An analytical technique for nonlinear oscillators based on the higher order odd polynomial functions

Abstract

An analytical technique based on the certain odd polynomial functions is introduced for solving nonlinear oscillatory problems, applicable to both autonomous and non-autonomous cases. A set of algebraic equations are found when the proposed technique is applied. The algebraic equations are linear and relatively straightforward for determining the unknown coefficients associated with the solution. An excellent agreement is found between approximated and numerical solutions, which prove that the technique is very efficient and easy to implement. The technique allows to obtain any desired order of approximate solutions. The results are shown in figures.

Keywords

analytical technique nonlinear oscillations approximate solution differential transform method

Introduction

Nonlinear oscillators have a great importance in the area of engineering applications. Many researchers continuously invest their time and effort to explore the behaviour of nonlinear damped forced oscillators. Analytical techniques, numerical techniques, or a combination of both are usually employed to determine the solutions. The classical perturbation methods^1–4 were widely used to find periodic solutions of weakly nonlinear oscillators (e.g. $\ddot{x} + x + ε f (x) = 0$ , where $ε ≪ 1$ is a nonlinear parameter). Later, several authors modified to classical perturbation methods^5–9 to solve strongly nonlinear problems where $ε = Ο (1)$ . By considering a finite Taylor series as an approximate solution, a semi-analytical technique, namely, the differential transform method (DTM)^10–15 was used for obtaining periodic solution of strongly nonlinear oscillators. However, a lot of terms are essential to obtain a desired solutions but it is a laborious task. Recently, a new analytical technique, similar to DTM^16,17 has been developed to solve strongly nonlinear autonomous oscillators with large oscillations.

Burton and Hamdan^18,19 made a modification on Lindstedt-Poincare method to study conservative system of strongly nonlinear oscillators with weak harmonic excitation. After that, the modified multiple time scale method,²⁰ the modified harmonic balance method²¹ were used to obtain periodic solutions of strongly nonlinear damped forced oscillators. However, in the cases of large excitations, the above methods do not yield satisfactory results. A modified Lindstedt-Poincaré method was introduced by Cheung et al.²² for certain the following strongly nonlinear oscillator

\ddot{x} + ω_{0}^{2} x + ε f (x, \dot{x}) + ε g (ν t) = 0, x (0) = 0, \dot{x} (0) = b,

(1.1)

where

ε ≪ ω_{0}^{2}

ω_{0}

is known as unperturbed frequency,

ε

is a constant,

f

is a nonlinear function satisfying

f (- x, - \dot{x}) = - f (x, \dot{x})

and

g

represents a periodic forcing term, typically a trigonometric sine or cosine function and

ν

is a frequency of external excitation. In the recent past, a new modified Lindstedt-Poincaré method was introduced by Alam et al.²³ for solving nonlinear damped forced oscillations equation (1.1). The obtained solutions are valid for large excitations. But the mathematical manipulations are laborious. In addition, several authors are continuously giving their knowledge and effort to significantly analyse the nonlinear damped force oscillators.^24–26 In this paper, an analytical technique based on the certain odd polynomial functions is introduced to solve nonlinear oscillatory problems whether it is autonomous or non-autonomous. The technique is straightforward, and the process of determining the solutions is relatively simple.

The method

Introducing a new variable $τ = ν t - θ$ , equation (1.1) is transformed to

ν^{2} x^{″} + ω_{0}^{2} x + ε f (x, ν x^{'}) + ε g (τ + θ) = 0, x (0) = 0, x^{'} (0) = \frac{b}{ν} = β

(2.1)

where the primes denote the differentiation with respect to

τ

. It is noted that a periodic solution is obtained with a phase difference

θ

.²³ In accordance to the proposed technique

x

and

θ

are expanded in powers of the polynomial function,

ψ_{n} (τ)

x (τ) = B_{1} ψ_{n} (τ) + B_{3} {ψ_{n}}^{3} (τ) + B_{5} {ψ_{n}}^{5} (τ) + \dots,

(2.2)

and

θ = θ_{0} + θ_{2} {ψ_{n}}^{2} (τ) + θ_{4} {ψ_{n}}^{4} (τ) + \dots,

(2.3)

where

ψ_{n} (τ)

is obtained by

ψ_{i} (τ) = \frac{{ψ_{i}}^{*}}{{ψ_{i}}^{*} (\frac{π}{2})}, {ψ_{i}}^{*} = \int_{π / 2 - x}^{π / 2} {ψ_{i - 1}}^{*} d τ, i = 1, 2, \dots, n; ψ_{0} = 1

