11485


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 21, No 1, 2023, pp. 21 - 29  

https://doi.org/10.22190/FUME230108006H 

© 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

A GOOD INITIAL GUESS FOR APPROXIMATING  

NONLINEAR OSCILLATORS  

BY THE HOMOTOPY PERTURBATION METHOD  

Ji-Huan He1,2,3, Chun-Hui He4,5, Abdulrahman Ali Alsolami5 

1National Engineering Laboratory for Modern Silk,  

College of Textile and Clothing Engineering, Soochow University, Suzhou, China 
2School of Mathematics and Information Science,  

Henan Polytechnic University, Jiaozuo, China  
3School of Science, Xi’an University of Architecture and Technology, Xi’an, China 

4School of Mathematics, China University of Mining and Technology,  

Xuzhou, Jiangsu, China  
5Department of Mathematics, Faculty of Science, King Abdulaziz University,  

Jeddah, Saudi Arabia 

Abstract. A good initial guess and an appropriate homotopy equation are two main 

factors in applications of the homotopy perturbation method. For a nonlinear oscillator, 

a cosine function is used in an initial guess. This article recommends a general approach 

to construction of the initial guess and the homotopy equation. Duffing oscillator is 

adopted as an example to elucidate the effectiveness of the method. 

Key words: Homotopy perturbation method, Nonlinear oscillator, Periodic solution 

1. INTRODUCTION  

The nonlinear vibration theory has triggered skyrocketing interest in both nonlinear 

science and engineering, from vibration isolators [1] to nano materials [2] and 

micro-electromechanical systems [3,4], and the homotopy perturbation method [5] has laid 

the foundation for fast and accurate insight into the frequency-amplitude relation of a 

nonlinear oscillator, which occurs anywhere in engineering and science [6]. The traditional 

perturbation method [6] is widely used for this purpose, however, it is only valid for the 

 
Received: January 08, 2023 / Accepted February 05, 2023  

Corresponding author: Ji-Huan He and Chun-Hui He 

Affiliation, National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, 
Soochow University, Suzhou, P. R. China 

E-mails: Hejihuan@suda.edu.cn  (J.H. He), mathew_he@yahoo.com (C.H. He) 

mailto:Hejihuan@suda.edu.cn
mailto:mathew_he@yahoo.com


22 JI.H. HE, C.H. HE, A.A. ALSOLAMI 

weak nonlinearity. Other effective analytical methods include the variational iteration 

method [7,8,9] and the variational approach [10,11,12].  

The homotopy perturbation method [5] is a universal method for nonlinear vibration 

systems, it has been considered as a strongly promising and unprecedented method for 

nonlinear problems [13,14], and many reliable modifications were recommended in 

literature, among which Li-He’s modification [15,16] is much attractive.  

The homotopy perturbation method is to decompose a nonlinear equation to infinite 

linear equations where Laplace transform can be applied, this modification is called as 

He-Laplace method [17]. The parameterized homotopy perturbation method [18], and the 

couple of the homotopy perturbation method with Kashuri Fundo transform [19] or the 

Lindstedt-Poincare technology [20] has also been caught much attention. The homotopy 

perturbation method is extremely effective for fractional calculus [21-24], machine 

learning [25-28] and imaging process [29-33]. Though the method is almost matured, there 

is still much space to further improvement. 

2. HOMOTOPY PERTURBATION METHOD  

The homotopy perturbation method [5] is a powerful tool to nonlinear vibration 

systems. We consider a nonlinear vibration equation in the form 

 ( ) 0u A u+ =  (1) 

where A is a nonlinear function of u, and A/u>0. 

In order to elucidate the solving process of the homotopy perturbation method, we can 

construct the following universal homotopy equation  

  2 2( ) ( ) 0 + + − + =u t u p u A u   (2) 

where ω is the frequency, p is the homotopy parameter. The homotopy perturbation 

method is to deform Eq. (2) gradually from p=0 to p=1. When p tends to zero, Eq. (2) 

results in a linearized oscillator, while when p tends to1, Eq. (2) turns to be the original one.  

The most important two factors of the homotopy perturbation method are: 1) how to 

choose a good initial guess with possible unknown parameters; and 2) how to establish an 

appropriate homotopy equation. All iteration methods are sensitive to the initial guess. A 

good choice of the initial guess leads to a fast convergence, while an inappropriate choice 

might result in a wrong result [34].  

