Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Haifa A Alyousef; M R Alharthi; Alvaro H Salas; S A El-Tantawy

doi:10.1088/1572-9494/ac7bdc

PDF(1616 KB)

Communications in Theoretical Physics ›› 2022, Vol. 74 ›› Issue (10) : 105002. DOI: 10.1088/1572-9494/ac7bdc

Mathematical Physics

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Author information +

History +

Abstract

The (un)forced (un)damped parametric pendulum oscillator (PPO) is analyzed analytically and numerically using some simple, effective, and more accurate techniques. In the first technique, the ansatz method is employed for analyzing the unforced damped PPO and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions. In the second approach, some approximations to (un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method. In the third approach, He's frequency-amplitude principle is applied for deriving some approximations to the (un)damped PPO. In the forth approach, He's homotopy technique is employed for analyzing the forced (un)damped PPO numerically. In the fifth approach, the p-solution Method, which is constructed based on Krylov–Bogoliúbov Mitropolsky method, is introduced for deriving an approximation to the forced damped PPO. In the final approach, the hybrid Padé-finite difference method is carried out for analyzing the damped PPO numerically. All proposed techniques are compared to the fourth-order Runge–Kutta (RK4) numerical solution. Moreover, the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations. The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science, e.g. waves and oscillations in plasma physics.

Key words

parametric pendulum equation / Ansatz method / He's frequency-amplitude principle / He's homotopy technique / Krylov–Bogoliúbov Mitropolsky method / the hybrid Padé-finite difference method

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

Haifa A Alyousef, M R Alharthi, Alvaro H Salas, et al. Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator[J]. Communications in Theoretical Physics, 2022, 74(10): 105002 https://doi.org/10.1088/1572-9494/ac7bdc

1. Introduction

The theory of linear oscillations has been successfully applied by many authors for modeling and analyzing the oscillatory devices. However, nonlinear behavior appears in a lot of real world phenomena [1–7]. As a result, researchers from various fields are exploring nonlinear systems and trying to model these systems and come up with explanations and solutions to some problems, whether in the manufacture of large small machines, as well as electronic chips. Accordingly, the nonlinear oscillation is one of the most popular and interesting topics among researchers because it has many different applications in sensing, automobiles, liquid and solid interaction, micro and nanoscale, space, bioengineering, and nonlinear oscillations in plasma. The complex pendulum is a model in the study of nonlinear oscillations and many other nonlinear phenomena in engineering, physics, and nonlinear dynamics [8–10]. The nonlinear oscillators can arise in several branches of science including the oscillations of a high-amplitude physical pendulum, nonlinear electrical circuits, image processing, open states of DNA, the movement of satellites, Bose–Einstein condensates, oscillations in different plasma models, and many others phenomena [11–15]. Moreover, the simple pendulum has been used as a physical model to solve several natural problems related to bifurcations, oscillations, and chaos such as nonlinear plasma oscillations [16–19], and many other oscillations in different fields of science [20–37].

There are few attempts are devoted to analyze one of the equations of motion to the nonlinear damped pendulum taking the friction forces into consideration [38]. However, in addition to the friction force, there are many other physical forces that can affect on the pendulum oscillations, e.g. the perturbed and periodic forces. For example, the following damped parametric pendulum oscillator (PPO) has been investigated only numerically via implicit discrete mappings [39]

\begin{array}{rcl} R \equiv \ddot{φ} + 2 β \dot{φ} + ϕ (t) \sin φ = 0, \end{array}

(1)

where

ϕ (t) = ω_{0}^{2} - Q_{0} \cos (γ t)

ω_{0} = \sqrt{g / l}

gives the eigenfrequency of the system,

β = μ / (2 m l)

represents the coefficient of the damping term, Q₀ = ϵω₂ is the excitation amplitude, ω₂ = γ²/l. Here, A simple mathematical parametric pendulum system is modeled by a point mass m, hanging at the end of a massless wire with length l and fixed to a supporting point ‘O', swinging to and from in a vertical plane. Motivated by the potential applications of the nonlinear oscillator model, thus, in the present study, the parametric pendulum equation (PPE) will be solved and investigated using some new analytical and numerical approaches. A new formula for the approximate analytical solution will be obtained in the form of Mathieu function. In addition, the hybrid Padé-finite difference method (Padé-FDM) is employed for anatomy equation (1) numerically. Due to the high-accuracy of the fourth-order Runge–Kutta (RK4), which is characterized by an error of the fourth-order, thus, we will take this method as a benchmark to measure the accuracy of the approximate solutions that we will obtain.

In this investigation, some approximations to the (un)forced (un)damped PPO using different approaches are obtained. In the first approach, an effective and high-accuracy approximation to the unforced damped PPO is obtained using the ansatz method in the form of Mathieu function. In the second approach, some approximations to (un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method. In the third approach, He's frequency-amplitude principle (He's-FAP) is devoted for deriving some approximations to the (un)damped PPO. In the forth approach, He's homotopy technique (He's-HT) is carried out for analyzing the forced (un)damped PPO. In the fifth approach, the p-solution Method which is constructed based on Krylov–Bogoliúbov Mitropolsky method is implemented for getting an approximation to the forced damped PPO. In the final approach, the hybrid Padé-FDM is performed for analyzing the damped PPO.

2. First approach for analyzing the damped PPO

Let us find an approximation to the i.v.p.

\begin{array}{rcl} {\begin{cases} R \approx \ddot{φ} + 2 β \dot{φ} + ϕ (t) (φ - \sum_{i = 3}^{\infty} λ_{i} φ^{i}) = 0, \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0}, \end{cases} \end{array}

(2)

where i = 3, 5, ⋯ , i.e. only odd numbers. Note that the values of λ_i can be estimated from the Chebyshev polynomial approximation [40].

Here, we try to find an analytical approximation to the i.v.p. (2) in the ansatz form

\begin{array}{rcl} φ = θ (g (t)), \end{array}

(3)

where g(t) is an undetermined time-dependent function which can be obtained later using the initial conditions (ICs) g(0) = 0 while θ ≡ θ(t) indicates the exact solution to the following i.v.p. for small θ

(\sin (θ) \approx θ,)

\begin{array}{rcl} {\begin{cases} \ddot{θ} + 2 β \dot{θ} + ϕ (t) θ = 0, \\ θ (0) = φ_{0} a n d θ^{'} (0) = {\dot{φ}}_{0} . \end{cases} \end{array}

(4)

The exact solution to the i.v.p. (4) reads

\begin{array}{rcl} \begin{array}{l} θ (t) = e^{- β t} (\frac{2 (β φ_{0} + c) M a t h i e u S [Z_{1}, Z_{2}, \frac{γ}{2} t]}{γ M a t h i e u S P r i m e [Z_{1}, Z_{2}, 0]} \\ + \frac{φ_{0} M a t h i e u C [Z_{1}, Z_{2}, \frac{γ}{2} t]}{M a t h i e u C [Z_{1}, Z_{2}, 0]}), \end{array} \end{array}

(5)

with

\begin{array}{rcl} Z_{1} = - \frac{4 (β^{2} - ω_{0}^{2})}{γ^{2}}, Z_{2} = \frac{2 Q_{0}}{γ^{2}} . \end{array}

(6)

Note that solution (5) is easy to obtain using DSolve command in MATHEMATICA software.

For determining the function g(t), we replace θ(t) by

θ (g (t))

in solution (5) to be

\begin{array}{rcl} \begin{array}{l} θ (g (t)) = e^{- β f (t)} (\frac{2 (β φ_{0} + {\dot{φ}}_{0}) M a t h i e u S [Z_{1}, Z_{2}, \frac{γ}{2} g (t)]}{γ M a t h i e u S P r i m e [Z_{1}, Z_{2}, 0]} \\ + \frac{φ_{0} M a t h i e u C [Z_{1}, Z_{2}, \frac{γ}{2} g (t)]}{M a t h i e u C [Z_{1}, Z_{2}, 0]}), \end{array} \end{array}

(7)

and inserting the obtained result in

R

(given in equation (2)), we get

\begin{array}{rcl} - 2 β g^{'} (t) + 2 β g^{'} {(t)}^{2} - β g^{''} (t) = 0. \end{array}

(8)

By solving the ode (8) using the IC g(0) = 0, we have

\begin{array}{rcl} g (t) = \frac{1}{2 β} (2 t β + l o g [1 + e^{ρ}] - l o g [1 + e^{ρ + 2 t β}]), \end{array}

(9)

here, ρ denotes the free integration constant. Observe that

g (t)

does not depend on the coefficient λ give in

R_{1}

. Thus, the obtained approximation does not depend on the coefficient λ. Consequently, our analytical approximation is considered an effective and gives high-accuracy. By substituting the value of

g (t)

