21 Hybrid Functions of Block-Pulse and Chebyshev CardinalFunctions Hybrid functions 120579
119899119898(119905) 119899 = 1 2 119873 119898 =
0 1 119872 minus 1 are defined on the interval [0 119905119891) as
120579119899119898
(119905)
=
119862119898(2119873
119905119891
119905 minus 2119899 + 1) 119905 isin [(119899 minus 1
119873) 119905119891119899
119873119905119891)
0 otherwise(4)
where 119899 and 119898 are the order of block-pulse functions andChebyshev cardinal functions respectively Here 119862
119898(119905) are
defined as [36ndash38]
119862119898(119905) =
119879119872
(119905)
119879119872119905
(119905119898) (119905 minus 119905
119898) 119898 = 0 1 119872 minus 1 (5)
where 119879119872(119905) is the first kind Chebyshev polynomial of order
119872 in [minus1 1] defined by
119879119872
(119905) = cos (119872arccos (119905)) (6)
subscript 119905 denotes 119905-differentiation and 119905119898 119898 = 0 1
119872minus1 are the zeros of 119879119872(119905) defined by cos((2119898+1)120587)2119872)
119898 = 0 1 119872 minus 1 with the Kronecker property
119862119898(119905119894) = 120575119898119894
= 1 if 119894 = 119898
0 if 119894 = 119898(7)
where 120575119898119894
is the Kronecker delta function
22 Function Approximation A function 119891(119905) defined overthe interval [0 119905
119891) may be expanded as
119891 (119905) =
infin
sum119899=1
infin
sum119898=0
119888119899119898
120579119899119898
(119905) (8)
If the infinite series in (8) is truncated then (8) can beexpressed as
119891 (119905) ≃
119873
sum119899=1
119872minus1
sum119898=0
119888119899119898
120579119899119898
(119905) = 119862119879Θ (119905) (9)
where
119862 = [11988810 119888
1119872minus1 11988820 119888
2119872minus1 119888
1198730 119888
119873119872minus1]119879
(10)
Θ (119905) = [12057910
(119905) 1205791119872minus1
(119905) 12057920
(119905)
1205792119872minus1
(119905) 1205791198730
(119905) 120579119873119872minus1
(119905)]119879
(11)
Journal of Difference Equations 3
In (10) and (11) 119888119899119898 119899 = 1 2 119873 119898 = 0 1 119872 minus 1
are the expansion coefficients of the function 119891(119905) in the119899th subinterval [((119899 minus 1)119873)119905
119891 (119899119873)119905
119891) and 120579
119899119898(119905) 119899 =
1 2 119873 119898 = 0 1 119872 minus 1 are defined as in (4) Withthe aid of (7) the coefficients 119888
119899119898can be obtained as
119888119899119898
= 119891(119905119891
2119873(119905119898
+ 2119899 minus 1)) (12)
23TheOperationalMatrix of Integration In this section theoperational matrix of integration is derived The integrationof the vector Θ(119905) defined in (11) can be approximated as
int119905
0
Θ (119904) 119889119904 ≃ 119875Θ (119905) (13)
where119875 is the119872119873times119872119873 operationalmatrix of integration forChebyshev cardinal functions The matrix 119875 can be obtainedby the following process Let
Case 3 If 119899 gt 119897 then (119897119873)119905119891lt (1199051198912119873)(119905
119898+ 2119899 minus 1) and we
obtain119888119899119898
= int(119897119873)119905119891
((119897minus1)119873)119905119891
119862119896(2119873
119905119891
119904 minus 2119897 + 1)119889119904 =119905119891
2119873int1
minus1
119862119896(V) 119889V
(18)
Comparing (13) (14) and (15) we obtain
119875 = (
119864 119867 119867 sdot sdot sdot 119867
0 119864 119867 sdot sdot sdot 119867
0 0 119864 sdot sdot sdot 119867
0 0 0 sdot sdot sdot 119864
) (19)
where 119864 and 119867 are 119872 times 119872 matrices that can be obtained asfollows
Let
119864 = (119890119894119895) 119867 = (ℎ
119894119895) (20)
then for 119894 119895 = 0 1 119872 minus 1 we have
119890119894119895=
119905119891
2119873int119905119895
minus1
119862119894(V) 119889V
ℎ119894119895=
119905119891
2119873int1
minus1
119862119894(V) 119889V
(21)
where 119905119895 119895 = 0 1 119872 minus 1 are the zeros of the first
kind Chebyshev polynomial of order 119872 It is noted that 119864 isthe operational matrix of integration for Chebyshev cardinalfunctions over interval [((119899 minus 1)119873)119905
119891 (119899119873)119905
119891)
Remark 1 To calculate the entries 119890119894119895and ℎ
119894119895 119894 119895 = 0 1
119872 minus 1 we have
119862119898(V) =
119879119872
(V)119879119872V (119905119898) (V minus 119905
119898)=
120573
119879119872V (119905119898)
times
119872minus1
prod119896=0119896 =119898
(V minus 119905119896)
(22)
where 120573 = 2119872minus1 is the coefficient of 119905
119872 in the Chebyshevpolynomial function 119879
119872(V) Using (22) we get
119890119894119895=
119905119891120573
2119873119879119872119905
(119905119894)int119905119895
minus1
119872minus1
prod119896=0119896 =119894
(V minus 119905119896) 119889V
ℎ119894119895=
119905119891120573
2119873119879119872119905
(119905119894)int1
minus1
119872minus1
prod119896=0119896 =119894
(V minus 119905119896) 119889V
(23)
for 119894 119895 = 0 1 119872 minus 1
24 The Operational Matrix of Product The following prop-erty of the product of two hybrid function vectors will also beused Let
Θ (119905)Θ119879
(119905) 119865 ≃ 119865Θ (119905) (24)
where
119865 = [11989110 119891
1119872minus1 11989120 119891
2119872minus1 119891
1198730 119891
119873119872minus1]119879
(25)
4 Journal of Difference Equations
and 119865 is an 119872119873 times 119872119873 product operational matrix To find119865 we apply the following procedure First by using (11) and(25) we obtain
Θ (119905)Θ119879
(119905) 119865 =
((((((((((((((((((((((((((
(
12057910
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
12057911
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198730
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198731
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
120579119873119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
))))))))))))))))))))))))))
)
(26)
Using (9) any function 120579119894119895(119905)120579119897119896(119905) 119894 119897 = 1 2 119873 119895 119896 =
therefore we find the 119872119873 times 119872119873 matrix 119865 as
119865 = diag [11989110 119891
1119872minus1 11989120
1198912119872minus1
1198911198730
119891119873119872minus1
] (30)