To show the importance of the homotopy equation in the solving process, if we 

construct the following homotopy equation for a nonlinear oscillator  

  2 2( ) ( ) 0 − + + =u t u p u A u   (3) 

we need an infinite iteration to obtain an approximate solution converging extremely 

slowly to the exact solution. This is because when p=0, Eq. (3) has no any property of 

oscillation, it should be emphasized that the initial guess must have the basic properties of 

the solution, the classic homotopy perturbation method always begins with  

 
0
( ) cos=u t A t  (4) 



 A Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method 23 

In this paper, we improve the homotopy equation instead of Eq. (2) and choose a better 

initial guess. Instead of Eq. (4), this paper uses the following initial guess  

 
0

0

( ) cos(2 1)
=

= +
N

i

i

u t a i t  (5) 

where  ai (i=0~N) are the constants satisfying the following identity  

 
0=

=
N

i

i

a A  (6) 

The initial guess given in Eq. (4), with an unknown frequency, is widely used in the 

homotopy perturbation method, while Eq. (5) contains more unknown constants and 

provides a more flexible approach to an accurate identification of the frequency. 

3. AN EFFECTIVE IMPROVEMENT OF THE HOMOTOPY PERTURBATION METHOD  

In this section, we adopt the well-known Duffing oscillator as an example to elucidate 

the solving process 

 
3

0u u u+ + =  (7) 

with initial conditions 

 (0) , (0) 0= =u A u  (8) 

The Duffing oscillator is always used as a good paradigm to elucidate the effectiveness 

and reliability of a method [35-39].  

We choose an initial guess in the form  

 
0
( ) cos cos 3= +u t a t b t   (9) 

Eq. (9) is the exact solution of the following linear oscillator with a forcing term 

 
2 2

0 0
( ) 8 cos 3 0 + + =u t u b t   , (0) =u A  and (0) 0 =u  (10) 

where a and b are unknown parameters. According to the initial conditions, the parameters 

a and b should satisfy the following identity  

 + =a b A  (11) 

Accordingly we recommend the following homotopy equation  

  2 2 2 3 2( ) 8 cos 3 (1 ) 8 cos 3 0 + + + − + − =u t u b t p u u b t        (12) 

It is obvious that when p=0, Eq. (12) becomes Eq. (10), whose solution is Eq. (9); when 

p=1, Eq. (9) becomes the original one. For b=0, Eq. (12) is the standard homotopy 

equation.  

According to the homotopy perturbation method, the solution is expanded as  

 
2

0 1 2
= + + +u u pu p u  (13) 



24 JI.H. HE, C.H. HE, A.A. ALSOLAMI 

Eq. (12) becomes  

 
 

2 2 2 2

0 1 2 0 1 2

2 2 2 3 2

0 1 2 0 1 2

( ) 8 cos 3

(1 )( ) ( ) 8 cos 3 0

  + + + + + + + +

+ − + + + + + + + − =

u pu p u u pu p u b t

p u pu p u u pu p u b t

  

   
 (14) 

Proceeding the standard solving process required by the perturbation method [5,6], we 

can obtain a series of linear differential equations. The first two equations are  

 
2 2

0 0
( ) 8 cos 3 0 + + =u t u b t   , 

0
(0) =u A  and 

0
(0) 0 =u  (15) 

 
2 2 3 2

1 1 0 0
( ) (1 ) 8 cos 3 0 + + − + − =u t u u u b t     ,

1
(0) 0=u  and 

1
(0) 0 =u  (16) 

The solution of Eq. (15) is Eq. (9). Using this result, Eq. (16) becomes 

 

2 2

1 1

3 2

( ) (1 )( cos cos 3 )

( cos cos 3 ) 8 cos 3 0

 + + − +

+ + − =

u t u a t b t

a t b t b t

   

    
 (17) 

Simplifying Eq. (17) yields the following equation  

 

2 2 3 2 2

1 1

3 2 3 2

2 2 2 3

3 3 3
( ) (1 ) ( ) cos

4 4 2

1 3 3
( ) 8 cos 3
4 2 4

3 3 3 1
( ) cos 5 cos 7 cos 9
4 4 4 4

 
 + + − + + + 

 

 
+ + + − + + 
 

+ + + +

u t u a a a b ab t

a a b b b b t

a b ab t ab t b t

   

  

     

 (18) 

No term of  tcosωt  should be involved in u1  for a periodic solution, so the coefficient of   

cosωt should be zero: 

 
2 3 2 23 3 3

(1 ) ( ) 0
4 4 2

− + + + =a a a b ab   (19) 

or 
2 2 2 2 2 23 3 3 3 3

1 ( ) 1 ( 2 ) 1 ( 2 )
4 4 2 4 4

= + + + = + + = + + −a ab b aA b A b bA     (20) 

From Eq. (7) we have  

 
3 2

(0) (0) (0) (1 )u u u A A  = − − = − +  (21) 

while Eq. (9) predicts 

 
2

(0) ( 9 )u a b  = − +  (22) 

We, therefore, have  

 
2 2

( 9 ) (1 )+ = +a b A A   (23) 

or 
2 2

2 (1 ) (1 )

9 8

+ +
= =

+ +

A A A A

a b A b

 
  (24) 



 A Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method 25 

For given A and ε, we can obtain the approximate frequency easily by solving Eqs. (20) 

and (24) simultaneously. 