(given in equation (9)) into solution (7), then the solution of the i.v.p. (2) is obtained

\begin{array}{rcl} \begin{array}{l} φ |_{A p p r o x (1)} = θ (g (t)) = \frac{\sqrt{1 + e^{ρ + 2 t β}}}{\sqrt{1 + e^{ρ}}} e^{- t β} \\ \times (\frac{2 (β φ_{0} + {\dot{φ}}_{0}) M a t h i e u S [Z_{1}, Z_{2}, \frac{γ}{2} g (t)]}{γ M a t h i e u S P r i m e [Z_{1}, Z_{2}, 0]} \\ + \frac{φ_{0} M a t h i e u C [Z_{1}, Z_{2}, \frac{γ}{2} g (t)]}{M a t h i e u C [Z_{1}, Z_{2}, 0]}), \end{array} \end{array}

(10)

For optimizing solution (10), the following choice is considered

\begin{array}{rcl} {\begin{cases} {\dot{φ}}_{0} \to (1 + e^{ρ}) {\dot{φ}}_{0}, \\ β \to F β, \end{cases} \end{array}

(11)

where both

(ρ, F)

give the free parameters of optimization. In this case the values of

g (t)

given in equation (9) and Z₁ given in equation (6) have the following new forms

\begin{array}{rcl} \begin{array}{rcl} \tilde{g} (t) & = & \frac{1}{2 F β} (2 t F β + l o g [1 + e^{ρ}] \\ - l o g [1 + e^{ρ + 2 t F β}]), \\ {\tilde{Z}}_{1} & = & - \frac{4 (F^{2} β^{2} - ω_{0}^{2})}{γ^{2}} . \end{array} \end{array}

(12)

Then the optimal analytical approximation to the i.v.p. (2) reads

\begin{array}{rcl} \begin{array}{l} φ |_{A p p r o x (2)} = θ (\tilde{g} (t)) = \frac{\sqrt{1 + e^{ρ + 2 F β t}}}{\sqrt{1 + e^{ρ}}} e^{- F β t} \\ \times (\frac{2 (F β φ_{0} + (1 + e^{ρ}) {\dot{φ}}_{0}) M a t h i e u S [{\tilde{Z}}_{1}, Z_{2}, \frac{γ}{2} \tilde{g} (t)]}{γ M a t h i e u S P r i m e [{\tilde{Z}}_{1}, Z_{2}, 0]} \\ + \frac{φ_{0} M a t h i e u C [{\tilde{Z}}_{1}, Z_{2}, \frac{γ}{2} \tilde{g} (t)]}{M a t h i e u C [{\tilde{Z}}_{1}, Z_{2}, 0]}) . \end{array} \end{array}

(13)

We can choose the default values of

(ρ, F) = (- T, 1)

where T represents the maximum value of t.

Both approximations (10) and (13) are, respectively, presented in figure 1 for

(β, ω_{0}, Q_{0}, γ, {\dot{φ}}_{0}) = (0.1, 1, 0.1, 1, 0.2)

and with different values to φ₀. Also, these approximations give excellent results compared to the RK4 numerical approximation as shown in figure 1. Also, the global maximum residual distance error (GMRDE) L_d according to the following relation:

\begin{array}{rcl} L_{d} = max_{0 ⩽ t ⩽ T} | A p p r o x i m a t i o n - R K 4 |, \end{array}

is estimated for the two approximations as follows

\begin{array}{rcl} \begin{array}{rcl} L_{d} |_{φ_{0} = \frac{π}{6}} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x i . (10) - R K 4 | \\ = & 0.0285408, \\ L_{d} |_{(φ_{0}, F) = (\frac{π}{6}, 1.06)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x i . (13) - R K 4 | \\ = & 0.0270288, \end{array} \end{array}

and

\begin{array}{rcl} \begin{array}{l} L_{d} |_{φ_{0} = \frac{π}{3}} = max_{0 ⩽ t ⩽ 30} | A p p r o x i . (10) \\ - R K 4 | = 0.177002, \\ L_{d} |_{(φ_{0}, F) = (\frac{π}{3}, 1.3)} = max_{0 ⩽ t ⩽ 30} | A p p r o x i . (13) \\ - R K 4 | = 0.156497 . \end{array} \end{array}

It is clear that the second optimal parameter

F

plays a vital role in the improving the accuracy of the second approximation (13). In this case, the accuracy of the second approximation (13) becomes better than the first one (10).

Figure 1. The profile of two-analytical approximations (10) and (13) to the i.v.p. (4) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Full size|PPT slide

3. Second approach for analyzing the (un)forced damped PPO

Let us consider the i.v.p.

\begin{array}{rcl} {\begin{cases} \ddot{φ} + 2 β \dot{φ} + ϕ (t) \sin φ = 0, \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0} . \end{cases} \end{array}

(14)

The solution to i.v.p. (14) is assumed to have the following ansatz form

\begin{array}{rcl} φ (t) = 2 \tan^{- 1} u (t) . \end{array}

(15)

Then

\begin{array}{rcl} R = \frac{2}{{(u {(t)}^{2} + 1)}^{2}} R_{u}, \end{array}

(16)

with

\begin{array}{rcl} R_{u} = (\begin{array}{c} - 2 u^{'} {(t)}^{2} u (t) + 2 β u^{'} (t) u {(t)}^{2} + u^{''} (t) u {(t)}^{2} \\ + u {(t)}^{3} ϕ (t) + u (t) ϕ (t) + 2 β u^{'} (t) + u^{''} (t) \end{array}) . \end{array}

(17)

Now, let the value of u(t) is given by

\begin{array}{rcl} u (t) = \exp (- β t) (c_{1} \cos w (t) + c_{2} \sin w (t)) . \end{array}

(18)

By substituting equations (18) into (17), we have

\begin{array}{rcl} R_{u} = S_{1} \cos w (t) + S_{2} \sin w (t) + h . o . t ., \end{array}

(19)

with

\begin{array}{rcl} \begin{array}{rcl} S_{1} & = & {\begin{array}{c} e^{- β t} [c_{2} w^{''} (t) - c_{1} (β^{2} - ϕ (t) + w^{'} {(t)}^{2})] \\ - \frac{1}{4} (c_{1}^{2} + c_{2}^{2}) e^{- 3 β t} [c_{1} (9 β^{2} - 3 ϕ (t) + 5 w^{'} {(t)}^{2}) - c_{2} (4 β w^{'} (t) + w^{''} (t))] \end{array}}, \\ S_{2} & = & {\begin{array}{c} - \frac{1}{4} (c_{1}^{2} + c_{2}^{2}) e^{- 3 β t} [c_{2} (9 β^{2} - 3 ϕ (t) + 5 w^{'} {(t)}^{2}) + c_{1} (4 β w^{'} (t) + w^{''} (t))] \\ - e^{- β t} [c_{2} (β^{2} - ϕ (t) + w^{'} {(t)}^{2}) + c_{1} w^{''} (t)] \end{array}}, \end{array} \end{array}

where ‘h.o.t.' indicates the higher-order terms.

Equating both S₁ and S₂ to zero and then eliminating w″(t) from the resulting system to obtain an ode for w(t). Solving the obtained ode gives the value of the frequency-amplitude formulation as follows

\begin{array}{rcl} \begin{array}{l} w (t) = \int_{0}^{t} \times \sqrt{\frac{4 e^{2 β τ} (ϕ (τ) - β^{2}) + 3 (c_{1}^{2} + c_{2}^{2}) (ϕ (τ) - 3 β^{2})}{4 e^{2 β τ} + 5 (c_{1}^{2} + c_{2}^{2})}} d τ, \end{array} \end{array}

(20)

where the constants c₁ and c₂ are found from the ICs:

c_{1} = \tan (φ_{0} / 2)

and c₂ is a solution to the following quartic equation

\begin{array}{rcl} W_{0} + W_{1} c_{2}^{2} + W_{2} c_{2}^{4} = 0, \end{array}

(21)

with

\begin{array}{rcl} \begin{array}{l} W_{0} = \frac{1}{2} (9 - \cos (φ_{0})) \sec^{6} (\frac{φ_{0}}{2}) (β \sin (φ_{0}) + {\dot{φ}}_{0})^{2}, \\ W_{1} = \sec^{4} (\frac{φ_{0}}{2}) \\ \times [\begin{matrix} β^{2} (13 + 8 \cos (φ_{0}) - 5 \cos (2 φ_{0})) - \\ 2 \cos^{2} (\frac{φ_{0}}{2}) (7 + \cos (φ_{0})) ϕ_{0} + 5 {\dot{φ}}_{0} (2 β \sin (φ_{0}) + {\dot{φ}}_{0}) \end{matrix}], \\ W_{2} = (36 β^{2} - 12 ϕ_{0}) c_{2}^{4} = 0, \end{array} \end{array}

where φ₀ = φ(0).