(2.4)

and

B_{1}

B_{3}

B_{5}, \dots

and

θ_{0}

θ_{2}

θ_{4}, \dots

are unknown constants depend on

ν

ε

and

ω_{0}

. To get the better approximate solutions, the value of

n

is considered more than

3

. In this paper,

n

is considered to be

7 .

Differentiating $x (τ)$ twice with respect to $τ$ , the following results are found (Appendix A):

x^{'} (τ) = (1 - \frac{300 {ψ_{n}}^{2}}{61 π^{2}} - \frac{45360 {ψ_{n}}^{4}}{3721 π^{4}}) (B_{1} + 3 B_{3} {ψ_{n}}^{2} + 5 B_{5} {ψ_{n}}^{4} + \dots),

(2.5)

\begin{array}{l} x^{″} (τ) = {(1 - \frac{300 {ψ_{n}}^{2}}{61 π^{2}} - \frac{45360 {ψ_{n}}^{4}}{3721 π^{4}})}^{2} (6 B_{3} ψ_{n} + 20 B_{5} {ψ_{n}}^{3} + \dots) \\ - (\frac{600 ψ_{n}}{61 π^{2}} + \frac{1440 {ψ_{n}}^{3}}{3721 π^{4}} + \frac{104064 {ψ_{n}}^{5}}{226981 π^{6}}) (B_{1} + 3 B_{3} {ψ_{n}}^{2} + 5 B_{5} {ψ_{n}}^{4} + \dots) \end{array}

(2.6)

From equation (2.4), it is found that $ψ_{n} = 0$ when $τ = 0$ . Applying the initial condition $x^{'} (0) = β$ in equation (2.5), $B_{1} = β$ is obtained. Now substituting $x (τ)$ , $x^{'} (τ)$ and $x^{″} (τ)$ together with $θ$ from equation (2.3), into equation (2.1) and equating the coefficients of ${(ψ_{n})}^{i}, i = 0, 1, 2, 3, \dots$ a set of linear algebraic equations are found which provide the values of unknown coefficients $B_{3}$ , $B_{5}, \dots$ and $θ_{0}$ , $θ_{2}$ , $θ_{4}, \dots$ in terms of $β$ , $ν$ , $ε$ and $ω_{0}$ .

From equation (2.4), it is also found that the maximum value of $ψ_{n} = 1$ when $τ = \frac{π}{2}$ . Certainly, $x (τ)$ has the maximum value known as amplitude of the oscillation, say $a > 0$ at $τ = \frac{π}{2}$ (If $b > 0$ as well as $β > 0$ , then $a > 0$ and if $b < 0$ as well as $β < 0$ , then $x (τ)$ becomes minimum, $- a$ at $τ = \frac{π}{2}$ ). Therefore, a relation among $a$ , $β$ , $ν$ , $ε$ and $ω_{0}$ is obtained from equation (2.2) as

a = β + B_{3} + B_{5} + \dots .

(2.7)

In general,

ν

ε

and

ω_{0}

are given. To obtain a periodic solution,

a

and

β

are to be determined. So, it requires another relation between

a

and

β

. Multiplying equation (2.1) by

2 x^{'}

and then integrating with respect to

τ

, it readily becomes¹⁶

{ν^{2} β}^{2} + ε F + ε G = ω_{0}^{2} a^{2},

(2.8)

where

F = \int_{0}^{π / 2} 2 x^{'} f d τ and G = \int_{0}^{π / 2} 2 x^{'} g (τ + θ) d τ .

(2.9)

Thus, the determination of

a

and

β

is clear. It is noted that equation (2.8) is well known as an energy form of equation (2.1).

Example

Let us consider a nonlinear damped forced oscillator

\ddot{x} + ω_{0}^{2} x + ε μ \dot{x} + ε x^{3} = ε E \sin (ν t) .