4. DISCUSSION 

In case ε<<1 , we can determine approximately the value of b from the following 

equation by the perturbation method 

 
2

2 23 (1 )
1 ( 2 )

4 8

+
+ + − =

+

A A
A b bA

A b


  (25) 

The perturbation solution for b reads         

 
31

32
=b A  (26) 

We, therefore, obtain  

 
2

2 2

2

1 3
1

1 4
1

4

+
= = +

+

A
A

A


 



 (27) 

Eq. (27) is just same as that solved by the classic homotopy perturbation method, and it 

is valid for ε<<1. Considering our small assumption of the parameter ε, we cannot fail to 

appreciate its harmony and intoxicating formula valid for all values of  ε>0 . 

Considering another case when ε tends to infinity, Eq. (25) becomes 

 
3

2 23
( 2 )

4 8
+ − =

+

A
A b bA

A b


  (28) 

Solving for b from Eq. (28) results in  

 0.04943=b A  (29) 

As a result, we have  

 
2

2 2 2(1 ) 1
0.7166

8 1.3955

+
= = =

+

A A
A A

A b


    (30) 

or 
1/ 2

0.8465= A   (31) 

while the exact frequency when ε tends to infinity is [40]:  

 
2 2

/ 2
2 0.5

0

= 0.8472
2 (1 0.5sin )

ex
A A

t dt



  

−
= =

−
 (32) 



26 JI.H. HE, C.H. HE, A.A. ALSOLAMI 

 

Fig. 1 Comparison between the exact solution (continuous line) and the approximate one 

(discontinuous line) for different cases: (a) A=1 and ε =1; (b) A=1 and  ε=100; (c) 

A=1 and  ε=10000. 



 A Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method 27 

The relative error is 0.08%. Fig.1 gives the comparison between the approximate 

solution and the exact one for different cases, showing an extremely high accuracy of the 

approximate solution from small to large ones. 

5. CONCLUSION  

We looked into the effect of the initial guess on the solution accuracy. Obviously, the 

obtained solution has a better accuracy than that by the classic homotopy perturbation 

[41,42], the approximate solutions obtained by other analytical methods, especially the 

frequency formulation [43, 44], can also be used as the initial guess, this idea can lead to a 

new modification of the homotopy perturbation method, and we will discuss it in a 

forthcoming article.  

As a conclusion, we give a new way to construction of a suitable homotopy equation for 

accurate estimate the periodic solution of a nonlinear vibration equation regardless of its 

nonlinearity strength. Though we use the method to solve nonlinear oscillators, it is also 

valid for other nonlinear problems. 

REFERENCES  

1. Ibrahim, R.A., 2008, Recent advances in nonlinear passive vibration isolators, Journal of Sound and 
Vibration, 314(3-5), pp. 371-452. 

2. Ke, L.L., Yang, J., Kitipornchai, S., 2010 , Nonlinear free vibration of functionally graded carbon 
nanotube-reinforced composite beams, Composite Structures, 92(3) , pp. 676-683. 

3. Yang, Q., 2023, A mathematical control for the pseudo-pull-in stability arising in a 
micro-electromechanical system, Journal of Low Frequency Noise, Vibration and Active Control, doi: 

10.1177/14613484221133603 
4. Feng, G.Q., Niu, J.Y., 2023, The analysis for the dynamic pull-in of a micro-electromechanical system, 

Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484221145588 
5. He, J.H., 1999, Homotopy perturbation technique, Computer Methods in Applied Mechanics and 

Engineering, 178(3-4), pp. 257-262. 

6. Nayfeh, A.H., 1973, Perturbation methods, New York, Wiley & Sons. 
7. Wang, S.Q., He, J.H., 2007, Variational iteration method for solving integro-differential equations, 

Physics letters A, 367(3), pp. 188-191. 
8. Odibat, Z.M., Momani, S., 2006, Application of variational iteration method to Nonlinear differential 

equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), 

pp. 27-34. 
9. Deng, S.X., Ge, X.X., 2022, The variational iteration method for Whitham-Broer-Kaup system with local 

fractional derivatives, Thermal Science, 26(3), pp. 2419-2426. 
10. Wang, S.Q., 2009, A variational approach to nonlinear two-point boundary value problems, Computers & 

Mathematics with Applications, 58(11), pp. 2452-2455. 