For investigating the accuracy of the obtained solution, we analyze this solution numerically using the following data:

(β, ω_{0}, Q_{0}, γ) = (0.1, 2, 0.1, 1)

with the ICs: φ₀ = 0 and

{\dot{φ}}_{0} = 0.1

which lead to

\begin{array}{rcl} {\begin{cases} φ^{''} + 0.2 φ^{'} + (2 - 0.1 \cos (t)) \sin (φ) = 0, \\ φ_{0} = 0 a n d {\dot{φ}}_{0} = 0.1 . \end{cases} \end{array}

(22)

Solution (15) to the i.v.p. (22) is compared with RK4 numerical approximation as illustrated in figure 2. Also, the GMRDE L_d is estimated for

(β, ω_{0}, Q_{0}, γ, φ_{0}, {\dot{φ}}_{0}) = (0.1, 2, 0.1, 1, 0, 0.1)

and

(β, ω_{0}, Q_{0}, γ, φ_{0}, {\dot{φ}}_{0}) = (0.1, 2, 0.1, 1, \frac{π}{6}, 0.1)

, respectively

\begin{array}{rcl} {L_{d} |}_{φ_{0} = 0} = max_{0 ⩽ t ⩽ 30} | A p p r o x . (15) - R K 4 | = 0.000427262 . \end{array}

and

\begin{array}{rcl} {L_{d} |}_{φ_{0} = \frac{π}{6}} = max_{0 ⩽ t ⩽ 30} | A p p r o x . (15) - R K 4 | = 0.00638336 . \end{array}

Now, let us consider the forced parametric pendulum

\begin{array}{rcl} {\begin{cases} N \equiv \ddot{φ} + 2 β \dot{φ} + ϕ (t) \sin φ - F (t) = 0, \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0}, \end{cases} \end{array}

(23)

where

F (t) = F_{0} \cos (ω t)

Figure 2. The profile of the approximation (15) to the i.v.p. (22) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Full size|PPT slide

The solution to i.v.p. (23) is assumed to have the following ansatz form

\begin{array}{rcl} φ (t) = θ + d_{1} \cos ω t + d_{2} \sin ω t, \end{array}

(24)

where the function θ ≡ θ(t) is a solution to the following unforced pendulum

\begin{array}{rcl} {\begin{cases} \ddot{θ} + 2 β \dot{θ} + ϕ (t) \sin θ = 0, \\ θ (0) = φ_{0} - d_{1} a n d θ^{'} (0) = {\dot{φ}}_{0} - d_{2} ω . \end{cases} \end{array}

(25)

For

\sin φ \approx φ - \frac{1}{6} φ^{3}

, we have

\begin{array}{rcl} \begin{array}{rcl} N & \approx & \ddot{φ} + 2 β \dot{φ} + (ω_{0}^{2} - Q_{0} \cos (γ t)) (φ - \frac{1}{6} φ^{3}) - F (t) \\ = & X_{1} \cos (ω t) + X_{2} \sin (ω t) + h . o . t . \end{array} \end{array}

(26)

with

\begin{array}{rcl} \begin{array}{rcl} X_{1} & = & 2 β ω d_{2} - F_{0} - \frac{1}{8} d_{1}^{3} ω_{0}^{2} + d_{1} (- ω^{2} + ω_{0}^{2} - \frac{1}{8} d_{2}^{2} ω_{0}^{2}) \\ + \frac{1}{8} d_{1} (- 8 + d_{1}^{2} + d_{2}^{2}) Q_{0} \cos (γ t) \\ - \frac{1}{2} d_{1} (ω_{0}^{2} - Q_{0} \cos (γ t)) θ {(t)}^{2}, \end{array} \end{array}

and

\begin{array}{rcl} \begin{array}{rcl} X_{2} & = & - 2 β ω d_{1} - ω^{2} d_{2} + ω_{0}^{2} d_{2} - \frac{1}{8} ω_{0}^{2} d_{1}^{2} d_{2} \\ - \frac{1}{8} ω_{0}^{2} d_{2}^{3} + \frac{1}{8} d_{2} (d_{1}^{2} + d_{2}^{2} - 8) Q_{0} \cos (γ t) \\ - \frac{1}{2} d_{2} (ω_{0}^{2} - Q_{0} \cos (γ t)) θ {(t)}^{2} . \end{array} \end{array}

We will choose the values of d₁ and d₂ so that

\begin{array}{rcl} {\begin{cases} 2 β ω d_{2} - F_{0} - \frac{1}{8} d_{1}^{3} ω_{0}^{2} + d_{1} (- ω^{2} + ω_{0}^{2} - \frac{1}{8} d_{2}^{2} ω_{0}^{2}) = 0, \\ - 2 β ω d_{1} - ω^{2} d_{2} + ω_{0}^{2} d_{2} - \frac{1}{8} ω_{0}^{2} d_{1}^{2} d_{2} - \frac{1}{8} ω_{0}^{2} d_{2}^{3} = 0. \end{cases} \end{array}

(27)

We choose the least in magnitude real roots to the system (27).

Solution (24) for

(β, ω_{0}, Q_{0}, γ, F_{0}, ω) = (0.1, 2, 0.1, 1, 0.1, 0.1)

and different values to

(φ_{0}, {\dot{φ}}_{0})

is introduced in figure 3. Also, the GMRDE L_d is estimated as follows

\begin{array}{rcl} {L_{d} |}_{φ_{0} = 0} = max_{0 ⩽ t ⩽ 30} | A p p r o x . (24) - R K 4 | = 0.00135514 . \end{array}

and

\begin{array}{rcl} {L_{d} |}_{φ_{0} = \frac{π}{6}} = max_{0 ⩽ t ⩽ 30} | A p p r o x . (24) - R K 4 | = 0.014404 . \end{array}

The exact match between the obtained approximation (24) and the RK4 numerical approximation was observed, as shown in figure 3. Moreover, the obtained approximation (24) is characterized by high-accuracy, as is evident from the GMRDE L_d.

Figure 3. The profile of the approximation (24) to the i.v.p. (23) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Full size|PPT slide

4. He's-FAP for analyzing the (un)damped pendulum oscillator

Let us consider the i.v.p.

\begin{array}{rcl} {\begin{cases} \ddot{φ} + 2 β \dot{φ} + ϕ (t) \sin φ = 0, \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0} . \end{cases} \end{array}

(28)

For β = 0 and

\sin φ \approx φ - \frac{1}{6} φ^{3} + \frac{1}{120} φ^{5},

the i.v.p. (28) reduces to

\begin{array}{rcl} {\begin{cases} \ddot{φ} + f (φ) = 0, \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0}, \end{cases} \end{array}

(29)

with

\begin{array}{rcl} f (φ) = ϕ (t) (φ - \frac{1}{6} φ^{3} + \frac{1}{120} φ^{5}) . \end{array}

(30)

He's principle states that [41–46]

\begin{array}{rcl} ω^{2} = w^{'} {(t)}^{2} = {\frac{f (φ)}{φ} |}_{φ = \frac{\sqrt{3}}{2} A} = ϕ (t) (1 - \frac{A^{2}}{8} + \frac{3 A^{4}}{640}) . \end{array}

(31)

For the damped case, we can replace A with

A \exp (- β t)

in relation (31) so that if φ = φ(t) is a solution, then we have

\begin{array}{rcl} φ (t) = A \exp (- β t) \cos (w (t) + B), \end{array}

(32)

with

\begin{array}{rcl} w^{'} {(t)}^{2} = ϕ (t) (1 - \frac{A^{2}}{8} \exp (- 2 β t) + \frac{3 A^{4}}{640} \exp (- 4 β t)), \end{array}

(33)

and the integration of relation (33) yields

\begin{array}{rcl} \begin{array}{l} w (t) = \int_{0}^{t} \times \sqrt{ϕ (τ) (1 - \frac{A^{2}}{8} \exp (- 2 β τ) + \frac{3 A^{4}}{640} \exp (- 4 β τ))} \times d τ + B . \end{array} \end{array}

(34)

The constants A and B are found from the ICs.

The approximation (32) for

(β, ω_{0}, Q_{0}, γ) = (0.1, 2, 0.1, 1)

and

(φ_{0}, {\dot{φ}}_{0}) = (\frac{π}{6}, 0.1)

, is presented in figure 4. Also, the GMRDE L_d for the approximation (32) is estimated as follows

\begin{array}{rcl} L_{d} = max_{0 ⩽ t ⩽ 30} | A p p r o x . (32) - R K 4 | = 0.00631863 . \end{array}

Figure 4. The profile of the approximation (32) to the i.v.p. (29) is plotted against the RK4 numerical approximation for $(φ_{0}, {\dot{φ}}_{0}) = (\frac{π}{6}, 0.1)$ .

Full size|PPT slide

5. He's-HT for analyzing the forced (un)damped PPO

In the beginning, let us consider the forced undamped oscillator

\begin{array}{rcl} {\begin{cases} \ddot{φ} + ϕ (t) \sin φ = F (t), \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0}, \end{cases} \end{array}

(35)

where

F (t) = F_{0} \cos (w t)

We will replace the i.v.p. (35) with the i.v.p.

\begin{array}{rcl} {\begin{cases} \ddot{φ} + (ω_{0}^{2} - Q_{0} \cos (γ t)) (φ - \frac{1}{6} φ^{3} + \frac{1}{120} φ^{5}) = F (t), \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0} . \end{cases} \end{array}

(36)

The following homotopy is considered [46–49]

\begin{array}{rcl} \begin{array}{l} H (ρ) = \ddot{φ} + ω_{0}^{2} φ + ρ (- Q_{0} \cos (γ t) φ - \frac{1}{6} ϕ (t) φ^{3} \\ + \frac{1}{120} ϕ (t) φ^{5} - F (t)) . \end{array} \end{array}

(37)

Assuming the solution is given by the following ansatz form

\begin{array}{rcl} φ (t) = A \cos ω t + ρ u + ρ^{2} v + \dots, \end{array}

(38)

with

\begin{array}{rcl} ω = \sqrt{ω_{0}^{2} + ρ ω_{1} + ρ^{2} ω_{2} + \dots}, \end{array}

(39)

where u ≡ u(ωt) and v ≡ v(ωt).