(3.1)

In terms of

τ

, equation (3.1) becomes

ν^{2} x^{″} + ω_{0}^{2} x + ε μ {ν x}^{'} + ε x^{3} = ε E \sin (τ + θ) .

(3.2)

The noted set of linear algebraic equations for $B_{3}$ , $B_{5}, \dots$ and $θ_{0}$ , $θ_{2}$ , $θ_{4}$ , $\dots$ of this oscillator are obtained as

\begin{array}{l} ε β μ ν = ε E s_{0}, - \frac{600 β ν^{2}}{61 π^{2}} + 6 ν^{2} B_{3} + β ω_{0}^{2} = ε E c_{0}, \\ - \frac{300 β μ ε ν}{61 π^{2}} + 3 ε μ ν B_{3} = ε E c_{0} θ_{2} - \frac{1}{2} ε E s_{0}, \\ ε β^{3} - \frac{1440 β ν^{2}}{3721 π^{4}} - \frac{5400 B_{3} ν^{2}}{61 π^{2}} + ω_{0}^{2} B_{3} + 20 ν^{2} B_{5} = \frac{100 E ε c_{0}}{61 π^{2}} - \frac{1}{6} E ε c_{0} - E ε θ_{2} s_{0}, \dots, \end{array}

(3.3)

where

c_{0} = \cos θ_{0}

and

s_{0} = \sin θ_{0}

. Solving equation (3.3), the coefficients

B_{3}

B_{5}, \dots

and

θ_{0}

θ_{2}

\dots

are found as

\begin{array}{l} θ_{0} = \sin^{- 1} \frac{β μ ν}{E}, B_{3} = \frac{100 β}{61 π^{2}} - \frac{β ω_{0}^{2}}{6 ν^{2}} + \frac{ε E c_{0}}{6 ν^{2}}, θ_{2} = - \frac{300 β μ ν s_{e 0}}{61 E π^{2}} + \frac{3 μ ν B_{3} s_{e 0}}{E} + \frac{t_{0}}{2}, \\ B_{5} = \frac{72 β}{3721 π^{4}} + \frac{270 B_{3}}{61 π^{2}} - \frac{ω_{0}^{2} B_{3}}{20 ν^{2}} - \frac{ε β^{3}}{20 ν^{2}} - \frac{ε E θ_{2} s_{0}}{20 ν^{2}} - \frac{E ε c_{0}}{120 ν^{2}} + \frac{5 E ε c_{0}}{61 π^{2} ν^{2}}, \\ θ_{4} = \frac{θ_{2}}{2} + \frac{5 B_{5} μ ν s_{e 0}}{E} - \frac{45360 β μ ν s_{e 0}}{3721 E π^{4}} - \frac{900 μ ν B_{3} s_{e 0}}{61 E π^{2}} - \frac{t_{0}}{24} + \frac{{100 t}_{0}}{61 π^{2}}, \end{array}

(3.4)

B_{7} = \frac{17344 β}{1588867 π^{6}} + \frac{2500 B_{5}}{427 π^{2}} + \frac{1440 B_{3}}{26047 π^{4}} - \frac{ω_{0}^{2} B_{5}}{42 ν^{2}} - \frac{ε β^{2} B_{3}}{14 ν^{2}} + \frac{E ε c_{0}}{5040 ν^{2}} - \frac{25 E ε c_{0}}{1281 π^{2} ν^{2}} - \frac{ε E θ_{4} s_{0}}{42 ν^{2}},

where

s_{e 0} = \sec θ_{0}

and

t_{0} = \tan θ_{0}

For this oscillator, equation (2.8) becomes

{ν^{2} β}^{2} + ε μ ν D + ε E S = ω_{0}^{2} a^{2} + ε a^{4} / 2,

(3.5)

where

D = \int_{0}^{π / 2} 2 {x^{'}}^{2} d τ = 2 β^{2} + 4 β B_{3} + \frac{18 B_{3}^{2}}{5} + 4 β B_{5} - \frac{18144 β^{2}}{3721 π^{4}} - \frac{200 β^{2}}{61 π^{2}} - \frac{720 β B_{3}}{61 π^{2}},

