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

Abstract

The (un)forced (un)damped parametric pendulum oscillator 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 parametric pendulum oscillator (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 (He's-FAP) is applied for deriving some approximations to the (un)damped PPO. In the forth approach, He's Homotopy technique (He's-HT) 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 (KBMM) is introduced for deriving an approximation to the forced damped PPO. In the final approach, the hybrid Padé-finite difference method (Padé-FDM) 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.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

“…The Multiple Scales Method (MSM) and Krýlov-Bogoliúbov-Mitropólsky method (KBMM) were employed to provide approximate solutions for a time Delay Duffing-Helmholtz equation [25]. Furthermore, both KBMM and MSM were used for analyzing and solving several nonlinear oscillators with strong nonlinearity [26][27][28][29][30][31][32].…”

Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach

“…and some other equations related to this oscillator have been analyzed and investigated using some different effective analytical and numerical techniques, such as the ansatz method [4], He's frequency-amplitude principle [4], He's homotopy perturbation method (HPM) [4], the Krylov-Bogoliúbov Mitropolsky (KBM) method [4], the 4th-order Runge Kutta (RK4), the hybrid Padé-finite difference method [4], the Chebyshev collocation method (CCM) [5], the Galerkin method [6], the ansatz method (AM) and He's frequency formulation [6]. Moreover, the AM and the HPT with the extended KBM were used in the study of the damped cubic nonlinearity Duffing-Mathieu-type oscillator [7].…”

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

“…The nonlinear differential equations (NDEs) including all their types, such as ordinary, partial, linear, and nonlinear types have a huge impact on scientific research because they can model a wide range of real-life phenomena, engineering, and physical problems [1][2][3][4][5][6][7][8][9][10]. For instance, Wazwaz [1] discussed a huge number of (non)linear partial differential equations (PDEs), (in)homogeneous PDEs, some systems of (non)linear PDEs, and one-dimensional and multidimensional PDEs by using many analytical and numerical methods, such as the Adomian decomposition method (ADM), modified ADM, the variational iteration method (VIM), the tanh method, the tanh-coth method, the sine-cosine method, Hirota's bilinear method, etc.…”

“…El-Dib [6] used the linearizing method for determining the displacement amplitude and approximate frequency to the third-order ordinary differential equations (ODEs). He [7] applied some asymptotic techniques such as parameter-expanding methods, variational approaches, the homotopy perturbation method (HPM), the parameterized perturbation technique (PT), ancient Chinese methods, iteration PT for analyzing both weakly and strongly NDEs. He and El-Dib [8] reduced the damped KGE to the Duffing equation (DE) by using suitable transformation and solved the DE by using the hybrid reducing rank method with the HPM.…”