Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method

Abdelhafez, H. M.

doi:10.3390/math4010011

Cited by 10 publications

(4 citation statements)

References 26 publications

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“…The nonlinear equation obtained by the analysis of the electric circuit can be reached to (5.4) by making it dimensionless by means of the transformations given in [1,8,19]. The solution of this problem is compared with the results of the modified differential transform method (MDTM) developed by Abdelhafez [1] in Table 4 and Figure 5.…”

Section: Examplementioning

confidence: 99%

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

Tarakçı¹,

Özel²,

Sezer³

2020

Turk J Math

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Nonlinear differential equations have many applications in different science and engineering disciplines. However, a nonlinear differential equation cannot be solved analytically and so must be solved numerically. Thus, we aim to develop a novel numerical algorithm based on Morgan-Voyce polynomials with collocation points and operational matrix method to solve nonlinear differential equations. In the our proposed method, the nonlinear differential equations including quadratic and cubic terms having the initial conditions are converted to a matrix equation. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB. The solution of this method for the convergence and efficiency was compared with the equations such as Van der Pol differential equation calculated by different methods.

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Section: Examplementioning

confidence: 99%

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

Tarakçı¹,

Özel²,

Sezer³

2020

Turk J Math

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“…Pollack Periodica 13, 2018, 3 To study nonlinear systems there are numerical, graphical (Cobweb plot [14]) and analytical methods (determination of the eigenvalues of the Jacobian determinant after linearization [15], perturbation methods [16]). With perturbation methods the approximate analytical solution of the Van der Pol oscillator and the Duffing-Holmes oscillator has already been determined [17]. As numerical simulations are very common nowadays in engineering numerical examination methods have been chosen for our research.…”

Section: F Hajdumentioning

confidence: 99%

Numerical examination of nonlinear oscillators

Hajdú

2018

Pollack Periodica

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The numerical examination of nonlinear oscillators is presented in this paper. First some methods of nonlinear system modeling are described then the numerical creation of phaseplane, bifurcation diagrams and Poincaré sections are expounded in detail. The next part of the paper is the numerical examination of nonlinear oscillators, like the Duffing-Holmes oscillator and a mechatronic semi-active suspension system. The paper concludes with further development tasks.

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“…Also, the homotopy analysis method (HAM) has been employed for analyzing the damped Dufng oscillator (DDO) [23]. Te Laplace transform, modifed diferential transform method (MDTM), and Padé approximants have been applied for analyzing and investigating the approximations to many NPOs such as the forced DDO and forced damped van der Pol oscillator [24]. Both damped and undamped Helmholtz-Dufng (H-D) oscillators have been studied and analyzed using the ansatz method with the moving boundary technique to obtain high-accurate approximations [25].…”

Section: Introductionmentioning

confidence: 99%

Novel Approximations to the Damped Parametric Driven Pendulum Oscillators

Alhejaili

Salas

El-Tantawy

2023

Journal of Mathematics

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The damped parametric driven nonlinear pendulum equation/oscillator (NPE), also known as the damped disturbed NPE, is examined, along with some associated oscillators for arbitrary angles with the vertical pivot. For analyzing and solving the current pendulum equation, we reduce this equation to the damped Duffing equation (DDE) with variable coefficients. After that, the DDE with variable coefficients is divided into two cases. In the first case, two analytical approximations to the damped undisturbed NPE are obtained. The first approximation is determined using the ansatz method while the second one is derived using He’s frequency formulation. In the second case, i.e., the damped disturbed NPE, three analytical approximations in terms of the trigonometric and Jacobi elliptic functions are derived and discussed using the ansatz method. The semianalytical solutions of the two mentioned cases are graphically compared with the 4th-order Runge–Kutta (RK4) approximations. In addition, the maximum error for all the derived approximations is estimated as compared with the RK4 approximation. The proposed approaches as well as the obtained solutions may greatly help in understanding the mysteries of various nonlinear phenomena that arise in different scientific fields such as fluid mechanics, plasma physics, engineering, and electronic chips.

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Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method

Cited by 10 publications

References 26 publications

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method

Numerical examination of nonlinear oscillators

Novel Approximations to the Damped Parametric Driven Pendulum Oscillators

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