(3.6)

and

S = \int_{0}^{π / 2} 2 x^{'} \sin (τ + θ) d τ = c_{0} (\frac{11 β}{12} + \frac{3 B_{3}}{2} + \frac{50 β}{61 π^{2}} + \frac{7 β θ_{2}}{15} + \frac{6 B_{3} θ_{2}}{5} + \frac{2 β θ_{4}}{5})

+ s_{0} (\frac{101 β}{60} + \frac{7 B_{3}}{5} + 2 B_{5} - \frac{40 β}{61 π^{2}} - \frac{b θ_{2}}{2})

(3.7)

Now substituting $B_{3}$ , $B_{5}, \dots$ into equation (2.7) and then solving together with equation (3.5), the values of $a$ and $β$ are obtained for every value of $ν$ .

In the case of $E = μ = 0$ , equation (3.1) turns into a cubic Duffing oscillator

\ddot{x} + ω_{0}^{2} x + ε x^{3} = 0 .

(3.8)

In terms of

τ

, equation (3.8) becomes

ν^{2} x^{″} + ω_{0}^{2} x + ε x^{3} = 0 .

(3.9)

The unknowns $B_{3}$ , $B_{5}, \dots$ are obtained of equation (3.9) as

\begin{array}{l} B_{3} = \frac{100 β}{61 π^{2}} - \frac{β ω_{0}^{2}}{6 ν^{2}}, B_{5} = \frac{72 β}{3721 π^{4}} + \frac{270 B_{3}}{61 π^{2}} - \frac{ω_{0}^{2} B_{3}}{20 ν^{2}} - \frac{ε β^{3}}{20 ν^{2}}, \\ B_{7} = \frac{17344 β}{1588867 π^{6}} + \frac{2500 B_{5}}{427 π^{2}} + \frac{1440 B_{3}}{26047 π^{4}} - \frac{ω_{0}^{2} B_{5}}{42 ν^{2}} - \frac{ε β^{2} B_{3}}{14 ν^{2}}, \end{array}

(3.10)

and the energy form of equation (3.9) is

{ν^{2} β}^{2} = ω_{0}^{2} a^{2} + ε a^{4} / 2 .

(3.11)

Solving equation (2.7) and equation (3.11) with these coefficients

B_{3}

B_{5}, \dots

, the values of

ν

and

β

are obtained for different values of the amplitude

a

Results and discussions

An analytical technique has been introduced for solving nonlinear oscillators, applicable to both autonomous and non-autonomous systems. In this paper, the nonlinear damped force oscillator as well as cubic Duffing oscillators are considered. The algebraic equations containing unknown coefficients are linear. In contrast, corresponding equations for HBM²¹ are nonlinear. This is a key advantage of the proposed technique. Several methods^20,27,28 have been introduced for solving damped forced oscillators and suitable results yielded only for small excitation. The present technique is suitable for both small as well as large excitations. Cheung et al.²² expanded $ν^{2}$ in equation (3.1) as a series in terms of $α$ rather than $ε$ , and shown the frequency-amplitude curve for large excitations, considering only the first harmonic. Although, only the first harmonic was considered, the mathematical calculations are still laborious. The present technique offers a more efficient alternative. Additionally, for various excitation, the frequency against amplitude curves (displayed in Figures 1–4) show a good agreement with numerical results obtained using the Runge-Kutta fourth-order method (RK4).

Figure 1.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $μ = 0.1, ω_{0} = 1, E = 10$ .

Figure 2.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $μ = 0.1, ω_{0} = 1, E = 20$ .

Figure 3.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $μ = 0.2, ω_{0} = 1, E = 1$ .

Figure 4.

Frequency versus amplitude curves of equation (3.1) between present results and the corresponding numerical one for $μ = 0.2, ω_{0} = 1, E = 20$ .

To further validate and demonstrate the accuracy of the current solutions, a comparison of time versus displacement for various system parameters is plotted in Figures 5–7 alongside the corresponding numerical results. The comparison reveals that the present solutions align remarkably well with the numerical solutions. In Figure 8, the 3-D error graph of the proposed method corresponding to the RK4 of equation (3.1) for the variations in parameters $E$ and $ν$ which shows the acceptable errors.