11. Shen, Y.Y., Huang, X.X., Kwak, K., et al., 2016, Subcarrier-pairing-based resource optimization for 
OFDM wireless powered relay transmissions with time switching scheme, IEEE Transactions on Signal 

Processing, 65(5), pp. 1130-1145. 
12. Khan, Y., Akbarzade, M., Kargar, A., 2012, Coupling of homotopy and the variational approach for a 

conservative oscillator with strong odd-nonlinearity, Scientia Iranica, 19(3), pp. 417-422. 

13. Biazar, J., Ghanbari, B., Porshokouhi, M.G., Porshokouhi, M.G., 2011, He's homotopy perturbation 
method: A strongly promising method for solving non-linear systems of the mixed Volterra-Fredholm 

integral equations, Computers & Mathematics with Applications, 61(4) , pp. 1016-1023. 
14. Saranya, K., Mohan, V., Kizek, R., Fernandez, C., Rajendran, L., 2018, Unprecedented homotopy 

perturbation method for solving nonlinear equations in the enzymatic reaction of glucose in a spherical 

matrix, Bioprocess and Biosystems Engineering, 41(2) , pp. 281-294. 



28 JI.H. HE, C.H. HE, A.A. ALSOLAMI 

15. Anjum, N., He, J.H., Ain, Q. T., Tian, D., 2021, Li-He’s modified homotopy perturbation method for 
doubly-clamped electrically actuated microbeams-based microelectromechanical system, Facta 

Universitatis Series Mechanical Engineering, 19(4), pp. 601-612. 
16. Ji, Q.P., Wang, J., Lu, L.X., Ge, C.F., 2021, Li-He's modified homotopy perturbation method coupled with 

the energy method for the dropping shock response of a tangent nonlinear packaging system, Journal of 

Low Frequency Noise, Vibration and Active Control,  40(2) , pp.675-682. 
17. Nadeem, M., Li, F.Q.,2019, He-Laplace method for nonlinear vibration systems and nonlinear wave 

equations Journal of Low Frequency Noise, Vibration and Active Control,  38(3-4) , pp. 1060-1074. 
18. Adamu, M.Y., Ogenyi, P., 2018, New approach to parameterized homotopy perturbation method, Thermal 

Science, 22(4), pp. 1865-1870. 

19. Peker, H.A., Cuha, F.A., 2022, Application of Kashuri Fundo transform  and homotopy perturbation 
method for fractional heat transfer and porous media equations, Thermal Science, 26(4), pp. 2877-2884. 

20. Alam, M.S., Sharif, N., Molla, M.H.U., 2022, Combination of modified Lindstedt-Poincare and homotopy 
perturbation methods, Journal of Low Frequency Noise, Vibration and Active Control, doi: 

10.1177/14613484221148049 

21. Li, Z.Y., Wang, M.C., Wang, Y.L., 2022, Solving a class of variable order nonlinear fractional integral 
differential equations by using reproducing kernel function, AIMS Mathematics, 7(7), pp. 12935-12951.         

22. Li, Z.Y., Chen, Q.T., Wang, Y.L., Li, X.Y., 2022, Solving two-sided fractional super-diffusive partial 
differential equations with variable coefficients in a class of new reproducing kernel spaces, Fractal and 
Fractional, 6(9), 492.    

23. Wang, K.L., 2023, Novel analytical approach to modified fractal gas dynamics model with the variable 
coefficients, ZAMM, https://doi.org/10.1002/zamm.202100391. 

24. Wang, K.L., Wei, C.F., 2023, Fractal soliton solutions for the fractal-fractional shallow water wave 
equation arising in ocean engineering, Alexandria Engineering Journal, 65, pp. 859-865. 

25. Wang, S.Q., Wang, X.Y., Shen, Y.Y., et. al, 2022, An ensemble-based densely-connected deep learning 
system for assessment of skeletal maturity, IEEE Transactions on Systems, Man, and Cybernetics: 
Systems, 52(1), pp. 426-437. 

26. Wang, S.Q., Shen, Y.Y., Shi, C.H., et. al, 2018, Skeletal maturity recognition using a fully automated 
system with convolutional neural networks, IEEE Access, 6, pp. 29979-29993. 