Inserting both equations (38) and (39) into equation (37), we final get

\begin{array}{rcl} H (ρ) = ρ Y_{1} + ρ^{2} Y_{2} + \dots, \end{array}

(40)

where Y₁ and Y₂ are defined in appendix (I). Equating to zero the coefficients

Y_{i} (i = 1, 2, 3, \dots)

and then solving the resulting ode system, we have

\begin{array}{rcl} ω_{1} = \frac{1}{192} A^{4} ω_{0}^{2} - \frac{1}{8} A^{2} ω_{0}^{2}, \end{array}

(41)

and

\begin{array}{rcl} u (τ) = \frac{1}{Z_{9}} \sum_{i = 1}^{8} Z_{i}, \end{array}

(42)

where the values of

Z_{i} (i = 1, 2, \dots 9)

are defined in appendix (II).

By substituting equation (41) into equation (39), the following frequency-amplitude formula is obtained

\begin{array}{rcl} ω = ω_{0} \sqrt{1 - \frac{1}{8} A^{2} + \frac{1}{192} A^{4}} . \end{array}

(43)

For the damped case, we can replace A with

A \exp (- β t)

solution (38) and in frequency-amplitude formula (39) or (43).

The approximation (38) for both unforced (F₀ = 0) and forced (F₀ = 0.1 & w = 0.1) damped PPO is introduced in figure 5 for

(β, ω_{0}, Q_{0}, γ) = (0.1, 2, 0.1, 1)

and with different values to

(φ_{0}, {\dot{φ}}_{0})

. Also, the GMRDE L_d for both unforced (F₀ = 0) and forced (F₀ = 0.1 & w = 0.1) are estimated as follow

\begin{array}{rcl} \begin{array}{rcl} {L_{d} |}_{(φ_{0}, F_{0}) = (0, 0)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (38) - R K 4 | \\ = & 0.00171708, \\ {L_{d} |}_{(φ_{0}, F_{0}) = (0, 0.1)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (38) - R K 4 | \\ = & 0.00249545 \end{array} \end{array}

and

\begin{array}{rcl} \begin{array}{rcl} {L_{d} |}_{(φ_{0}, F_{0}) = (\frac{π}{6}, 0)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (38) - R K 4 | \\ = & 0.0236368, \\ {L_{d} |}_{(φ_{0}, F_{0}) = (\frac{π}{6}, 0.1)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (38) - R K 4 | \\ = & 0.022029 \end{array} \end{array}

Figure 5. The profile of the approximation (38) to the (un)forced i.v.p. (36) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Full size|PPT slide

6. The p-solution Method

Now, we proceed for analyzing the following the forced damped i.v.p.

\begin{array}{rcl} {\begin{cases} \ddot{φ} + 2 β \dot{φ} + ϕ (t) \sin φ = F (t), \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0}, \end{cases} \end{array}

(44)

for

\sin φ \approx φ - \frac{1}{6} φ^{3} + \frac{1}{120} φ^{5}

and

F (t) = F_{0} \cos (w t)

, using the p-solution Method. To do that, the forced damped i.v.p. (44) in the form of p- problem is considered

\begin{array}{rcl} {\begin{cases} \ddot{φ} + ω_{0}^{2} φ + p (2 β \dot{φ} - Q_{0} \cos (γ t) φ - \frac{1}{6} ϕ (t) φ^{3} + \frac{1}{120} ϕ (t) φ^{5} - F (t)) = 0, \\ φ (0) = φ_{0} a n d φ^{'} (0) = {\dot{φ}}_{0} . \end{cases} \end{array}

(45)

The solution of equation (45) is assumed to be

\begin{array}{rcl} φ_{p} = φ_{p} (t) = a \cos (ψ) + \sum_{n = 1}^{N} p^{n} u_{n} (a, ψ) + O (p^{N + 1}), \end{array}

(46)

where each u_n ≡ u_n(a, ψ) is a periodic function of ψ, and the functions

(a, ψ) \equiv (a (t), ψ (t))

are given by

\begin{array}{rcl} \frac{d a}{d t} \equiv \dot{a} = \sum_{n = 1}^{N} p^{n} A_{n} (a) + O (p^{N + 1}), \end{array}

(47)

\begin{array}{rcl} \frac{d ψ}{d t} \equiv \dot{ψ} = ω_{0} + \sum_{n = 1}^{N} p^{n} ψ_{n} (a) + O (p^{N + 1}) . \end{array}

(48)

Based on this technique, the solution of equation (45) until p², reads

\begin{array}{rcl} φ (t) = a \cos (ψ) + {p u}_{1} (a, ψ) + p^{2} u_{2} (a, ψ) + \dots, \end{array}

(49)

where the values of u₁(a, ψ) and u₂(a, ψ) are defined in appendix (II). The odes for determining the functions

(a, ψ)

read

\begin{array}{rcl} \dot{a} = - β a (t) p + \frac{β ϕ (t)}{16 ω_{0}^{2}} a {(t)}^{3} [\frac{1}{12} a {(t)}^{2} - 1] p^{2}, \end{array}

(50)

and

\begin{array}{rcl} \dot{ψ} = ω_{0} + p ψ_{1} + p^{2} ψ_{2}, \end{array}

(51)

with

\begin{array}{rcl} \begin{array}{rcl} ψ_{1} & = & \frac{a {(t)}^{4} ϕ (t) ((5 p - 8) ω_{0}^{2} - 9 {p Q}_{0} \cos (γ t))}{3072 ω_{0}^{3}} \\ - \frac{a {(t)}^{2} ϕ (t) ({p Q}_{0} \cos (γ t) + 2 ω_{0}^{2})}{32 ω_{0}^{3}} - \frac{Q_{0} \cos (γ t)}{2 ω_{0}}, \end{array} \end{array}

and

\begin{array}{rcl} \begin{array}{rcl} ψ_{2} & = & - \frac{β^{2}}{2 ω_{0}} - \frac{Q_{0}^{2} \cos^{2} (γ t)}{8 ω_{0}^{3}} + \frac{ϕ (t)^{2}}{9216 ω_{0}^{3}} a {(t)}^{6} \\ - \frac{11 a {(t)}^{8} ϕ (t)^{2}}{8847360 ω_{0}^{3}} . \end{array} \end{array}

Observe that this method can be applied for any oscillator, conservative, autonomous or not. The results are different from that obtained using He's-HT. On the other hand, He's-HT is hard to apply for a nonconservative oscillators. Even if the application is possible, the results are too large and cumbersome, as we saw in the previous section. Moreover, the solutions obtained using the p-method are simpler. The solution to the original problem is obtained for p = 1.

The approximation (49) for both unforced (F₀ = 0) and forced (F₀ = 0 & w = 3) damped PPO is introduced in figure 6 for

(β, ω_{0}, Q_{0}, γ, {\dot{φ}}_{0}) = (0.1, 2, 0.1, 1, 0.1)

and with different values to

(φ_{0}, F_{0})

. Also, the GMRDE L_d for both unforced (F₀ = 0) and forced (F₀ = 0.1 & w = 0.1) are estimated as follow

\begin{array}{rcl} \begin{array}{rcl} {L_{d} |}_{(φ_{0}, F_{0}) = (0, 0)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (49) - R K 4 | \\ = & 0.000522143, \\ {L_{d} |}_{(φ_{0}, F_{0}) = (0, 0.1)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (49) - R K 4 | \\ = & 0.000780554, \end{array} \end{array}

and

\begin{array}{rcl} \begin{array}{rcl} {L_{d} |}_{(φ_{0}, F_{0}) = (\frac{π}{6}, 0)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (49) - R K 4 | \\ = & 0.0059618, \\ {L_{d} |}_{(φ_{0}, F_{0}) = (\frac{π}{6}, 0.1)} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (49) - R K 4 | \\ = & 0.00629445 . \end{array} \end{array}

Figure 6. The profile of the approximation (46) to the (un)forced i.v.p. (44) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Full size|PPT slide

7. Finite different method for analyzing the damped PPO

Suppose we are given the following i.v.p.

\begin{array}{rcl} {\begin{cases} \ddot{x} + F (x, \dot{x}, t) = 0, \\ x (0) = x_{0} a n d x^{'} (0) = {\dot{x}}_{0} . \end{cases} \end{array}

(52)

Note here, for simplicity only we put

x (t) = φ (t)

and

F (φ, \dot{φ}, t) = 2 β \dot{φ} + ϕ (t) \sin (φ)