Figure 5.

Time versus displacement curves of equation (3.1) between present solution and the corresponding numerical one for $μ = 0.1, ω_{0} = 1, ν = 6, E = 10$ with the initial condition $x (0) = 0, \dot{x} (0) = 35.62212$ .

Figure 6.

Time versus displacement curves of equation (3.1) between present solution and the corresponding numerical one for $μ = 0.1, ω_{0} = 1, ν = 7, E = 20$ with the initial condition $x (0) = 0, \dot{x} (0) = 48.78188$ .

Figure 7.

Time versus displacement curves of equation (3.1) between present solution and the corresponding numerical one for $μ = 0.2, ω_{0} = 1, ν = 5, E = 20$ with the initial condition $x (0) = 0, \dot{x} (0) = 25.32035$ .

Figure 8.

3-D error graph of the proposed method corresponding to the RK4 of equation (3.1) for the variations in parameters $E$ and $ν$ .

The comparison of the approximated and existing amplitudes corresponding to the exact ones of equation (3.1) for various parameters

μ = 0.2, ω_{0} = 1, E = 20

are shown in Table 1. The proposed method provides the almost similar results as compared with the existing ones for small frequencies. Notably, the proposed method gives excellent results as compared to the existing results for large frequencies.

Table 1.

Comparison of the approximated and existing amplitudes corresponding to the exact ones of equation (3.1) for various parameters $μ = 0.2, ω_{0} = 1, E = 20$ .

Frequency $ν$	$a_{e x a c t}$	Cheng et al.²²	Zahangir et al.²³	Present study
Frequency $ν$	$a_{e x a c t}$	$E r (%)$	$E r (%)$	$E r (%)$
3	4.30342	4.29997 (0.08)	4.29602 (0.17)	4.31492 (0.26)
4	5.17308	5.17961 (0.12)	5.17598 (0.06)	5.18546 (0.24)
5	6.17686	6.19106 (0.22)	6.18330 (0.10)	6.18869 (0.19)
6	7.24929	7.27236 (0.31)	7.25971 (0.14)	7.25914 (0.13)
7	8.35674	8.39161 (0.42)	8.36976 (0.16)	8.36323 (0.08)
8	9.48082	9.53244 (0.54)	9.49665 (0.17)	9.48103 (0.002)
9	10.6063	10.6862 (0.75)	10.6307 (0.23)	10.5985 (0.07)

For the cubic Duffing oscillator equation (3.8), the unknown coefficients

B_{3}

B_{5}, \dots

in equation (3.10) and the energy expression in equation (3.11) are relatively simple. The frequencies of this oscillator for various amplitudes

a

can be easily determined and are well approximated by numerical frequencies, as shown in Table 2. Additionally, a comparison of the time versus displacement curve for the oscillator in equation (3.8) is plotted in Figure 9. This comparison demonstrates that the present analytical technique is also applicable to the cubic Duffing oscillator and provides excellent results.

Table 2.

Comparison of the present and exact frequency of the cubic Duffing oscillator for various amplitude.

Amplitude $a$	$ν_{e x a c t}$	$ν_{p r e s e n t}$
Amplitude $a$	$ν_{e x a c t}$	$E r (%)$
1	1.317776	1.317396 (0.03)
2	1.976016	1.974730 (0.06)
5	4.357461	4.351904 (0.12)
10	8.533586	8.519908 (0.16)
100	84.72748	84.58856 (0.16)

Figure 9.

Time versus displacement curves of equation (3.8) between present solution and the corresponding numerical one for $ω_{0} = 1, a = 100,$ with the initial condition $x (0) = 0, \dot{x} (0) = 7083.38678$ .

Conclusion

An analytical technique is presented for solving nonlinear damped forced oscillators, as well as for force free undamped case. The solution procedure of the present technique is simple. Also, it is suitable for large excitations under various damping conditions. The results obtained in the present technique are surprisingly good agreement with corresponding numerical one. However, the proposed analytical technique should continue to develop for applying to the quadratic nonlinear force oscillators as well as Van der Pol nonlinear force oscillators. To sum up, it is said that, the proposed technique has a great effectiveness for solving nonlinear oscillators, particularly for the nonlinear damped forced oscillatory systems arising in nonlinear science and engineering.

