M. Y.Adamu and P.Ogenyi, Nonlinear Sci. Lett. A, Vol.8, No.2, pp.240-243, June 2017
Parameterized homotopy perturbation method M. Y. Adamu1, P. Ogenyi2
Department of Mathematical Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria Email:
[email protected] To cite this article M Y Adamu, P Ogenyi, Parameterized homotopy perturbation method, Nonlinear Sci. Lett. A, 8(2)(2017),240-243. Copyright 2017 Asian Academic Publisher Ltd. Journal Homepage: www.NonlinearScience.com
Abstract
This paper presents a modification of the homotopy perturbation method, the parameterized homotopy perturbation method, by introducing a new parameter, alpha, which can be optimally determined, and when it is equal to unity, it turns to its classic version. Duffing equation is used as an example to elucidate the solution process. Keywords: Homotopy perturbation method, Duffing oscillator, nonlinear differential equation. 1. Introduction
Homotopy perturbation method (HPM) is one of the most powerful method for nonlinear problems[1]. The method was first introduced by He in later 1990s [2,3], which is to combine the basic ideas of homotopy in topology and the classical perturbation techniques to continuously deform difficult problems into simple ones that are easily handled. It provides us with a convenient way of obtaining analytic or approximated solutions for wide variety of problems arising in different fields. The method has been developing very fast, the main progress was the homotopy perturbation method with two expanding parameters [4] and the homotopy perturbation method with an auxiliary term [5]. The homotopy parameter involved in the method plays an important role in various applications. In order to take full advantages of the perturbation methods, the homotopy parameter, which is used as an expanding parameter, should be as small as possible. In this paper, a parameterized homotopy perturbation method is proposed to shin new light on analytical solutions with an extremely high accuracy. 2. Basic Idea of the Parameterized Homotopy Perturbation Method For the basic idea we consider a general nonlinear oscillator: λ(u) − f(x) = 0, x ∈
with boundary conditions:
(0) =
∈ , (0) =
∈
(1) (2)
where is a differential operator, and are the known boundary conditions, ( ) is a known analytical function. Dividing λ(u) into two components, a linear part (L) and nonlinear part (N), we can rewrite Eq.(1) in the form 240
M. Y.Adamu and P.Ogenyi, Nonlinear Sci. Lett. A, Vol.8, No.2, pp.240-243, June 2017
( )+
( )– ( ) = 0
(3)
For a nonlinear oscillator, we construct a homotopy in the form: or
( , ) = (1 −
( , )=
)[ ( )– (
( )– (
)+
(
)] +
[λ(u) – ( ) −
]=
)+
( ( )– ( ) −
)=
1
(
1
−2 )
(4)
−2
(5)
where = , p∈ [0, ] is the embedding parameter, is the initial approximate of equation (1) which satisfies the boundary conditions (2). It is obvious that ( , 0) = ( )– ( ) = 0 (6) , = λ(u) – ( ) = 0 (7)
The transition of from 0 to is that of ( , ) from ( ) to ( ). This is generally a continuous deformation which is the central idea of homotopy in topology and ( )– ( ) is homotopic to λ(u)– ( ). Now we use p as an expanding parameter and assume the solution of equations (4) or (5) can be written in the form: = + (8) Now, setting = , the approximate solution of equation (3) becomes: = lim = + (9)
Consider the Duffing oscillator:
with initial conditions: Rewriting equation (10) we have: +
+
+
−ω
+
(0) = ,
+ω
=0
(0) = 0
+
[(1 − ω ) +
Setting the initial solution in the form
and ignoring the higher order of p, we have: or
+ω
+ [(1 − ω )
+ [(1 − ω ) + 3 4 No secular term in requires +ω
Considering where
]= (
−
= − 1
+
−
− 1] cos ω +
1+3 4
]=(
−
1
,
1
1 1 ( − 1)
1
−2 )
− 2)
is a free parameter. We assume it is a function of , it is obvious that 241
∈ 0,
(13) (14)
cos 3ω = (
4
−1 =(
(12)
−2 )
=
(1 − ω ) + 3 4 = 1, Eq. (17) becomes =
(11)
−
Constructing a homotopy of equation (12), we have: +ω
(10)
(15) 1
−2 )
(16) (17) (18)
M. Y.Adamu and P.Ogenyi, Nonlinear Sci. Lett. A, Vol.8, No.2, pp.240-243, June 2017
= 1 when = 0.