According to the FDM, we make use the following backward finite difference formulas for the first- and second-derivatives

\begin{array}{rcl} \begin{array}{rcl} x^{'} (t_{k}) & = & \frac{- 2 x_{k - 3} + 9 x_{k - 2} - 18 x_{k - 1} + 11 x_{k}}{6 Δ t}, \\ x^{''} (t_{k}) & = & \frac{- x_{k - 3} + 4 x_{k - 2} - 5 x_{k - 1} + 2 x_{k}}{Δ t^{2}} . \end{array} \end{array}

(53)

In order to obtain suitable values to x₁, x₂, and x₃, we make use the Padé-approximate near t = 0 as follows

\begin{array}{rcl} \begin{array}{l} P a d \overset{´}{e} (t) = x_{0} + t \\ \times ({\dot{x}}_{0} - \frac{3 t F (x_{0}, {\dot{x}}_{0}, 0)^{2}}{\begin{array}{c} 2 F (x_{0}, {\dot{x}}_{0}, 0) (F^{(0, 1, 0)} (x_{0}, {\dot{x}}_{0}, 0) t + 3) \\ - 2 t (F^{(0, 0, 1)} (x_{0}, {\dot{x}}_{0}, 0) + {\dot{x}}_{0} F^{(1, 0, 0)} (x_{0}, {\dot{x}}_{0}, 0)) \end{array}}) . \end{array} \end{array}

(54)

Accordingly, we get

\begin{array}{rcl} {\begin{cases} x_{1} = P a d \overset{´}{e} (t_{1}), \\ x_{2} = P a d \overset{´}{e} (t_{2}), \\ x_{3} = P a d \overset{´}{e} (t_{3}) . \end{cases} \end{array}

(55)

Now, for k = 4, 5, ⋯ , we have

\begin{array}{rcl} \begin{array}{l} F (z, \frac{- 2 x_{k - 3} + 9 x_{k - 2} - 18 x_{k - 1} + 11 z}{6 d t}, t_{k}) \\ + \frac{- x_{k - 3} + 4 x_{k - 2} - 5 x_{k - 1} + 2 z}{{d t}^{2}} = 0. \end{array} \end{array}

(56)

This is a transcendental equation for z = x_k. We may solve this equation using the Newton–Raphson or other suitable method, taking z = x_k−1 as initial value. This way, the FDM gives us the advantage of solving the ode recursively. Thus, the discretization form to the i.v.p. (2) reads

\begin{array}{rcl} \begin{array}{l} \frac{- φ_{k - 3} + 4 φ_{k - 2} - 5 φ_{k - 1} + 2 z}{{d t}^{2}} \\ + 2 β \frac{- 2 φ_{k - 3} + 9 φ_{k - 2} - 18 φ_{k - 1} + 11 z}{6 d t} \\ + (ω_{0}^{2} - Q_{0} \cos (γ k Δ t)) \sin (z) = 0, \\ x_{0} = φ_{0}, φ_{1} = P a d \overset{´}{e} (t_{1}), φ_{2} = P a d \overset{´}{e} (t_{2}), \\ a n d φ_{3} = P a d \overset{´}{e} (t_{3}), z = φ_{k}, \\ t_{k} = k Δ t f o r k = 4, 5, 6, \dots . \end{array} \end{array}

(57)

(β, ω_{0}, Q_{0}, γ, {\dot{φ}}_{0}) = (0.1, 1, 0.1, 1, 0.2)

and for different values to φ₀ (here φ₀ = 0 and φ₀ = π/6), the comparison between the analytical approximation (13) and both Padé-FDM and RK4 numerical approximations is displayed in figure 7. It is found that the numerical approximation using the Padé-FDM is better than the analytical approximation (13). However, in general, both analytical and numerical approximations give highly-accurate and acceptable results. Moreover, the comparison between all proposed techniques at

(β, ω_{0}, Q_{0}, γ, φ_{0}, {\dot{φ}}_{0}) = (0.1, 1, 0.1, 1, π / 6, 0.2)

is carried out as follows

\begin{array}{rcl} \begin{array}{rcl} L_{d} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x i . (10) - R K 4 | = 0.0285408, \\ L_{d} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x i . (13) - R K 4 | = 0.0270288, \\ L_{d} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x i . (15) - R K 4 | = 0.0257492, \\ L_{d} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x i . (32) - R K 4 | = 0.0293165, \\ L_{d} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (38) - R K 4 | = 0.0244667, \\ L_{d} & = & max_{0 ⩽ t ⩽ 30} | A p p r o x . (49) - R K 4 | = 0.0279174, \\ L_{d} & = & max_{0 ⩽ t ⩽ 30} | P a d \overset{´}{e} - F D M - R K 4 | = 0.0000314009 . \end{array} \end{array}

Figure 7. The profile of the numerical solutions using both Padé-FDM and RK4 method and the optimal analytical approximation (13) to the i.v.p. (2) for $(φ_{0}, {\dot{φ}}_{0}) = (0, 0.2)$ and $(φ_{0}, {\dot{φ}}_{0}) = (π / 6, 0.2)$ is considered.

Full size|PPT slide

8. Conclusions

The (un)forced (un)damped PPE have been investigated analytical and numerical using some different approaches. The ansatz method was devoted for deriving some analytical approximations to the damped PPE in the form angular Mathieu functions. Using some suitable assumptions, the obtained analytical approximation has been improved based on two-optimal parameters. In the second approach, the ansatz method was applied for deriving some approximations to the (un)forced damped PPE in the form of trigonometric functions using the ansatz method. In the third approach, He's-FAP was implemented for getting some approximations to the (un)damped PPE. In the forth approach, He's-HT was employed for analyzing the forced (un)damped PPE. In the fifth approach, the p-solution Method was implemented for deriving an approximation to the forced damped PPE. In the final approach, the hybrid Padé-FDM was performed for analyzing the damped PPE. During the numerical simulation, some different cases for small and large angle with the vertical pivot have been discussed. It was found that the accuracy of both first and second formulas for the analytical approximations become identical for small angle, but for large angle the accuracy of second formula becomes better. Furthermore, both analytical and numerical approximations were compared with each other and it turned out that the numerical approximation using Padé-FDM is more accurate than the other approaches. The analytical and numerical techniques that were used in this study can be extended to investigate many nonlinear oscillators.

Acknowledgments

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project (Grant No. PNURSP2022R17), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Taif University Researchers supporting project number (TURSP-2020/275), Taif University, Taif, Saudi Arabia.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Author contributions

All authors contributed equally and approved the final manuscript.

Appendix

Appendix (I): the coefficients of equation (40)

\begin{array}{rcl} Y_{1} = [\begin{array}{c} - \frac{1}{120} A^{5} Q_{0} \cos^{5} (τ) \cos (\frac{γ τ}{ω_{0}}) + \frac{1}{6} A^{3} Q_{0} \cos^{3} (τ) \cos (\frac{γ τ}{ω_{0}}) \\ + ω_{0}^{2} (\frac{1}{120} A^{5} \cos^{5} (τ) - \frac{1}{6} A^{3} \cos^{3} (τ) + u^{''} (τ) + u (τ)) \\ - {A Q}_{0} \cos (τ) \cos (\frac{γ τ}{ω_{0}}) - A ω_{1} \cos (τ) - F_{0} \cos (\frac{τ w}{ω_{0}}) \end{array}], \end{array}

and

\begin{array}{rcl} Y_{2} = \frac{1}{48 ω_{0}^{2}} [\begin{array}{c} 2 ω_{0}^{2} (\begin{matrix} (\begin{matrix} A^{2} ω_{0}^{2} \cos^{2} (τ) (A^{2} \cos^{2} (τ) - 12) - \\ Q_{0} (A^{4} \cos^{4} (τ) - 12 A^{2} \cos^{2} (τ) + 24) \cos (\frac{γ τ}{ω_{0}}) \end{matrix}) u (τ) \\ - 24 A ω_{2} \cos (τ) + 24 ω_{0}^{2} (v^{''} (τ) + v (τ)) \end{matrix}) \\ + ω_{1} (\begin{matrix} ω_{0}^{2} (\begin{matrix} A^{5} (- τ) \sin (τ) \cos^{4} (τ) + 12 A^{3} τ \sin (τ) \cos^{2} (τ) \\ + 24 τ u^{(3)} (τ) + 48 u^{''} (τ) + 24 τ u^{'} (τ) \end{matrix}) + \\ {A Q}_{0} τ \sin (τ) (A^{4} \cos^{4} (τ) - 12 A^{2} \cos^{2} (τ) + 24) \cos (\frac{γ τ}{ω_{0}}) \end{matrix}) \\ + 24 A τ ω_{1}^{2} \sin (τ) \end{array}], \end{array}

where ω₀t = τ.