Lemma 2 The functions 119862119895(119905) 119895 = 0 1 119872 minus 1 are
orthogonal with respect to 119908(119905) = 1radic1 minus 1199052 on [minus1 1] andsatisfy the orthogonality condition
⟨119862119894(119905) 119862
119895(119905)⟩119908
= int1
minus1
119862119894(119905) 119862119895(119905)
radic1 minus 1199052119889119905 =
120587
119872 119894119891 119895 = 119894
0 119894119891 119895 = 119894
(31)
The proof of this lemma is presented in [39]
Remark 3 Since 120579119899119898
(119905) consists of block-pulse functions andChebyshev cardinal functions which are both complete andorthogonal the set of hybrid of block-pulse functions andChebyshev cardinal functions is a complete orthogonal set inthe Hilbert space 119871
2[0 119905119891)
Lemma 4 Let MN vectors 119862 and 119862119901be hybrid functions
coefficients of 119906(119905) and 119906119901(119905) respectively If
119862 = [11988810 119888
1119872minus1 11988820 119888
2119872minus1 119888
1198730 119888
119873119872minus1]119879
(32)
then
119862119901≃ [119888119901
10 119888
119901
1119872minus1 119888119901
20 119888
119901
2119872minus1 119888
119901
1198730 119888
119901
119873119872minus1]119879
(33)
where 119901 ge 1 is a positive integer
Proof When 119901 = 1 (33) follows at once from 119906119901(119905) = 119906(119905)
Suppose that (33) holds for119901 we will deduce it for119901+1 Since119906119901+1
(119905) = 119906(119905)119906119901(119905) from (24) and (30) we have
where 119862119901can be calculated in a similar way to matrix 119865 in
(24) Now using (33) we obtain
119862119901+1
= 119862119879119862119901= [119888119901+1
10 119888
119901+1
1119872minus1 119888119901+1
20
119888119901+1
2119872minus1 119888
119901+1
1198730 119888
119901+1
119873119872minus1]119879
(35)
Therefore (33) holds for 119901 + 1 and the lemma isestablished
3 Hybrid Functions Method to SolveDuffing-Harmonic Oscillator
In this section by using the results obtained in the previoussection about hybrid functions an effective and accuratemethod for solving Duffing-harmonic oscillator (1) is pre-sented
Consider the following nonlinear Duffing-harmonicoscillator
Table 1 Comparison of various approximate angular frequencies with exact angular frequency
119860 120596HBM 120596EBM 120596Tiw 120596exHFM
119873 119872 119905119891
120596HFM
001 000866 000866 000866 000847 10 5 750 000846
005 004326 004326 004327 004232 10 5 150 004236
01 008628 008627 008624 008439 10 5 75 008446
05 039736 039638 039423 038737 10 5 165 038757
10 065465 065164 064359 063678 15 5 10 063673
50 097435 097343 096731 096698 14 10 65 096699
100 099340 099314 099095 099092 14 10 65 099093
A
09
08
07
06
05
04
03
02
01
0
0 2 4 6 8 10
A
120596
120596
HFMHBMEBM
TiwExact
A
120596
HFMHBMEBM
TiwExact
0940
0935
0930
0925
0920
3 305 310 315 320 325
0976
0975
0974
0973
0972
0971
0970
0969
0968
0967500 505 510 515 520 525
A
120596
HFMHBMEBM
TiwExact
0990
0989
0988
0987
8 805 810 815 820 825
Figure 1 Comparison of the approximate frequencies with corresponding exact frequency
where Θ(119905) is defined in (11) and 119880 is a vector with 119872119873
unknowns as follows
119880 = [11990610 119906
1119872minus1 11990620 119906
2119872minus1 119906
1198730 119906
119873119872minus1]119879
(40)
By expanding 119906(0) = 119860 and 119891(119905) = 1 in terms of hybridfunctions we get
119906 (0) = 119860 = A119879Θ (119905) (41)
119891 (119905) = 1 = 119864119879Θ (119905) (42)
whereA = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119860 119860 119860]
119879 and 119864 = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 1 1]
119879 Integrating(39) from 0 to 119905 and using (41) we obtain
1199061015840
(119905) = int119905
0
119880119879Θ (119904) 119889119904 + 119906
1015840
(0) ≃ 119880119879119875Θ (119905)
119906 (119905) = int119905
0
1199061015840
(119904) 119889119904 + 119906 (0)
≃ int119905
0
119880119879119875Θ (119904) 119889119904 +A
119879Θ (119905) ≃ 119880
1198791198752Θ (119905) +A
119879Θ (119905)
= U119879Θ (119905)
(43)
6 Journal of Difference Equations
0010
0005
0
minus0005
minus0010
100 200 400300 600500 700
t
t t t
t t
u(t)
u(t)
u(t)
u(t)
u(t)
u(t)
A = 001 N = 10M = 5 A = 005 N = 10M = 5
004
002
0
minus002
minus004
50 100 150
010
005
0
minus005
minus010
10 20 30 40 50 60 70
A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
04
02
0
minus02
minus04
2 4 6 8 10 12 14 16
1
05
0
minus05
minus1
2 4 6 8 10
4
2
0
minus2
minus4
1 2 3 4 5 6
Figure 2 Plots of displacement 119906 versus time 119905 Solid line HFM solid circle RK4
whereU = (1198801198791198752+A119879)
119879 and 119875 is the operational matrix ofintegration given in (13) Using Lemma 4 the functions 1199062(119905)and 119906
3(119905) can be expanded as
1199062
(119905) ≃ U119879
2Θ (119905) (44)
1199063
(119905) ≃ U119879
3Θ (119905) (45)
Therefore by using (39) and (42)ndash(45) the right side of (38)can be approximated as
(1 + 1199062
(119905)) 11990610158401015840
(119905) + 1199063
(119905)
≃ (119864119879+U119879
2)Θ (119905) Θ
119879
(119905) 119880⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Θ(119905)
+U119879
3Θ (119905)
≃ [(119864119879+U119879
2) +U
119879
3]Θ (119905) = 0
(46)
where can be calculated in a similar way tomatrix119865 in (24)Since the above equation is satisfied for every 119905 isin [0 119905
119891) we
can get
(119864119879+U119879
2) +U
119879
3= 0 (47)
This is a system of algebraic equations with 119872119873 equationsand 119872119873 unknowns which can be solved by Newtonrsquositeration method to obtain the unknown vector 119880
Table 2 Error percentage comparison between previous results andHFM with various 119860
119860Error percentage
HBM EBM Tiw HFM001 2242 2242 2242 0118
005 2223 2219 2212 0095
01 2239 2225 2197 0083
05 2579 2326 1771 0052
10 2807 2334 1070 0008
50 0763 0667 0034 0001
100 0250 0224 0003 0001
Remark 5 The approximate period and frequency of thehybrid functions method (HFM) can be obtained as follows
119879HFM = 120572 120596HFM =2120587
120572 (48)
where120572 is the first positive root of equation119906(119905)minus119860 = 0 Herewe use the famous Newtonrsquos iteration method for finding aproper approximation 120572 of nonlinear equation 119906(119905) minus 119860 = 0in the following form