Footnotes

ORCID iDs

M. Z. Alam

Md. Alal Hosen

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

References

Nayfeh

. Perturbation method. John Wiley & Sons, 1973.

Nayfeh

Mook

. Nonlinear oscillations. John Wiley & Sons, 1979.

Krylov

Bogolyubov

. Introduction to nonlinear mechanics. Princeton University Press, 1947.

Bogoliubov

Mitropolskii

. Asymptotic methods in the theory of nonlinear oscillations. Gordan and Breach, 1961.

. Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations-I: expansion of a constant. Int J Non Lin Mech 2002; 38: 309–314.

Alam

Yeasmin

, et al. A modified Krylov-Bogoliubov-Mitropolskii method for solving damped nonlinear oscillators with large oscillation. Int J Non Lin Mech 2023; 155: 104459.

Alam

Yeasmin

Ahamed

. Generalization of the modified Lindstedt-Poincare method for solving some strong nonlinear oscillators. Ain Shams Eng J 2019; 10: 195–201.

Beléndez

Hernández

Beléndez

, et al. Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators. Chaos Solitons Fractals 2008; 37: 770–780.

. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999; 178: 257–262.

10.

Di Matteo

Pirrotta

. Generalized differential transform method for nonlinear boundary value problem of fractional order. Commun Nonlinear Sci Numer Simul 2015; 29: 88–101.

11.

Arikoglu

Ozkol

. Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals 2007; 34: 1473–1481.

12.

El-Shahed

. Application of differential transform method to non-linear oscillatory systems. Commun Nonlinear Sci Numer Simul 2008; 13: 1714–1720.

13.

Momani

Erturk

. Solutions of non-linear oscillators by the modified differential transform method. Comput Math Appl 2008; 55: 833–842.

14.

Alquran

. Applying differential transform method to nonlinear partial differential equations: a modified approach. Appl Appl Math 2012; 7: 155–163.

15.

Biazar

Eslami

Islam

. Differential transform method for nonlinear parabolic-hyperbolic partial differential equations. Appl Appl Math 2010; 5: 396–406.

16.

Alam

Huq

Hasan

, et al. A new technique for solving a class of strongly nonlinear oscillatory equations. Chaos Solitons Fractals 2021; 152: 111362.

17.

Huq

Hasan

Alam

. An approximation technique for solving nonlinear oscillators. J Low Freq Noise Vib Act Control 2024; 43: 239–249.

18.

Burton

. A perturbation method for certain nonlinear oscillators. Int J Non Lin Mech 1984; 19: 397–407.

19.

Burton

Rahman

. On the multi-scale analysis of strongly non-linear forced oscillators. Int J Non Lin Mech 1986; 21: 135–146.

20.

Razzak

Alam

Sharif

. Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems. Results Phys 2018; 8: 231–238.

21.

Ullah

Rahman

Uddin

. A modified harmonic balance method for solving forced vibration problems with strong nonlinearity. J Low Freq Noise Vib Act Control 2021; 40: 1096–1104.

22.

Cheung

Chen

Lau

. A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators. Int J Non Lin Mech 1991; 26: 367–378.

23.

Alam

Hosen

Alam

. A new modified Lindstedt-Poincare method for nonlinear damped forced oscillations. Results Phys 2023; 51: 106673.

24.

Moatimid

Amer

. Dynamical analysis of a damped harmonic forced duffing oscillator with time delay. Sci Rep 2023; 13: 6507.

25.

El-Dib

. The masking technique for forced nonlinear oscillator stability behavior analysis using the non-perturbative approach. J Low Freq Noise Vib Act Control 2024; 43(4): 1481–1497.

26.

Adamu

Abdullahi

. Frequency-amplitude relationship in damped and forced nonlinear oscillators with irrational nonlinearities. J Comput Appl Mech 2025; 56(2): 307–317.

27.

Alex

Oscar

Daniel

, et al. Exact steady-state solution of fractals damped, and forced systems. Results Phys 2021; 28: 104580.

28.

Gamal

Marwa

. Forced nonlinear oscillator in a fractal space. Facta Universitatis 2022; 20: 1–20.