Appendix (II): The values of

Z_{i} (i = 1, 2, \dots, 9)

\begin{array}{rcl} \begin{array}{rcl} Z_{1} & = & 1200 Q Ω^{2} \cos^{4} (τ) \sin (τ) \\ \times \sin (\frac{Ω τ}{ω_{0}}) ω_{0} (Ω^{2} - 4 ω_{0}^{2}) (ω_{0}^{2} - w^{2}) A^{5}, \\ Z_{2} & = & Ω \cos^{5} (τ) (Ω^{2} - 4 ω_{0}^{2}) (w^{2} - ω_{0}^{2}) \\ \times [Ω^{4} - 52 ω_{0}^{2} Ω^{2} + 576 ω_{0}^{4} - 24 Q \\ \times \cos (\frac{Ω τ}{ω_{0}}) (Ω^{2} + 24 ω_{0}^{2})] A^{5}, \end{array} \end{array}

\begin{array}{rcl} \begin{array}{rcl} Z_{3} & = & 240 Q Ω^{2} \cos^{2} (τ) \sin (τ) \sin (\frac{Ω τ}{ω_{0}}) ω_{0} (w^{2} - ω_{0}^{2}) \\ \times [\begin{array}{c} - 9 (A^{2} - 16) Ω^{2} \\ + 16 (19 A^{2} - 324) ω_{0}^{2} + 20 A^{2} \cos (2 τ) ω_{0}^{2} \end{array}] A^{3}, \end{array} \end{array}

\begin{array}{rcl} \begin{array}{rcl} Z_{4} & = & 15 (A^{2} - 16) Ω \cos (3 τ) (w^{2} - ω_{0}^{2}) \\ \times (Ω^{6} - 56 ω_{0}^{2} Ω^{4} + 784 ω_{0}^{4} Ω^{2} - 2304 ω_{0}^{6}) A^{3}, \end{array} \end{array}

\begin{array}{rcl} \begin{array}{rcl} Z_{5} & = & 5 Ω \cos^{3} (τ) (w^{2} - ω_{0}^{2}) \\ \times [\begin{array}{c} - A^{2} Ω^{6} + A^{2} \cos (2 τ) Ω^{6} \\ - 24 Q \cos (\frac{Ω τ}{ω_{0}}) (\begin{matrix} A^{2} \cos (2 τ) (Ω^{4} + 20 ω_{0}^{2} Ω^{2} - 96 ω_{0}^{4}) \\ - 16 (Ω^{4} + (3 A^{2} - 28) ω_{0}^{2} Ω^{2} + 12 (A^{2} - 24) ω_{0}^{4}) \end{matrix}) \end{array}] A^{3}, \end{array} \end{array}

\begin{array}{rcl} \begin{array}{rcl} Z_{6} & = & - 10 \sin (τ) ω_{0} (ω_{0}^{2} - w^{2}) \\ \times [\begin{array}{c} 8 A^{4} Ω \cos^{3} (τ) \sin (τ) ω_{0} (7 Ω^{4} - 98 ω_{0}^{2} Ω^{2} + 288 ω_{0}^{4}) \\ - 3 Q \sin (\frac{Ω τ}{ω_{0}}) (\begin{matrix} - 3 Ω^{4} A^{4} + 9216 ω_{0}^{4} A^{4} - 412 Ω^{2} ω_{0}^{2} A^{4} \\ + Ω^{2} \cos (4 τ) (11 Ω^{2} - 4 ω_{0}^{2}) A^{4} - 192 Ω^{4} A^{2} \\ - 221184 ω_{0}^{4} A^{2} + 13056 Ω^{2} ω_{0}^{2} A^{2} \\ + 8 Ω^{2} \cos (2 τ) ((A^{2} - 24) Ω^{2} + 4 (216 - 13 A^{2}) ω_{0}^{2}) A^{2} \\ + 3072 Ω^{4} + 1769472 ω_{0}^{4} - 159744 Ω^{2} ω_{0}^{2} \end{matrix}) \end{array}] A, \end{array} \end{array}

\begin{array}{rcl} \begin{array}{l} Z_{7} = - \frac{5}{8} Ω \cos (τ) (w^{2} - ω_{0}^{2}) \\ \times [\begin{array}{c} 4 \cos (2 τ) (Ω^{6} - 56 ω_{0}^{2} Ω^{4} + 784 ω_{0}^{4} Ω^{2} - 2304 ω_{0}^{6}) A^{4} \\ - \cos (4 τ) (Ω^{6} - 56 ω_{0}^{2} Ω^{4} + 784 ω_{0}^{4} Ω^{2} - 2304 ω_{0}^{6}) A^{4} \\ - 3 (\begin{matrix} A^{4} Ω^{6} - 56 A^{4} ω_{0}^{2} Ω^{4} - 4 A^{4} Q \cos (τ (\frac{Ω}{ω_{0}} + 4)) Ω^{4} \\ - 32 A^{4} Q \cos (τ (\frac{Ω}{ω_{0}} + 2)) Ω^{4} + 768 A^{2} Q \cos (τ (\frac{Ω}{ω_{0}} + 2)) Ω^{4} \\ - 56 A^{4} Q \cos (\frac{Ω τ}{ω_{0}}) Ω^{4} + 1536 A^{2} Q \cos (\frac{Ω τ}{ω_{0}}) Ω^{4} \\ - 24576 Q \cos (\frac{Ω τ}{ω_{0}}) Ω^{4} + 784 A^{4} ω_{0}^{4} Ω^{2} \\ + 1664 A^{4} Q \cos (τ (\frac{Ω}{ω_{0}} + 2)) ω_{0}^{2} Ω^{2} - 21504 A^{2} Q \cos (τ (\frac{Ω}{ω_{0}} + 2)) ω_{0}^{2} Ω^{2} \\ - 80 A^{4} Q \cos (τ (\frac{Ω}{ω_{0}} + 4)) ω_{0}^{2} Ω^{2} + 384 A^{4} Q \cos (τ (\frac{Ω}{ω_{0}} + 4)) ω_{0}^{4} \\ + 3488 A^{4} Q \cos (\frac{Ω τ}{ω_{0}}) ω_{0}^{2} Ω^{2} - 116736 A^{2} Q \cos (\frac{Ω τ}{ω_{0}}) ω_{0}^{2} Ω^{2} \\ + 1277952 Q \cos (\frac{Ω τ}{ω_{0}}) ω_{0}^{2} Ω^{2} - 2304 A^{4} ω_{0}^{6} \\ + 12288 A^{4} Q \cos (τ (\frac{Ω}{ω_{0}} + 2)) ω_{0}^{4} - 221184 A^{2} Q \cos (τ (\frac{Ω}{ω_{0}} + 2)) ω_{0}^{4} \\ - 99072 A^{4} Q \cos (\frac{Ω τ}{ω_{0}}) ω_{0}^{4} + 2211840 A^{2} Q \cos (\frac{Ω τ}{ω_{0}}) ω_{0}^{4} - 14155776 Q \cos (\frac{Ω τ}{ω_{0}}) ω_{0}^{4} \\ - 4 A^{4} Q \cos (τ (4 - \frac{Ω}{ω_{0}})) (Ω^{4} + 20 ω_{0}^{2} Ω^{2} - 96 ω_{0}^{4}) \\ - 32 A^{2} Q \cos (τ (2 - \frac{Ω}{ω_{0}})) ((A^{2} - 24) Ω^{4} + 4 (168 - 13 A^{2}) ω_{0}^{2} Ω^{2} - 384 (A^{2} - 18) ω_{0}^{4}) \end{matrix}) \end{array}] A \end{array} \end{array}

\begin{array}{rcl} \begin{array}{rcl} Z_{8} & = & - 46080 γ Ω \cos (\frac{w τ}{ω_{0}}) (Ω^{6} - 56 ω_{0}^{2} Ω^{4} \\ + 784 ω_{0}^{4} Ω^{2} - 2304 ω_{0}^{6}), \end{array} \end{array}

\begin{array}{rcl} \begin{array}{rcl} Z_{9} & = & 46080 Ω (w^{2} - ω_{0}^{2}) (Ω^{6} - 56 ω_{0}^{2} Ω^{4} \\ + 784 ω_{0}^{4} Ω^{2} - 2304 ω_{0}^{6}) . \end{array} \end{array}