120572119899+1
= 120572119899minus
119865 (120572119899)
1198651015840 (120572119899) 119899 = 0 1 (49)
where 119865(119905) = 119906(119905) minus 119860 = (U119879 minus A119879)Θ(119905) and 1205720is initial
approximation
Journal of Difference Equations 7
25
20
15
10
05
0
25
20
15
10
05
0
times10minus7
times10minus7
times10minus6
times10minus6
times10minus6
times10minus7
0 0100 50 100 150200 300 400 500 600 700
120591i
120591i 120591i 120591i
120591i 120591i
14
12
10
80
60
40
20
00 10 20 30 40 50 60 70
90
80
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
20
18
16
14
12
10
08
06
04
02
0 02 214 436 658 10
70
60
50
40
30
20
10
0
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
In (10) and (11) 119888119899119898 119899 = 1 2 119873 119898 = 0 1 119872 minus 1
are the expansion coefficients of the function 119891(119905) in the119899th subinterval [((119899 minus 1)119873)119905
119891 (119899119873)119905
119891) and 120579
119899119898(119905) 119899 =
1 2 119873 119898 = 0 1 119872 minus 1 are defined as in (4) Withthe aid of (7) the coefficients 119888
119899119898can be obtained as
119888119899119898
= 119891(119905119891
2119873(119905119898
+ 2119899 minus 1)) (12)
23TheOperationalMatrix of Integration In this section theoperational matrix of integration is derived The integrationof the vector Θ(119905) defined in (11) can be approximated as
int119905
0
Θ (119904) 119889119904 ≃ 119875Θ (119905) (13)
where119875 is the119872119873times119872119873 operationalmatrix of integration forChebyshev cardinal functions The matrix 119875 can be obtainedby the following process Let
Case 3 If 119899 gt 119897 then (119897119873)119905119891lt (1199051198912119873)(119905
119898+ 2119899 minus 1) and we
obtain119888119899119898
= int(119897119873)119905119891
((119897minus1)119873)119905119891
119862119896(2119873
119905119891
119904 minus 2119897 + 1)119889119904 =119905119891
2119873int1
minus1
119862119896(V) 119889V
(18)
Comparing (13) (14) and (15) we obtain
119875 = (
119864 119867 119867 sdot sdot sdot 119867
0 119864 119867 sdot sdot sdot 119867
0 0 119864 sdot sdot sdot 119867
0 0 0 sdot sdot sdot 119864
) (19)
where 119864 and 119867 are 119872 times 119872 matrices that can be obtained asfollows
Let
119864 = (119890119894119895) 119867 = (ℎ
119894119895) (20)
then for 119894 119895 = 0 1 119872 minus 1 we have
119890119894119895=
119905119891
2119873int119905119895
minus1
119862119894(V) 119889V
ℎ119894119895=
119905119891
2119873int1
minus1
119862119894(V) 119889V
(21)
where 119905119895 119895 = 0 1 119872 minus 1 are the zeros of the first
kind Chebyshev polynomial of order 119872 It is noted that 119864 isthe operational matrix of integration for Chebyshev cardinalfunctions over interval [((119899 minus 1)119873)119905
119891 (119899119873)119905
119891)
Remark 1 To calculate the entries 119890119894119895and ℎ
119894119895 119894 119895 = 0 1
119872 minus 1 we have
119862119898(V) =
119879119872
(V)119879119872V (119905119898) (V minus 119905
119898)=
120573
119879119872V (119905119898)
times
119872minus1
prod119896=0119896 =119898
(V minus 119905119896)
(22)
where 120573 = 2119872minus1 is the coefficient of 119905
119872 in the Chebyshevpolynomial function 119879
119872(V) Using (22) we get
119890119894119895=
119905119891120573
2119873119879119872119905
(119905119894)int119905119895
minus1
119872minus1
prod119896=0119896 =119894
(V minus 119905119896) 119889V
ℎ119894119895=
119905119891120573
2119873119879119872119905
(119905119894)int1
minus1
119872minus1
prod119896=0119896 =119894
(V minus 119905119896) 119889V
(23)
for 119894 119895 = 0 1 119872 minus 1
24 The Operational Matrix of Product The following prop-erty of the product of two hybrid function vectors will also beused Let
Θ (119905)Θ119879
(119905) 119865 ≃ 119865Θ (119905) (24)
where
119865 = [11989110 119891
1119872minus1 11989120 119891
2119872minus1 119891
1198730 119891
119873119872minus1]119879
(25)
4 Journal of Difference Equations
and 119865 is an 119872119873 times 119872119873 product operational matrix To find119865 we apply the following procedure First by using (11) and(25) we obtain
Θ (119905)Θ119879
(119905) 119865 =
((((((((((((((((((((((((((
(
12057910
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
12057911
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198730
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198731
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
120579119873119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
))))))))))))))))))))))))))
)
(26)
Using (9) any function 120579119894119895(119905)120579119897119896(119905) 119894 119897 = 1 2 119873 119895 119896 =
therefore we find the 119872119873 times 119872119873 matrix 119865 as
119865 = diag [11989110 119891
1119872minus1 11989120
1198912119872minus1
1198911198730
119891119873119872minus1
] (30)
Lemma 2 The functions 119862119895(119905) 119895 = 0 1 119872 minus 1 are
orthogonal with respect to 119908(119905) = 1radic1 minus 1199052 on [minus1 1] andsatisfy the orthogonality condition
⟨119862119894(119905) 119862
119895(119905)⟩119908
= int1
minus1
119862119894(119905) 119862119895(119905)
radic1 minus 1199052119889119905 =
120587
119872 119894119891 119895 = 119894
0 119894119891 119895 = 119894
(31)
The proof of this lemma is presented in [39]
Remark 3 Since 120579119899119898
(119905) consists of block-pulse functions andChebyshev cardinal functions which are both complete andorthogonal the set of hybrid of block-pulse functions andChebyshev cardinal functions is a complete orthogonal set inthe Hilbert space 119871
2[0 119905119891)
Lemma 4 Let MN vectors 119862 and 119862119901be hybrid functions
coefficients of 119906(119905) and 119906119901(119905) respectively If
119862 = [11988810 119888
1119872minus1 11988820 119888
2119872minus1 119888
1198730 119888
119873119872minus1]119879
(32)
then
119862119901≃ [119888119901
10 119888
119901
1119872minus1 119888119901
20 119888
119901
2119872minus1 119888
119901
1198730 119888
119901
119873119872minus1]119879
(33)
where 119901 ge 1 is a positive integer
Proof When 119901 = 1 (33) follows at once from 119906119901(119905) = 119906(119905)
