In this paper, a Green’s function based iterative algorithm is proposed to solve strong nonlinear oscillators. The method’s essential part is based on finding an appropriate Green’s function that will be incorporated into a linear integral operator. An application of fixed point iteration schemes such as Picard’s or Mann’s will generate an iterative formula that gives reliable approximations to the true periodic solutions that characterize these kinds of equations. The applicability and stability of the method will be tested through numerical examples. Since exact solutions to these equations usually do not exist, the proposed method will be tested against other popular numerical methods such as the modified homotopy perturbation, the modified differential transformation, and the fourth-order Runge–Kutta methods.

References

References
1.
Nayfeh
,
A.
,
1973
,
Perturbation Methods
,
Wiley
,
New York
.
2.
Nayfeh
,
A.
, and
Mook
,
D.
,
1979
,
Nonlinear Oscillations
,
Wiley
,
New York
.
3.
Bogolioubov
,
N. N.
, and
Mitropolsky
,
Y. A.
,
1961
,
Asymptotic Methods in the Theory of Nonlinear Oscillations
,
Gordon and Breach
,
New York
.
4.
He
,
J. H.
,
2000
, “
A Coupling Method of a Homotopy Technique and Perturbation Technique for Nonlinear Problems
,”
Int. J. Non-Linear Mech.
,
35
(
1
), pp.
37
43
.
5.
Momani
,
S.
,
Erjaee
,
G. H.
, and
Alnasr
,
M. H.
,
2009
, “
The Modified Homotopy Perturbation Method for Solving Strongly Nonlinear Oscillators
,”
Comput. Math. Appl.
,
58
(11–12), pp.
2209
2220
.
6.
Momani
,
S.
, and
Ertürk
,
V. S.
,
2008
, “
Solutions of Non-Linear Oscillators by the Modified Differential Transform Method
,”
Comput. Math. Appl.
,
55
(
4
), pp.
833
842
.
7.
Ghosh
,
S.
,
Roy
,
A.
, and
Roy
,
D.
,
2007
, “
An Adaptation of Adomian Decomposition for Numeric–Analytic Integration of Strongly Nonlinear and Chaotic Oscillators
,”
Comput. Methods Appl. Mech. Eng.
,
196
(4–6), pp.
1133
1153
.
8.
Cai
,
J.
,
Wu
,
X.
, and
Li
,
Y. P.
,
2005
, “
An Equivalent Nonlinearization Method for Strongly Nonlinear Oscillations
,”
Mech. Res. Commun.
,
32
(
5
), pp.
553
560
.
9.
Ju
,
P.
, and
Xue
,
X.
,
2014
, “
Global Residue Harmonic Balance Method to Periodic Solutions of a Class of Strongly Nonlinear Oscillators
,”
Appl. Math. Model.
,
38
(
24
), pp.
6144
6152
.
10.
Lakrad
,
F.
,
2002
, “
Periodic Solutions of Strongly Non-Linear Oscillators by the Multiple Scales Method
,”
Sound Vib.
,
258
(
4
), pp.
677
700
.
11.
Khuri
,
S. A.
, and
Sayfy
,
A.
,
2015
, “
A Novel Fixed Point Scheme: Proper Setting of Variation Iteration Method for BVPs
,”
Appl. Math. Lett.
,
48
, pp.
75
84
.
12.
Cveticanin
,
L.
,
2006
, “
Homotopy Perturbation Method for Pure Nonlinear Differential Equations
,”
Chaos Solitons Fractals
,
30
(
5
), pp.
1221
1230
.
13.
Cveticanin
,
L.
,
2004
, “
Vibrations of the Nonlinear Oscillator With Quadratic Nonlinearity
,”
Physica A
,
341
, pp.
123
135
.
14.
Baldwin
,
D.
,
Goktab
,
U.
,
Herman
,
W.
,
Hong
,
L.
,
Martino
,
R. S.
, and
Miller
,
J. C.
,
2004
, “
Symbolic Computation of Exact Solutions Expressible in Hyperbolic and Elliptic Functions for Nonlinear PDEs
,”
J. Symbolic Comput.
,
37
(
6
), pp.
669
705
.
15.
Heydari
,
M.
,
Hosseini
,
S. M.
,
Loghmani
,
G. B.
, and
Ganji
,
D.
,
2011
, “
Solution of Strongly Nonlinear Oscillators Using Modified Variational Iterational Method
,”
Int. J. Nonlinear Dyn. Eng. Sci.
,
3
(
1
), pp.
33
45
.
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