Appendix (II): The values of both u₁(a, ψ) and u₂(a, ψ):

\begin{array}{rcl} \begin{array}{l} u_{1} (a, ψ) = \frac{1}{23040 ω_{0}^{2}} [a^{3} ϕ (t) \cos (ψ) (a^{2} (14 \cos (2 ψ) \\ + \cos (4 ψ) - 7) - 240 \cos (2 ψ) + 120)] + \frac{F (t)}{ω_{0}^{2}}, \end{array} \end{array}

and

\begin{array}{rcl} \begin{array}{l} u_{2} (a, ψ) = \frac{1}{4246732800 ω_{0}^{4}} \\ \times [\begin{array}{c} a^{9} ϕ {(t)}^{2} (- 5280 \cos (3 ψ) + 160 \cos (5 ψ) + 95 \cos (7 ψ) + 3 \cos (9 ψ)) \\ - 1440 a^{7} ϕ {(t)}^{2} (- 164 \cos (3 ψ) + 4 \cos (5 ψ) + \cos (7 ψ)) \\ + 7680 a^{5} ϕ (t) [\begin{matrix} 5 ω_{0} (\begin{matrix} 4 β (27 \sin (3 ψ) + \sin (5 ψ)) \\ - 63 ω_{0} \cos (3 ψ) + 3 ω_{0} \cos (5 ψ) \end{matrix}) \\ - 3 Q_{0} \cos (γ t) (\cos (5 ψ) - 165 \cos (3 ψ)) \end{matrix}] \\ + 1474560 a^{4} ϕ (t) F (t) (20 \cos (2 ψ) + \cos (4 ψ) - 45) \\ - 11059200 a^{3} ϕ (t) (3 ε ω_{0} \sin (3 ψ) + 2 Q_{0} \cos (γ t) \cos (3 ψ)) \\ - 353894400 a^{2} ϕ (t) F (t) (\cos (2 ψ) - 3) + 4246732800 F (t) Q_{0} \cos (γ t) \end{array}] . \end{array} \end{array}

References

List( Publishing order | Descend order by publishing year | Descend order by cited within ) Chart analysis

1	Baskonus H M Hasan B Abdulkadir S T 2017 Investigation of various travelling wave solutions to the extended (2+1)-dimensional quantum ZK equation Eur. Phys. J. Plus 132 482 https://doi.org/10.1140/epjp/i2017-11778-y Cited in this article [1]

2	Abdulkadir S T Hasan B Baskonus H M 2021 On the exact solutions to some system of complex nonlinear models Appl. Math. Nonlinear Sci. 6 29 42 https://doi.org/10.2478/amns.2020.2.00007

3	Hasan B Nesligül A E Miraç K Abdulkadir S T 2019 New solitary wave structures to the (3+1) dimensional Kadomtsev–Petviashvili and Schrödinger equation J. Ocean Eng. Sci. 4 373 378 https://doi.org/10.1016/j.joes.2019.06.002

4	Onur Alp İ Hasan B Abdulkadir S T Mehmet B H 2019 On the new wave behavior of the Magneto-Electro-Elastic(MEE) circular rod longitudinal wave Int. J. Optim. Control: Theories Appl. (IJOCTA) 10 1 8 https://doi.org/10.11121/ijocta.01.2020.00837

5	Abdulkadir S T 2020 Three-component coupled nonlinear Schrödinger equation: optical soliton and modulation instability analysis Phys. Scr. 95 065201 https://doi.org/10.1088/1402-4896/ab7c77

6	Abdulkadir S T Ibrahim N R Bashir M B 2019 Dark and singular solitons to the two nonlinear Schrödinger equations Optik 186 423 430 https://doi.org/10.1016/j.ijleo.2019.04.023

7	Abdulkadir S T Abdullahi Y Abdon A 2020 New lump, lump-kink, breather waves and other interaction solutions to the (3+1)-dimensional soliton equation Commun. Theor. Phys. 72 085004 https://doi.org/10.1088/1572-9494/ab8a21 Cited in this article [1]

8	Nayfeth N Mook D T 1973 Non-Linear Oscillations New York John Wiley https://www.wiley.com/en-us/Nonlinear+Oscillations-p-9780471121428 Cited in this article [1]

9	Albalawi W Salas A H El-Tantawy S A Youssef A Abd A-R 2021 Approximate analytical and numerical solutions to the damped pendulum oscillator: Newton–Raphson and moving boundary methods J. Taibah Univ. Sci. 15 479 https://doi.org/10.1080/16583655.2021.1989739

10	El-Tantawy S A Salas Alvaro H Alharthi M R 2021 On the analytical solutions of the forced damping duffing equation in the form of weierstrass elliptic function and its applications Math. Probl. Eng. 2021 6678102 https://doi.org/10.1155/2021/6678102 Cited in this article [1]

11	Wazwaz A-M 2009 Partial Differential Equations and Solitary Waves Theory, Higher Education Beijing, Berlin Springer Cited in this article [1]

12	Yang Z-J Zhang S-M Li X-L Pang Z-G 2018 Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation Appl. Math. Lett. 82 64 https://doi.org/10.1016/j.aml.2018.02.018

13	Song L-M Yang Z-J Li X-L Zhang S-M 2018 Controllable Gaussian-shaped soliton clusters in strongly nonlocal media Opt. Express 26 19182 https://doi.org/10.1364/OE.26.019182

14	Song L-M Yang Z-J Zhang S-M Li X-L 2019 Spiraling anomalous vortex beam arrays in strongly nonlocal nonlinear media Phys. Lett. A 99 063817

15	Song L-M Yang Z-J Pang Z-G Li X-L Zhang S-M 2019 Interaction theory of mirror-symmetry soliton pairs in nonlocal nonlinear Schrödinger equation Appl. Math. Lett. 90 42 https://doi.org/10.1016/j.aml.2018.10.008 Cited in this article [1]

16	O'Neil T 1965 Collisionless damping of nonlinear plasma oscillations Phys. Fluids 8 2255 https://doi.org/10.1063/1.1761193 Cited in this article [1]

17	Tantawy S A Salas Alvaro H Alharthi M R 2021 A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuits Commun. Theor. Phys. 73 035501 https://doi.org/10.1088/1572-9494/abda1b

18	Salas Alvaro H El-Tantawy S A Alharthi M R 2021 Novel solutions to the (un) damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method Phys. Scr. 96 104003 https://doi.org/10.1088/1402-4896/ac0c57

19	Salas Alvaro H El-Tantawy S A 2021 Anal. Solut. Some Strong Nonlinear Oscillators https://doi.org/10.5772/intechopen.97677 Cited in this article [1]

20	Barkham P G D Soudack A C 1969 An extension to the method of Kryloff and Bogoliuboff Int. J. Control. 10 377 https://doi.org/10.1080/00207176908905841 Cited in this article [1]

21	Aljahdaly Noufe H El-Tantawy S A 2021 On the Multistage Differential Transformation Method for Analyzing Damping Duffing Oscillator Applications to Plasma Physics Mathematics 9 432 https://doi.org/10.3390/math9040432

22	Salas Alvaro H Castillo H Jairo E Mosquera D P 2020 A New Approach for Solving the Undamped Helmholtz oscillator for the given arbitrary initial conditions and Its physical applications J. Math. Probl. Eng. 2020 7876413

23	Turkyilmazoglu M 2012 An effective approach for approximate analytical solutions of the damped Duffing equation Phys. Scr. 86 015301 https://doi.org/10.1088/0031-8949/86/01/015301

24	Abdelhafez H M 2016 Solution of Excited Non-Linear oscillators under damping effects using the modified differential transform method Mathematics 4 11 https://doi.org/10.3390/math4010011

25	Collins I F 1973 On the theory of rigid/perfectly plastic plates under uniformly distributed loads Acta Mech. 18 233 https://doi.org/10.1007/BF01178556

26	Alyousef Haifa A Salas Alvaro H Alkhateeb Sadah A El-Tantawy S A 2022 some novel analytical approximations to the (Un) damped duffing–mathieu oscillators J. Math. 2022 2715767

27	Zajaczkowski J 1981 Destabilizing effect of Coulomb friction on vibration of a beam supported at an axially oscillating mount J. Sound Vib. 79 575 https://doi.org/10.1016/0022-460X(81)90467-3

28	Chang S S 1983 The general solutions of the doubly periodic cracks Eng. Fract. Mech. 18 887 https://doi.org/10.1016/0013-7944(83)90133-9

29	Grozev D Shivarova A Boardman A D 1987 Envelope solitons of surface waves in a plasma column J. Plasma Phys. 38 427 https://doi.org/10.1017/S0022377800012691

30	Manevich A I 1994 Interaction of coupled modes accompanying non-linear flexural vibrations of a circular ring J. Appl. Math. Mech. 58 1061 https://doi.org/10.1016/0021-8928(94)90122-8

31	Rand R H Kinsey R J Mingori D L 1992 Dynamics of spinup through resonance Int. J. Non Linear Mech. 27 489 https://doi.org/10.1016/0020-7462(92)90015-Y

32	Hu W Scheeres D J 2000 Spacecraft Motion About Slowly Rotating Asteroids J. Guid. Control Dyn. 25 765–65

33	Lestari W Hanagud S 2001 Nonlinear vibration of buckled beams: some exact solutions Int. J. Solids Struct. 38 4741 https://doi.org/10.1016/S0020-7683(00)00300-0

34	Liu S Fu Z Liu S Zhao Q 2001 Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations Phys. Lett. A 289 69 https://doi.org/10.1016/S0375-9601(01)00580-1

35	He J-H Moatimid G M Zekry M H 2022 Forced nonlinear oscillator in s fractal space Facta Univer. Ser.: Mech. Eng. 20 1 20 https://doi.org/10.22190/FUME220118004H

36	He J-H Amer T S Elnaggar S Galal A A 2021 Periodic Property and Instability of a Rotating Pendulum System Axioms 10 191 https://doi.org/10.3390/axioms10030191