Suppose that (33) holds for119901 we will deduce it for119901+1 Since119906119901+1
(119905) = 119906(119905)119906119901(119905) from (24) and (30) we have
where 119862119901can be calculated in a similar way to matrix 119865 in
(24) Now using (33) we obtain
119862119901+1
= 119862119879119862119901= [119888119901+1
10 119888
119901+1
1119872minus1 119888119901+1
20
119888119901+1
2119872minus1 119888
119901+1
1198730 119888
119901+1
119873119872minus1]119879
(35)
Therefore (33) holds for 119901 + 1 and the lemma isestablished
3 Hybrid Functions Method to SolveDuffing-Harmonic Oscillator
In this section by using the results obtained in the previoussection about hybrid functions an effective and accuratemethod for solving Duffing-harmonic oscillator (1) is pre-sented
Consider the following nonlinear Duffing-harmonicoscillator
Table 1 Comparison of various approximate angular frequencies with exact angular frequency
119860 120596HBM 120596EBM 120596Tiw 120596exHFM
119873 119872 119905119891
120596HFM
001 000866 000866 000866 000847 10 5 750 000846
005 004326 004326 004327 004232 10 5 150 004236
01 008628 008627 008624 008439 10 5 75 008446
05 039736 039638 039423 038737 10 5 165 038757
10 065465 065164 064359 063678 15 5 10 063673
50 097435 097343 096731 096698 14 10 65 096699
100 099340 099314 099095 099092 14 10 65 099093
A
09
08
07
06
05
04
03
02
01
0
0 2 4 6 8 10
A
120596
120596
HFMHBMEBM
TiwExact
A
120596
HFMHBMEBM
TiwExact
0940
0935
0930
0925
0920
3 305 310 315 320 325
0976
0975
0974
0973
0972
0971
0970
0969
0968
0967500 505 510 515 520 525
A
120596
HFMHBMEBM
TiwExact
0990
0989
0988
0987
8 805 810 815 820 825
Figure 1 Comparison of the approximate frequencies with corresponding exact frequency
where Θ(119905) is defined in (11) and 119880 is a vector with 119872119873
unknowns as follows
119880 = [11990610 119906
1119872minus1 11990620 119906
2119872minus1 119906
1198730 119906
119873119872minus1]119879
(40)
By expanding 119906(0) = 119860 and 119891(119905) = 1 in terms of hybridfunctions we get
119906 (0) = 119860 = A119879Θ (119905) (41)
119891 (119905) = 1 = 119864119879Θ (119905) (42)
whereA = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119860 119860 119860]
119879 and 119864 = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 1 1]
119879 Integrating(39) from 0 to 119905 and using (41) we obtain
1199061015840
(119905) = int119905
0
119880119879Θ (119904) 119889119904 + 119906
1015840
(0) ≃ 119880119879119875Θ (119905)
119906 (119905) = int119905
0
1199061015840
(119904) 119889119904 + 119906 (0)
≃ int119905
0
119880119879119875Θ (119904) 119889119904 +A
119879Θ (119905) ≃ 119880
1198791198752Θ (119905) +A
119879Θ (119905)
= U119879Θ (119905)
(43)
6 Journal of Difference Equations
0010
0005
0
minus0005
minus0010
100 200 400300 600500 700
t
t t t
t t
u(t)
u(t)
u(t)
u(t)
u(t)
u(t)
A = 001 N = 10M = 5 A = 005 N = 10M = 5
004
002
0
minus002
minus004
50 100 150
010
005
0
minus005
minus010
10 20 30 40 50 60 70
A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
04
02
0
minus02
minus04
2 4 6 8 10 12 14 16
1
05
0
minus05
minus1
2 4 6 8 10
4
2
0
minus2
minus4
1 2 3 4 5 6
Figure 2 Plots of displacement 119906 versus time 119905 Solid line HFM solid circle RK4
whereU = (1198801198791198752+A119879)
119879 and 119875 is the operational matrix ofintegration given in (13) Using Lemma 4 the functions 1199062(119905)and 119906
3(119905) can be expanded as
1199062
(119905) ≃ U119879
2Θ (119905) (44)
1199063
(119905) ≃ U119879
3Θ (119905) (45)
Therefore by using (39) and (42)ndash(45) the right side of (38)can be approximated as
(1 + 1199062
(119905)) 11990610158401015840
(119905) + 1199063
(119905)
≃ (119864119879+U119879
2)Θ (119905) Θ
119879
(119905) 119880⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Θ(119905)
+U119879
3Θ (119905)
≃ [(119864119879+U119879
2) +U
119879
3]Θ (119905) = 0
(46)
where can be calculated in a similar way tomatrix119865 in (24)Since the above equation is satisfied for every 119905 isin [0 119905
119891) we
can get
(119864119879+U119879
2) +U
119879
3= 0 (47)
This is a system of algebraic equations with 119872119873 equationsand 119872119873 unknowns which can be solved by Newtonrsquositeration method to obtain the unknown vector 119880
Table 2 Error percentage comparison between previous results andHFM with various 119860
119860Error percentage
HBM EBM Tiw HFM001 2242 2242 2242 0118
005 2223 2219 2212 0095
01 2239 2225 2197 0083
05 2579 2326 1771 0052
10 2807 2334 1070 0008
50 0763 0667 0034 0001
100 0250 0224 0003 0001
Remark 5 The approximate period and frequency of thehybrid functions method (HFM) can be obtained as follows
119879HFM = 120572 120596HFM =2120587
120572 (48)
where120572 is the first positive root of equation119906(119905)minus119860 = 0 Herewe use the famous Newtonrsquos iteration method for finding aproper approximation 120572 of nonlinear equation 119906(119905) minus 119860 = 0in the following form
120572119899+1
= 120572119899minus
119865 (120572119899)
1198651015840 (120572119899) 119899 = 0 1 (49)
where 119865(119905) = 119906(119905) minus 119860 = (U119879 minus A119879)Θ(119905) and 1205720is initial
approximation
Journal of Difference Equations 7
25
20
15
10
05
0
25
20
15
10
05
0
times10minus7
times10minus7
times10minus6
times10minus6
times10minus6
times10minus7
0 0100 50 100 150200 300 400 500 600 700
120591i
120591i 120591i 120591i
120591i 120591i
14
12
10
80
60
40
20
00 10 20 30 40 50 60 70
90
80
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
20
18
16
14
12
10
08
06
04
02
0 02 214 436 658 10
70
60
50
40
30
20
10
0
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
and 119865 is an 119872119873 times 119872119873 product operational matrix To find119865 we apply the following procedure First by using (11) and(25) we obtain
Θ (119905)Θ119879
(119905) 119865 =
((((((((((((((((((((((((((
(
12057910
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
12057911
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198730
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
1205791198731
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
120579119873119872minus1
(119905)
infin
sum119899=1
infin
