Analytical solution of the damped Helmholtz–Duffing equation

Elı́as-Zúñiga, Alex

doi:10.1016/j.aml.2012.06.030

Cited by 34 publications

(42 citation statements)

References 7 publications

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“…Furthermore, we must notice that our derived solution can be reduced to that developed in [15] if G ¼ 0 and use the parametric relation A ¼ 8m 2 =9, which is applied to obtain the exact solution of the damped Duffing oscillator.…”

Section: Discussionmentioning

confidence: 99%

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Solution of the damped cubic–quintic Duffing oscillator by using Jacobi elliptic functions

Elı́as-Zúñiga

2014

Applied Mathematics and Computation

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Section: Discussionmentioning

confidence: 99%

“…By following the elliptic balance procedure described in [14,15], we may see that Eq. (3) holds for all time t if and only if:…”

Section: Introductionmentioning

confidence: 99%

Solution of the damped cubic–quintic Duffing oscillator by using Jacobi elliptic functions

Elı́as-Zúñiga

2014

Applied Mathematics and Computation

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“…An analytical solution for the differential equation Figure 1: Pendulum nonlinear equation of motion of a pendulum has been derived using the Jacobi elliptic function [14]. The analytical solution of the damped Helmholtz-Duffing oscillator has also been derived [15]. The exact analytical formulas for the period of a simple pendulum have been obtained in terms of the Jacobi elliptic function [16] [17].…”

Section: Pendulummentioning

confidence: 99%

Pendulum analysis by leaf functions and hyperbolic leaf functions

Shinohara

2019

ams

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The mathematical model representing the equation of motion of a pendulum is nonlinear. Solutions that satisfy the equation cannot be represented by elementary functions, such as trigonometric functions. To solve such problems, it is common to linearize the nonlinear equations and derive approximate numerical solutions and exact solutions. Applying such linearization is limited to cases in which the angle of the pendulum is relatively small. In cases where the angle of the pendulum is large, various methods have been presented that rely on numerical solutions and the exact solutions based on Jacobian elliptic functions, to name one example. On the other hand, the author has been studying a certain type differential equation. The second derivative of a function is equal to the term multiplied by −n(or n) for terms whose function is raised to 2n − 1. Curves based on solving this differential equation are constructed as regular waves with periods. The author has termed the function satisfying this differential equation the leaf function (or the hyperbolic leaf function). In this paper, we attempt to apply this leaf function (or hyperbolic leaf function) to undamped pendulums and overdamped pendulums for large angles.

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“…where Eqs. (9) and (10) is used. Our approximation that the Hubble expansion rate H is a constant is valid if |Ḣ/H 2 | ≪ 1.…”

Section: Rapid Roll Inflation With a Quartic Potentialmentioning

confidence: 99%

Towards an analytic solution of rapid roll inflation with a quartic potential

Lin

2019

Chinese Journal of Physics

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In this paper, I present an analytic solution to the equation of motion of the inflaton field in a model of rapid roll inflation with a quartic potential by assuming that the Hubble parameter is a constant during inflation. The result is obtained by using Jacobi elliptic functions. The spectral index ns and the running spectral index n ′ s is obtained without using rapid roll or extended slow roll approximations. The cosmological consequences of the model is discussed.

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Analytical solution of the damped Helmholtz–Duffing equation

Cited by 34 publications

References 7 publications

Solution of the damped cubic–quintic Duffing oscillator by using Jacobi elliptic functions

Solution of the damped cubic–quintic Duffing oscillator by using Jacobi elliptic functions

Pendulum analysis by leaf functions and hyperbolic leaf functions

Towards an analytic solution of rapid roll inflation with a quartic potential

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