37	He C H Tian D Moatimid G M Salman H F Zekry M H 2021 Stability analysis and controller J. Low Freq. Noise, Vib. Active Control 4 244 https://doi.org/10.1177/14613484211026407journals.sagepub.com/home/lfn Cited in this article [1]

38	Johannessen K 2014 An analytical solution to the equation of motion for the damped nonlinear pendulum Eur. J. Phys. 35 035014 https://doi.org/10.1088/0143-0807/35/3/035014 Cited in this article [1]

39	Yu Y G Luo A C J 2019 Bifurcation dynamics of a damped parametric pendulum synthesis lectures on mechanical engineering Synth. Lectures Mech. Eng. 3 1 98 https://doi.org/10.2200/S00956ED1V01Y201910MEC022 Cited in this article [1]

40	Salas Alvaro H Wedad A Alharthi M R El-Tantawy S A 2022 Some Novel solutions to a quadratically damped pendulum oscillator: analytical and numerical approximations Complexity 2022 7803798 Cited in this article [1]

41	He J-H 2006 Some asymptotic methods for strongly nonlinear equations Int J. Mod. Phys. B 20 1141 1199 https://doi.org/10.1142/S0217979206033796 Cited in this article [1]

42	He J-H 2008 An improved amplitude-frequency formulation for nonlinear oscillators Int. J. Nonlinear Sci. Numer. Simul. 9 211 212 https://doi.org/10.1515/IJNSNS.2008.9.2.211

43	He J-H 2008 Comment on He's frequency formulation for nonlinear oscillators Eur. J. Phys. 29 19 22

44	Ren Z-Y 2017 Theoretical basis of He's frequency–amplitude formulation for nonlinear oscillators Nonlinear. Sci. Lett. A 9 86 90

45	He J-H 2017 Amplitude-Frequency Relationship for Conservative Nonlinear Oscillators with Odd Nonlinearities Int. J. Appl. Comput. Math. 3 1557 1560 https://doi.org/10.1007/s40819-016-0160-0

Alyousef Haifa

Salas Alvaro

Alharthi

M R

El-Tantawy

S A

2022 Galerkin method, ansatz method, and He's frequency formulation for modeling the forced damped parametric driven pendulum oscillators J. Low Freq. Noise, Vib. Active Control

https://doi.org/10.1177/14613484221101235

Cited in this article [2]

47	He J-H El-Dib Yusry O 2021 Homotopy perturbation method with three expansions for Helmholtz-Fangzhu oscillator Int. J. Mod. Phys. B 35 2150244 https://doi.org/10.1142/S0217979221502441

48	He J-H El-Dib Yusry O Mady Amal A 2021 Homotopy Perturbation Method for the Fractal Toda Oscillator Fractal Fract. 5 93 https://doi.org/10.3390/fractalfract5030093

49	El-Dib Y O 2017 Multiple scales homotopy perturbation method for nonlinear oscillators Nonlinear Sci. Lett. A 8 352 364 Cited in this article [1]

RIGHTS & PERMISSIONS

PDF(1616 KB)

143

Accesses

Citation

Detail

Sections

Recommended

Abstract
Key words
Cite this article
1. Introduction
2. First approach for analyzing the damped PPO
Figure 1. The profile of two-analytical approximations (10) and (13) to the i.v.p. (4) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .
3. Second approach for analyzing the (un)forced damped PPO
Figure 2. The profile of the approximation (15) to the i.v.p. (22) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .
Figure 3. The profile of the approximation (24) to the i.v.p. (23) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .
4. He's-FAP for analyzing the (un)damped pendulum oscillator
Figure 4. The profile of the approximation (32) to the i.v.p. (29) is plotted against the RK4 numerical approximation for $(φ_{0}, {\dot{φ}}_{0}) = (\frac{π}{6}, 0.1)$ .
5. He's-HT for analyzing the forced (un)damped PPO
Figure 5. The profile of the approximation (38) to the (un)forced i.v.p. (36) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .
6. The p-solution Method
Figure 6. The profile of the approximation (46) to the (un)forced i.v.p. (44) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .
7. Finite different method for analyzing the damped PPO
Figure 7. The profile of the numerical solutions using both Padé-FDM and RK4 method and the optimal analytical approximation (13) to the i.v.p. (2) for $(φ_{0}, {\dot{φ}}_{0}) = (0, 0.2)$ and $(φ_{0}, {\dot{φ}}_{0}) = (π / 6, 0.2)$ is considered.
8. Conclusions
Acknowledgments
Conflicts of interest
Author contributions
Appendix
References
RIGHTS & PERMISSIONS

Received	Revised	Accepted	Published
2022-03-02	2022-06-06	2022-06-24	2022-09-26
Issue Date
2022-09-26

Please choose a citation manager

Content to export

Abstract

Key words

Cite this article

1. Introduction

2. First approach for analyzing the damped PPO

Figure 1. The profile of two-analytical approximations (10) and (13) to the i.v.p. (4) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

3. Second approach for analyzing the (un)forced damped PPO

Figure 2. The profile of the approximation (15) to the i.v.p. (22) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Figure 3. The profile of the approximation (24) to the i.v.p. (23) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

4. He's-FAP for analyzing the (un)damped pendulum oscillator

Figure 4. The profile of the approximation (32) to the i.v.p. (29) is plotted against the RK4 numerical approximation for $(φ_{0}, {\dot{φ}}_{0}) = (\frac{π}{6}, 0.1)$ .

5. He's-HT for analyzing the forced (un)damped PPO

Figure 5. The profile of the approximation (38) to the (un)forced i.v.p. (36) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

6. The p-solution Method

Figure 6. The profile of the approximation (46) to the (un)forced i.v.p. (44) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

7. Finite different method for analyzing the damped PPO

Figure 7. The profile of the numerical solutions using both Padé-FDM and RK4 method and the optimal analytical approximation (13) to the i.v.p. (2) for $(φ_{0}, {\dot{φ}}_{0}) = (0, 0.2)$ and $(φ_{0}, {\dot{φ}}_{0}) = (π / 6, 0.2)$ is considered.

8. Conclusions

Acknowledgments

Conflicts of interest

Author contributions

Appendix

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

RIGHTS & PERMISSIONS

模态框（Modal）标题

Please choose a citation manager

Content to export

Abstract

Key words

Cite this article

1. Introduction

2. First approach for analyzing the damped PPO

Figure 1. The profile of two-analytical approximations (10) and (13) to the i.v.p. (4) is plotted against the RK4 numerical approximation for different values to the ICs: (φ0,φ˙0).

3. Second approach for analyzing the (un)forced damped PPO

Figure 2. The profile of the approximation (15) to the i.v.p. (22) is plotted against the RK4 numerical approximation for different values to the ICs: (φ0,φ˙0).

Figure 3. The profile of the approximation (24) to the i.v.p. (23) is plotted against the RK4 numerical approximation for different values to the ICs: (φ0,φ˙0).

4. He's-FAP for analyzing the (un)damped pendulum oscillator

Figure 4. The profile of the approximation (32) to the i.v.p. (29) is plotted against the RK4 numerical approximation for (φ0,φ˙0)=(π6,0.1).

5. He's-HT for analyzing the forced (un)damped PPO

Figure 5. The profile of the approximation (38) to the (un)forced i.v.p. (36) is plotted against the RK4 numerical approximation for different values to the ICs: (φ0,φ˙0).

6. The p-solution Method

Figure 6. The profile of the approximation (46) to the (un)forced i.v.p. (44) is plotted against the RK4 numerical approximation for different values to the ICs: (φ0,φ˙0).

7. Finite different method for analyzing the damped PPO

Figure 7. The profile of the numerical solutions using both Padé-FDM and RK4 method and the optimal analytical approximation (13) to the i.v.p. (2) for (φ0,φ˙0)=(0,0.2) and (φ0,φ˙0)=(π/6,0.2) is considered.

8. Conclusions

Acknowledgments

Conflicts of interest

Author contributions

Appendix

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

RIGHTS & PERMISSIONS

Figure 1. The profile of two-analytical approximations (10) and (13) to the i.v.p. (4) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Figure 2. The profile of the approximation (15) to the i.v.p. (22) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Figure 3. The profile of the approximation (24) to the i.v.p. (23) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Figure 4. The profile of the approximation (32) to the i.v.p. (29) is plotted against the RK4 numerical approximation for $(φ_{0}, {\dot{φ}}_{0}) = (\frac{π}{6}, 0.1)$ .

Figure 5. The profile of the approximation (38) to the (un)forced i.v.p. (36) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Figure 6. The profile of the approximation (46) to the (un)forced i.v.p. (44) is plotted against the RK4 numerical approximation for different values to the ICs: $(φ_{0}, {\dot{φ}}_{0})$ .

Figure 7. The profile of the numerical solutions using both Padé-FDM and RK4 method and the optimal analytical approximation (13) to the i.v.p. (2) for $(φ_{0}, {\dot{φ}}_{0}) = (0, 0.2)$ and $(φ_{0}, {\dot{φ}}_{0}) = (π / 6, 0.2)$ is considered.