sum119898=0
119891119899119898
120579119899119898
(119905)
))))))))))))))))))))))))))
)
(26)
Using (9) any function 120579119894119895(119905)120579119897119896(119905) 119894 119897 = 1 2 119873 119895 119896 =
therefore we find the 119872119873 times 119872119873 matrix 119865 as
119865 = diag [11989110 119891
1119872minus1 11989120
1198912119872minus1
1198911198730
119891119873119872minus1
] (30)
Lemma 2 The functions 119862119895(119905) 119895 = 0 1 119872 minus 1 are
orthogonal with respect to 119908(119905) = 1radic1 minus 1199052 on [minus1 1] andsatisfy the orthogonality condition
⟨119862119894(119905) 119862
119895(119905)⟩119908
= int1
minus1
119862119894(119905) 119862119895(119905)
radic1 minus 1199052119889119905 =
120587
119872 119894119891 119895 = 119894
0 119894119891 119895 = 119894
(31)
The proof of this lemma is presented in [39]
Remark 3 Since 120579119899119898
(119905) consists of block-pulse functions andChebyshev cardinal functions which are both complete andorthogonal the set of hybrid of block-pulse functions andChebyshev cardinal functions is a complete orthogonal set inthe Hilbert space 119871
2[0 119905119891)
Lemma 4 Let MN vectors 119862 and 119862119901be hybrid functions
coefficients of 119906(119905) and 119906119901(119905) respectively If
119862 = [11988810 119888
1119872minus1 11988820 119888
2119872minus1 119888
1198730 119888
119873119872minus1]119879
(32)
then
119862119901≃ [119888119901
10 119888
119901
1119872minus1 119888119901
20 119888
119901
2119872minus1 119888
119901
1198730 119888
119901
119873119872minus1]119879
(33)
where 119901 ge 1 is a positive integer
Proof When 119901 = 1 (33) follows at once from 119906119901(119905) = 119906(119905)
Suppose that (33) holds for119901 we will deduce it for119901+1 Since119906119901+1
(119905) = 119906(119905)119906119901(119905) from (24) and (30) we have
where 119862119901can be calculated in a similar way to matrix 119865 in
(24) Now using (33) we obtain
119862119901+1
= 119862119879119862119901= [119888119901+1
10 119888
119901+1
1119872minus1 119888119901+1
20
119888119901+1
2119872minus1 119888
119901+1
1198730 119888
119901+1
119873119872minus1]119879
(35)
Therefore (33) holds for 119901 + 1 and the lemma isestablished
3 Hybrid Functions Method to SolveDuffing-Harmonic Oscillator
In this section by using the results obtained in the previoussection about hybrid functions an effective and accuratemethod for solving Duffing-harmonic oscillator (1) is pre-sented
Consider the following nonlinear Duffing-harmonicoscillator
Table 1 Comparison of various approximate angular frequencies with exact angular frequency
119860 120596HBM 120596EBM 120596Tiw 120596exHFM
119873 119872 119905119891
120596HFM
001 000866 000866 000866 000847 10 5 750 000846
005 004326 004326 004327 004232 10 5 150 004236
01 008628 008627 008624 008439 10 5 75 008446
05 039736 039638 039423 038737 10 5 165 038757
10 065465 065164 064359 063678 15 5 10 063673
50 097435 097343 096731 096698 14 10 65 096699
100 099340 099314 099095 099092 14 10 65 099093
A
09
08
07
06
05
04
03
02
01
0
0 2 4 6 8 10
A
120596
120596
HFMHBMEBM
TiwExact
A
120596
HFMHBMEBM
TiwExact
0940
0935
0930
0925
0920
3 305 310 315 320 325
0976
0975
0974
0973
0972
0971
0970
0969
0968
0967500 505 510 515 520 525
A
120596
HFMHBMEBM
TiwExact
0990
0989
0988
0987
8 805 810 815 820 825
Figure 1 Comparison of the approximate frequencies with corresponding exact frequency
where Θ(119905) is defined in (11) and 119880 is a vector with 119872119873
unknowns as follows
119880 = [11990610 119906
1119872minus1 11990620 119906
2119872minus1 119906
1198730 119906
119873119872minus1]119879
(40)
By expanding 119906(0) = 119860 and 119891(119905) = 1 in terms of hybridfunctions we get
119906 (0) = 119860 = A119879Θ (119905) (41)
119891 (119905) = 1 = 119864119879Θ (119905) (42)
whereA = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119860 119860 119860]
119879 and 119864 = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 1 1]
119879 Integrating(39) from 0 to 119905 and using (41) we obtain
1199061015840
(119905) = int119905
0
119880119879Θ (119904) 119889119904 + 119906
1015840
(0) ≃ 119880119879119875Θ (119905)
119906 (119905) = int119905
0
1199061015840
(119904) 119889119904 + 119906 (0)
≃ int119905
0
119880119879119875Θ (119904) 119889119904 +A
119879Θ (119905) ≃ 119880
1198791198752Θ (119905) +A
119879Θ (119905)
= U119879Θ (119905)
(43)
6 Journal of Difference Equations
0010
0005
0
minus0005
minus0010
100 200 400300 600500 700
t
t t t
t t
u(t)
u(t)
u(t)
u(t)
u(t)
u(t)
A = 001 N = 10M = 5 A = 005 N = 10M = 5
004
002
0
minus002
minus004
50 100 150
010
005
0
minus005
minus010
10 20 30 40 50 60 70
A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
04
02
0
minus02
minus04
2 4 6 8 10 12 14 16
1
05
0
minus05
minus1
2 4 6 8 10
4
2
0
minus2
minus4
1 2 3 4 5 6
Figure 2 Plots of displacement 119906 versus time 119905 Solid line HFM solid circle RK4
whereU = (1198801198791198752+A119879)
119879 and 119875 is the operational matrix ofintegration given in (13) Using Lemma 4 the functions 1199062(119905)and 119906
3(119905) can be expanded as
1199062
(119905) ≃ U119879
2Θ (119905) (44)
1199063
(119905) ≃ U119879
3Θ (119905) (45)
Therefore by using (39) and (42)ndash(45) the right side of (38)can be approximated as
(1 + 1199062
(119905)) 11990610158401015840
(119905) + 1199063
(119905)
≃ (119864119879+U119879
2)Θ (119905) Θ
119879
(119905) 119880⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Θ(119905)
+U119879
3Θ (119905)
≃ [(119864119879+U119879
2) +U
119879
3]Θ (119905) = 0
(46)
where can be calculated in a similar way tomatrix119865 in (24)Since the above equation is satisfied for every 119905 isin [0 119905
119891) we
can get
(119864119879+U119879
2) +U
119879
3= 0 (47)
This is a system of algebraic equations with 119872119873 equationsand 119872119873 unknowns which can be solved by Newtonrsquositeration method to obtain the unknown vector 119880
Table 2 Error percentage comparison between previous results andHFM with various 119860
119860Error percentage
HBM EBM Tiw HFM001 2242 2242 2242 0118
005 2223 2219 2212 0095
01 2239 2225 2197 0083
05 2579 2326 1771 0052
10 2807 2334 1070 0008
