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

1 Department of physics,College of Science,Princess Nourah bint Abdulrahman University,P.O.Box 84428,Riyadh 11671,Saudi Arabia

2 Department of Mathematics and Statistics,College of Science,Taif University,P.O.Box 11099,Taif 21944,Saudi Arabia

3 Department of Mathematics and Statistics,Universidad Nacional de Colombia,FIZMAKO Research Group,Sede Manizales,Colombia

4 Department of Physics,Faculty of Science,Port Said University,Port Said 42521,Egypt

5 Research Center for Physics(RCP),Department of Physics,Faculty of Science and Arts,Al-Mikhwah,Al-Baha University,Saudi Arabia

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.

Keywords:parametric pendulum equation,Ansatz method,He’s frequency-amplitude principle,He’s homotopy technique,Krylov–Bogoliúbov Mitropolsky method,the hybrid Padé-finite difference method

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 highamplitude 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]

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 frequencyamplitude 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.

where i=3,5,…,i.e.only odd numbers.Note that the values of λican 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 w

here 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(s in(θ)≈θ,)

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

with

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

For determining the function g(t),we replace θ(t)byθ(g(t))in solution(5)to be

and inserting the obtained result inR(given in equation(2)),we get

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

here,ρ denotes the free integration constant.Observe thatg(t)does not depend on the coefficient λ give inR1.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 ofg(t)(given in equation(9))into solution(7),then the solution of the i.v.p.(2)is obtained

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

where both(ρ,F)give the free parameters of optimization.In this case the values ofg(t)given in equation(9)and Z1given in equation(6)have the following new forms

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

It is clear that the second optimal parameterF 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).

3.Second approach for analyzing the(un)forced 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 0,0).

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

Equating both S1and S2to 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

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:(φ0,0).

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 Ld.

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

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:(φ0,0).

Figure 4.The profile of the approximation(32)to the i.v.p.(29)is plotted against the RK4 numerical approximation for.

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 0,0).

6.The p-solution Method

7.Finite different method for analyzing the damped PPO

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 0,0).

8.Conclusions

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,φ˙ 0)=(0 ,0.2)and(φ 0,0)=(π 6,0.2)is considered.

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)

and

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