27. Hu, S.Y.,  Yuan, J.P., Wang, S.Q., 2019, Cross-modality synthesis from MRI to PET using adversarial 
U-net with different normalization, 2019 International Conference on Medical Imaging Physics and 
Engineering, doi: 10.1109/ICMIPE47306.2019.9098219 

28. Wu, K., Shen, Y.Y., Wang, S.Q., 2018, 3D convolutional neural network for regional precipitation 
nowcasting, Journal of Image and Signal Processing, 7(4), pp. 200-212. 

29. Wang, S.Q., Li, X., Cui, J.L., et al., 2015, Prediction of myelopathic level in cervical spondylotic 
myelopathy using diffusion tensor imaging, Journal of Magnetic Resonance Imaging, 41(6), pp. 
1682-1688. 

30. Yu, W., Lei, B.Y., Shen, Y.Y., et. al, 2021, Morphological feature visualization of Alzheimer's disease via 
Multidirectional Perception GAN, IEEE Transactions on Neural Networks and Learning Systems, doi: 
10.1109/TNNLS.2021.3118369 

31. Hu, S.Y., Lei, B.Y., Wang, S.Q., et. al, 2021, Bidirectional mapping generative adversarial networks for 
brain MR to PET synthesis, IEEE Transactions on Medical Imaging, 41(1), pp. 145-157. 

32. Yu, W., Lei, B.Y., Shen, Y.Y., et. al, 2021, Morphological feature visualization of Alzheimer's disease via 
Multidirectional Perception GAN, IEEE Transactions on Neural Networks and Learning Systems, doi: 
10.1109/TNNLS.2021.3118369 

33. Hu, S.Y., Yu, W., Chen, Z., et al., 2020, Medical image reconstruction using generative adversarial 
network for Alzheimer disease assessment with class-imbalance problem, IEEE 6th International 

Conference on Computer and Communications (ICCC), doi: 10.1109/ICCC51575.2020.9344912 

34. Machina, A.,  Edwards, R., van den Driessche, P., 2013, Sensitive dependence on initial conditions in gene 
networks, Chaos, 23(2), 025101. 

35. El-Dib, Y. O., Elgazery, N. S., Mady, A. A., Alyousef, H.A., 2022, On the modeling of a parametric 
cubic–quintic nonconservative Duffing oscillator via the modified homotopy perturbation method, 

Zeitschrift für Naturforschung A, 77(5), pp. 475-486. 

36. Vahidi, A.R.,  Babolian, E.,  Azimzadeh, Z., 2018, An Improvement to the Homotopy Perturbation Method 
for Solving Nonlinear Duffing's Equations, Bulletin of the Malaysian Mathematical Sciences Society, 

41(2) , pp. 1105-1117. 
37. Azimzadeh, Z., Vahidi, A.R.,  Babolian, E., 2012, Exact solutions for non-linear Duffing's equations by 

He's homotopy perturbation method, Indian Journal of Physics, 86(8) , pp. 721-726. 



 A Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method 29 

38. El-Dib, Y.O., 2023, Properties of complex damping Helmholtz-Duffing oscillator arising in fluid 
mechanics, Journal of Low Frequency Noise, Vibration and Active Control, doi: 

10.1177/14613484221138560 
39. Aljahdaly, N.H.,  Alharbi, M.A ., 2022, Semi-analytical solution of non-homogeneous Duffing oscillator 

equation by the Pade differential transformation algorithm, Journal of Low Frequency Noise, Vibration 

and Active Control, 41(4), pp.1454-1465. 
40. Belendez, T., Belendez, F.J., Martinez, C., et al., 2016, Exact solution for the unforced Duffing oscillator 

with cubic and quintic nonlinearities, Nonlinear Dynamics, 86(3), pp. 1687–1700,  
41. Aljahdaly, N.H., Shah, R.S., Naeem, M., Arefin, M.A., 2022, A Comparative Analysis of Fractional 

Space-Time Advection-Dispersion Equation via Semi-Analytical Methods, Journal of Function Spaces, 

2022, 4856002, doi:10.1155/2022/4856002 

42. Dubey, V.P., Kumar, D., Singh, J., Alshehri, A.M., Dubey, S., 2022, Analysis of local fractional 
Klein-Gordon equations arising in relativistic fractal quantum mechanics, Waves in Random and 
Complex Media, 2022, doi:10.1080/17455030.2022.2112993 

43. Qie, N., Houa, W.-F., He, J.-H., 2021, The fastest insight into the large amplitude vibration of a string, Reports in 
Mechanical Engineering, 2(1), pp. 1-5. 

44. Tian, Y., 2022, Frequency formula for a class of fractal vibration system, Reports in Mechanical Engineering, 3(1), 
pp. 55-61.