50 0763 0667 0034 0001
100 0250 0224 0003 0001
Remark 5 The approximate period and frequency of thehybrid functions method (HFM) can be obtained as follows
119879HFM = 120572 120596HFM =2120587
120572 (48)
where120572 is the first positive root of equation119906(119905)minus119860 = 0 Herewe use the famous Newtonrsquos iteration method for finding aproper approximation 120572 of nonlinear equation 119906(119905) minus 119860 = 0in the following form
120572119899+1
= 120572119899minus
119865 (120572119899)
1198651015840 (120572119899) 119899 = 0 1 (49)
where 119865(119905) = 119906(119905) minus 119860 = (U119879 minus A119879)Θ(119905) and 1205720is initial
approximation
Journal of Difference Equations 7
25
20
15
10
05
0
25
20
15
10
05
0
times10minus7
times10minus7
times10minus6
times10minus6
times10minus6
times10minus7
0 0100 50 100 150200 300 400 500 600 700
120591i
120591i 120591i 120591i
120591i 120591i
14
12
10
80
60
40
20
00 10 20 30 40 50 60 70
90
80
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
20
18
16
14
12
10
08
06
04
02
0 02 214 436 658 10
70
60
50
40
30
20
10
0
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
Table 1 Comparison of various approximate angular frequencies with exact angular frequency
119860 120596HBM 120596EBM 120596Tiw 120596exHFM
119873 119872 119905119891
120596HFM
001 000866 000866 000866 000847 10 5 750 000846
005 004326 004326 004327 004232 10 5 150 004236
01 008628 008627 008624 008439 10 5 75 008446
05 039736 039638 039423 038737 10 5 165 038757
10 065465 065164 064359 063678 15 5 10 063673
50 097435 097343 096731 096698 14 10 65 096699
100 099340 099314 099095 099092 14 10 65 099093
A
09
08
07
06
05
04
03
02
01
0
0 2 4 6 8 10
A
120596
120596
HFMHBMEBM
TiwExact
A
120596
HFMHBMEBM
TiwExact
0940
0935
0930
0925
0920
3 305 310 315 320 325
0976
0975
0974
0973
0972
0971
0970
0969
0968
0967500 505 510 515 520 525
A
120596
HFMHBMEBM
TiwExact
0990
0989
0988
0987
8 805 810 815 820 825
Figure 1 Comparison of the approximate frequencies with corresponding exact frequency
where Θ(119905) is defined in (11) and 119880 is a vector with 119872119873
unknowns as follows
119880 = [11990610 119906
1119872minus1 11990620 119906
2119872minus1 119906
1198730 119906
119873119872minus1]119879
(40)
By expanding 119906(0) = 119860 and 119891(119905) = 1 in terms of hybridfunctions we get
119906 (0) = 119860 = A119879Θ (119905) (41)
119891 (119905) = 1 = 119864119879Θ (119905) (42)
whereA = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119860 119860 119860]
119879 and 119864 = [
119872119873
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 1 1]
119879 Integrating(39) from 0 to 119905 and using (41) we obtain
1199061015840
(119905) = int119905
0
119880119879Θ (119904) 119889119904 + 119906
1015840
(0) ≃ 119880119879119875Θ (119905)
119906 (119905) = int119905
0
1199061015840
(119904) 119889119904 + 119906 (0)
≃ int119905
0
119880119879119875Θ (119904) 119889119904 +A
119879Θ (119905) ≃ 119880
1198791198752Θ (119905) +A
119879Θ (119905)
= U119879Θ (119905)
(43)
6 Journal of Difference Equations
0010
0005
0
minus0005
minus0010
100 200 400300 600500 700
t
t t t
t t
u(t)
u(t)
u(t)
u(t)
u(t)
u(t)
A = 001 N = 10M = 5 A = 005 N = 10M = 5
004
002
0
minus002
minus004
50 100 150
010
005
0
minus005
minus010
10 20 30 40 50 60 70
A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
04
02
0
minus02
minus04
2 4 6 8 10 12 14 16
1
05
0
minus05
minus1
2 4 6 8 10
4
2
0
minus2
minus4
1 2 3 4 5 6
Figure 2 Plots of displacement 119906 versus time 119905 Solid line HFM solid circle RK4
whereU = (1198801198791198752+A119879)
119879 and 119875 is the operational matrix ofintegration given in (13) Using Lemma 4 the functions 1199062(119905)and 119906
3(119905) can be expanded as
1199062
(119905) ≃ U119879
2Θ (119905) (44)
1199063
(119905) ≃ U119879
3Θ (119905) (45)
Therefore by using (39) and (42)ndash(45) the right side of (38)can be approximated as
(1 + 1199062
(119905)) 11990610158401015840
(119905) + 1199063
(119905)
≃ (119864119879+U119879
2)Θ (119905) Θ
119879
(119905) 119880⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Θ(119905)
+U119879
3Θ (119905)
≃ [(119864119879+U119879
2) +U
119879
3]Θ (119905) = 0
(46)
where can be calculated in a similar way tomatrix119865 in (24)Since the above equation is satisfied for every 119905 isin [0 119905
119891) we
can get
(119864119879+U119879
2) +U
119879
3= 0 (47)
This is a system of algebraic equations with 119872119873 equationsand 119872119873 unknowns which can be solved by Newtonrsquositeration method to obtain the unknown vector 119880
Table 2 Error percentage comparison between previous results andHFM with various 119860
119860Error percentage
HBM EBM Tiw HFM001 2242 2242 2242 0118
005 2223 2219 2212 0095
01 2239 2225 2197 0083
05 2579 2326 1771 0052
10 2807 2334 1070 0008
50 0763 0667 0034 0001
100 0250 0224 0003 0001
Remark 5 The approximate period and frequency of thehybrid functions method (HFM) can be obtained as follows
119879HFM = 120572 120596HFM =2120587
120572 (48)
where120572 is the first positive root of equation119906(119905)minus119860 = 0 Herewe use the famous Newtonrsquos iteration method for finding aproper approximation 120572 of nonlinear equation 119906(119905) minus 119860 = 0in the following form
120572119899+1
= 120572119899minus
119865 (120572119899)
1198651015840 (120572119899) 119899 = 0 1 (49)
where 119865(119905) = 119906(119905) minus 119860 = (U119879 minus A119879)Θ(119905) and 1205720is initial
approximation
Journal of Difference Equations 7
25
20
15
10
05
0
25
20
15
10
05
0
times10minus7
times10minus7
times10minus6
times10minus6
times10minus6
times10minus7
0 0100 50 100 150200 300 400 500 600 700
120591i
120591i 120591i 120591i
120591i 120591i
14
12
10
80
60
40
20
00 10 20 30 40 50 60 70
90
80
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
20
18
16
14
12
10
08
06
04
02
0 02 214 436 658 10
70
60
50
40
30
20
10
0
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
04
02
0
minus02
minus04
2 4 6 8 10 12 14 16
1
05
0
minus05
minus1
2 4 6 8 10
4
2
0
minus2
minus4
1 2 3 4 5 6
Figure 2 Plots of displacement 119906 versus time 119905 Solid line HFM solid circle RK4