(19)
When tends to infinity, we have the following exact frequency[5,6]: 2 = 6.743 That means when tends to infinity, we have 1+3 4
−
1
1−
1
=
(20)
2 6.743
(21)
According to Eqs. (19) and (21), we identify in the form: 11.367 = ( + 1)11.367 − where = + 1.
(22)
Table 1 gives the values of for different values of . Table 2 compares of our result with exact frequency, showing remarkable accuracy. Table1. the values of 0.01 0.5002975
for different values of
0.1 0.50299197
1 0.5316228
when
10 1.2340693
= 1.
100 -0.101043
10000 -0.0008419827
1 x 106 -0.000008405814
Table2. Comparison among our result, exact frequency, and that by the classic homotopy perturbation method when = 1.
ex
0.01 0.09321836 0.09321840 1.00374299
0.1 0.29478235 0.29478250 1.03682208
1 0.93218364 0.93218414 1.32287565
10 2.94782350 2.94782508 2.91547594
100 9.32183640 9.32184142 8.71779788
10000 93.2183640 93.2184142 86.6083136
1 x 106 932.183640 932.184142 866.025981
3. Conclusion In this paper, we successfully solved Duffing oscillator using a new modification of the homotopy parameter. The new technique is proved to be powerful and efficient. The results depend on the values of the alpha, which can be identified optimally. References
[1] Babolian, E., Azizi, A., & Saeidian, J., Some Notes on Using the Homotopy Perturbation Method for Solving Time-Dependent Differential Equations. Mathematical and Computer Modelling, 50(2009), 213-224. [2] He, J.H., Homotopy Perturbation Technique. Computational Methods in Applied Mechanics and Engineering, 178(1999), 257-262. [3] He. J. H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Nonlinear Mechanics, 35(1) (2000), 37-43. [4] He, J.H., A note on the homotopy perturbation method, Thermal Science, 14(2) (2010),565-568. [5] He, J.H., Homotopy perturbation method with two expanding parameters, Indian Journal of Physics, 88(2) (2014), 193-196. 242
M. Y.Adamu and P.Ogenyi, Nonlinear Sci. Lett. A, Vol.8, No.2, pp.240-243, June 2017
[6] He, J.H., Homotopy Perturbation Method with an Auxiliary Term, Abstract and Applied Analysis, 2012, Article Number 857612. Biographical Notes
M. Y. Adamu received the B.Sc. degree in Mathematics in 1997 from University of Maiduguri, Nigeria, M.Sc. and Ph.D degrees in Mathematics in 2005 and 2013 respectively from Abubakar Tafawa Balewa University (ATBU), Bauchi, Nigeria. His current research mainly covers analytical-numerical methods for nonlinear differential equations. Dr. M. Y. Adamu is author or coauthor of 23 research articles published in both local and international journals.
P. Ogenyi. is a graduate of Abubakar Tafawa Balewa University(ATBU), Bauchi, Nigeria. He received the B.Tech. Degree in Mathematics in 2016. He is currently undergoing National Youth Service in Abia State. P. Ogenyi is aspiring to pursue MSc. in Applied Mathematics upon completion of the service. He coauthored 1 research article with Dr. M.Y Adamu. P. Ogenyi represented Abubakar Tafawa Balewa University for National Mathematics Competition for University Student held in Abuja in 2015. He has received some merit awards both from National Association of Mathematical Sciences Students (NAMSS) and Fellowship of Christian Science Students (FCSS) ATBU chapter. Competing Interests
All authors declare that there are no competing interests regarding the publication of this paper.
Authors’ Contributions
The idea of the parameterized homotopy perturbation method was originally suggested by Adamu; Both authors contributed equally to calculation.
243