whereU = (1198801198791198752+A119879)
119879 and 119875 is the operational matrix ofintegration given in (13) Using Lemma 4 the functions 1199062(119905)and 119906
3(119905) can be expanded as
1199062
(119905) ≃ U119879
2Θ (119905) (44)
1199063
(119905) ≃ U119879
3Θ (119905) (45)
Therefore by using (39) and (42)ndash(45) the right side of (38)can be approximated as
(1 + 1199062
(119905)) 11990610158401015840
(119905) + 1199063
(119905)
≃ (119864119879+U119879
2)Θ (119905) Θ
119879
(119905) 119880⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Θ(119905)
+U119879
3Θ (119905)
≃ [(119864119879+U119879
2) +U
119879
3]Θ (119905) = 0
(46)
where can be calculated in a similar way tomatrix119865 in (24)Since the above equation is satisfied for every 119905 isin [0 119905
119891) we
can get
(119864119879+U119879
2) +U
119879
3= 0 (47)
This is a system of algebraic equations with 119872119873 equationsand 119872119873 unknowns which can be solved by Newtonrsquositeration method to obtain the unknown vector 119880
Table 2 Error percentage comparison between previous results andHFM with various 119860
119860Error percentage
HBM EBM Tiw HFM001 2242 2242 2242 0118
005 2223 2219 2212 0095
01 2239 2225 2197 0083
05 2579 2326 1771 0052
10 2807 2334 1070 0008
50 0763 0667 0034 0001
100 0250 0224 0003 0001
Remark 5 The approximate period and frequency of thehybrid functions method (HFM) can be obtained as follows
119879HFM = 120572 120596HFM =2120587
120572 (48)
where120572 is the first positive root of equation119906(119905)minus119860 = 0 Herewe use the famous Newtonrsquos iteration method for finding aproper approximation 120572 of nonlinear equation 119906(119905) minus 119860 = 0in the following form
120572119899+1
= 120572119899minus
119865 (120572119899)
1198651015840 (120572119899) 119899 = 0 1 (49)
where 119865(119905) = 119906(119905) minus 119860 = (U119879 minus A119879)Θ(119905) and 1205720is initial
approximation
Journal of Difference Equations 7
25
20
15
10
05
0
25
20
15
10
05
0
times10minus7
times10minus7
times10minus6
times10minus6
times10minus6
times10minus7
0 0100 50 100 150200 300 400 500 600 700
120591i
120591i 120591i 120591i
120591i 120591i
14
12
10
80
60
40
20
00 10 20 30 40 50 60 70
90
80
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
20
18
16
14
12
10
08
06
04
02
0 02 214 436 658 10
70
60
50
40
30
20
10
0
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
A = 001 N = 10M = 5 A = 005 N = 10M = 5 A = 01 N = 10M = 5
A = 05 N = 10M = 5 A = 10 N = 15M = 5 A = 50 N = 14M = 10
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Δ(120591
i)
Figure 3 Plots of error values Δ(119903119894) with 120573 = 50
4 Results and Discussions
In this section we illustrate the accuracy of the hybridfunctions method (HFM) by comparing the approximatesolutions previously obtained with the exact angular fre-quency 120596ex All the results obtained here are computed usingthe Intel Pentium 5 22GHz processor and using Maple 17with 64-digit precision
The exact angular frequency 120596ex of the Duffing-harmonic oscillator was found by Lim and Wu in [32] as
120596ex =120587
2(int1205872
0
119860 cos 119905119889119905radic1198602cos2119905 + ln[1 minus 1198602cos2119905(1 + 1198602)]
)
minus1
(50)
By using alternative form (38) and applying the harmonicbalance method (HBM) [13] Mickens [31] obtained the firstapproximate angular frequency
120596HBM = (31198602
4 + 31198602)
12
(51)
Ozis and Yildirim [35] obtained the angular frequency usingthe energy balance method (EBM) in the following form
120596EBM = (1 minus2
1198602ln(
1 + 1198602
1 + (11986022)))
12
(52)
Ganji et al in [34] obtained the same approximation as thatin (52) Using a single-term approximate solution 119906(119905) =
119860 cos(120596119905) to (36) and the Ritz procedure [40] Tiwari et al[41] obtained an approximate angular frequency as follows
120596Tiw = (1 +2
1198602(
1
radic1 + 1198602minus 1))
12
(53)
The computed results for the HFM frequency 120596HFM withexact frequency 120596ex [32] HBM frequency 120596HBM [31] EBMfrequency 120596EBM [35] and Tiwarirsquos frequency 120596Tiw [41] arelisted in Tables 1 and 2 Table 2 shows that the maximumpercentage error between 120596HFM and exact frequency 120596ex is0118 Comparison of the exact frequency 120596ex obtained by(50) with 120596HBM 120596EBM 120596Tiw and 120596HFM is shown in Figure 1for 0 le 119860 le 10 30 le 119860 le 325 50 le 119860 le 525 and80 le 119860 le 825
8 Journal of Difference Equations
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
From Figure 1 and Tables 1 and 2 it can be observedthat the hybrid functions method (HFM) yields excellentapproximate frequencies for both small and large amplitudes
Figures 2 and 3 give a comparison between the presentHFM results and the numerical results obtained by using the4th order Runge-Kutta method (RK4) with time step Δ119905 =
0001 where
Δ (120591119894) =
1003816100381610038161003816119906HFM (120591119894) minus 119906RK4 (120591119894)
= 119894120573Δ119905 119894 = 0 1 119905119891(120573Δ119905) and 120573 isin N It can be
seen from these figures that the solutions obtained by theproposed procedure are in good agreement with the RK4based solutions
5 Conclusion
In this paper we presented a numerical scheme based onhybrid block-pulse functions and Chebyshev cardinal func-tions for solving Duffing-harmonic oscillator This algorithmreduces the solution of Duffing-harmonic oscillator differen-tial equation to the solution of a system of algebraic equationsin matrix formThemerit of this method is that the system ofequations obtained for the solution does not need to considercollocation points this means that the system of equations isobtained directly A comparative study between HBM [31]EBM [35] Tiwarirsquos method [41] and the proposed methodwas discussed in Section 4 The obtained results showed thatthe HFM is accurate capable and effective technique for thesolution of the Duffing-harmonic oscillator Further researchcan concentrate on other strongly nonlinear oscillators andmore complicated cases
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are very grateful to both reviewers for carefullyreading the paper and for their comments and suggestionswhich have improved the paper
References
[1] A H Nayfeh Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
[2] A H Nayfeh and D T Mook Nonlinear Oscillations JohnWiley New York NY USA 1979
[3] N Krylov and N Bogolioubov Introduction to Nonlinear Me-chanics Princeton University Press Princeton NJ USA 1943
[4] N N Bogolioubov and Y A Mitropolsky Asymptotic Methodsin theTheory ofNonlinearOscillations Gordon andBreachNewYork NY USA 1961
[5] J He ldquoModified LindstedtndashPoincaremethods for some stronglynon-linear oscillations part II a new transformationrdquo Interna-tional Journal of Non-Linear Mechanics vol 37 no 2 pp 315ndash320 2002
[6] J H He ldquoModified straightforward expansionrdquoMeccanica vol34 no 4 pp 287ndash289 1999
[7] J He ldquoPreliminary report on the energy balance for nonlinearoscillationsrdquo Mechanics Research Communications vol 29 no2-3 pp 107ndash111 2002
[8] J H He ldquoIteration perturbation method for strongly nonlinearoscillationsrdquo Journal of Vibration and Control vol 7 no 5 pp631ndash642 2001
[9] J H He ldquoBookkeeping parameter in perturbation methodsrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 2 no 3 pp 257ndash264 2001
[10] J He ldquoSome asymptotic methods for strongly nonlinear equa-tionsrdquo International Journal of Modern Physics B vol 20 no 10pp 1141ndash1199 2006
[11] J He ldquoMax-min approach to nonlinear oscillatorsrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 9 no 2 pp 207ndash210 2008
[12] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[13] R E Mickens Oscillations in Planar Dynamics Systems WorldScientific Singapore 1996
[14] G Adomian Solving Frontier Problems of PhysicsThe Composi-tion Method Kluwer Academic Publishers Boston Mass USA1994
[15] J He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] M Heydari S M Hosseini G B Loghmani and D D GanjildquoSolution of strongly nonlinear oscillators using modified var-iational iteration methodrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 3 pp 33ndash45 2011
[17] M Heydari G B Loghmani and S M Hosseini ldquoAn improvedpiecewise variational iteration method for solving stronglynonlinear oscillatorsrdquoComputational and AppliedMathematics2014
[18] J He ldquoHomotopy perturbation techniquerdquo Computer Methodsin AppliedMechanics and Engineering vol 178 no 3-4 pp 257ndash262 1999
[19] S Momani and V S Erturk ldquoSolutions of non-linear oscillatorsby the modified differential transform methodrdquo Computers ampMathematics with Applications vol 55 no 4 pp 833ndash842 2008
[20] S Momani G H Erjaee and M H Alnasr ldquoThe modifiedhomotopy perturbation method for solving strongly nonlinearoscillatorsrdquo Computers amp Mathematics with Applications vol58 no 11-12 pp 2209ndash2220 2009
[21] M Razzaghi J Nazarzadeh and K Y Nikravesh ldquoA collocationmethod for optimal control of linear systems with inequalityconstraintsrdquo Mathematical Problems in Engineering vol 3 no6 pp 503ndash515 1998
[22] M Razzaghi and H Marzban ldquoDirect method for variationalproblems via hybrid of block-pulse and Chebyshev functionsrdquoMathematical Problems in Engineering vol 6 no 1 pp 85ndash972000
[23] M Razzaghi and H R Marzban ldquoA hybrid domain analysisfor systems with delays in state and controlrdquo MathematicalProblems in Engineering Theory Methods and Applications vol7 no 4 pp 337ndash353 2001
[24] H R Marzban and M Razzaghi ldquoSolution of time-varyingdelay systems by hybrid functionsrdquoMathematics andComputersin Simulation vol 64 no 6 pp 597ndash607 2004
Journal of Difference Equations 9
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
[25] KMaleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[26] H R Marzban and M Razzaghi ldquoSolution of multi-delaysystems using hybrid of block-pulse functions and Taylorseriesrdquo Journal of Sound and Vibration vol 292 no 3ndash5 pp954ndash963 2006
[27] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions andLagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009
[28] H R Marzban H R Tabrizidooz and M Razzaghi ldquoHybridfunctions for nonlinear initial-value problemswith applicationsto Lane-Emden type equationsrdquo Physics Letters A vol 372 no37 pp 5883ndash5886 2008
[29] M Razzaghi ldquoOptimization of time delay systems by hybridfunctionsrdquoOptimization and Engineering vol 10 no 3 pp 363ndash376 2009
[30] H R Marzban H R Tabrizidooz and M Razzaghi ldquoA com-posite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 3pp 1186ndash1194 2011
[31] R E Mickens ldquoMathematical and numerical study of theDuffing-harmonic oscillatorrdquo Journal of Sound and Vibrationvol 244 no 3 pp 563ndash567 2001
[32] C W Lim and B S Wu ldquoA new analytical approach to theDuffing-harmonic oscillatorrdquo Physics Letters A vol 311 no 4-5 pp 365ndash373 2003
[33] C W Lim B S Wu andW P Sun ldquoHigher accuracy analyticalapproximations to the Duffing-harmonic oscillatorrdquo Journal ofSound and Vibration vol 296 no 4-5 pp 1039ndash1045 2006
[34] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[35] T Ozis and A Yildirim ldquoDetermination of the frequency-amplitude relation for a Duffing-harmonic oscillator by theenergy balance methodrdquo Computers ampMathematics with Appli-cations vol 54 no 7-8 pp 1184ndash1187 2007
[36] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2000
[37] M Heydari Z Avazzadeh and G B Loghmani ldquoCheby-shev cardinal functions for solving Volterra-Fredholm integro-differential equations using operational matricesrdquo Iranian Jour-nal of Science and Technology A vol 36 no 1 pp 13ndash24 2012
[38] M Heydari G B Loghmani and S M Hosseini ldquoOperationalmatrices of Chebyshev cardinal functions and their applicationfor solving delay differential equations arising in electrodynam-ics with error estimationrdquo Applied Mathematical Modelling vol37 no 14-15 pp 7789ndash7809 2013
[40] H N Abramson ldquoNonlinear vibrationrdquo in Shock and VibrationHandbook C M Harris Ed McGraw-Hill New York NYUSA 1988
[41] S B Tiwari B Nageswara Rao N Shivakumar Swamy K S SaiandH RNataraja ldquoAnalytical study on aDuffing-harmonic os-cillatorrdquo Journal of Sound and Vibration vol 285 no 4-5 pp1217ndash1222 2005
