A simple cubication method for approximate solution of nonlinear Hamiltonian oscillators

2019 ◽  
Vol 48 (3) ◽  
pp. 241-254 ◽  
Author(s):  
Akuro Big-Alabo
Keyword(s):  
Periodic Solution ◽  
Approximate Solution ◽  
Present Method ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Static Equilibrium ◽  
Elementary Functions ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

A new cubication method is proposed for periodic solution of nonlinear Hamiltonian oscillators. The method is formulated based on quasi-static equilibrium of the original oscillator and the undamped cubic Duffing oscillator. The cubication constants derived from the present cubication method are always based on elementary functions and are simpler than the constants derived by other cubication methods. The present method was verified using three common examples of strongly nonlinear oscillators and was found to give reasonably accurate results. The method can be used to introduce nonlinear oscillators in relevant undergraduate physics and mechanics courses.

scholarly journals Approximate Potentials with Applications to Strongly Nonlinear Oscillators with Slowly Varying Parameters

2003 ◽  
Vol 10 (5-6) ◽  
pp. 379-386 ◽  
Author(s):  
Jianping Cai ◽  
Y.P. Li
Keyword(s):  
Present Method ◽  
Nonlinear Oscillator ◽  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Oscillatory System ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Elapsed Time ◽  
Strongly Nonlinear Oscillators ◽  
Strongly Nonlinear Oscillator

A method of approximate potential is presented for the study of certain kinds of strongly nonlinear oscillators. This method is to express the potential for an oscillatory system by a polynomial of degree four such that the leading approximation may be derived in terms of elliptic functions. The advantage of present method is that it is valid for relatively large oscillations. As an application, the elapsed time of periodic motion of a strongly nonlinear oscillator with slowly varying parameters is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method.

Computation of the adiabatic invariant of the symmetric nonlinear oscillator $$d^2y \over dt^2+Q(\epsilon t)y^n=R(\epsilon t)y^m$$

1998 ◽  
Vol 9 (2) ◽  
pp. 187-194
Author(s):  
J. HU
Keyword(s):  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Adiabatic Invariant ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

scholarly journals Iterative Multistep Reproducing Kernel Hilbert Space Method for Solving Strongly Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Banan Maayah ◽  
Samia Bushnaq ◽  
Shaher Momani ◽  
Omar Abu Arqub
Keyword(s):  
Hilbert Space ◽  
Series Solution ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Nonlinear Oscillators ◽  
Strongly Nonlinear ◽  
Runge Kutta Method ◽  
Strongly Nonlinear Oscillators ◽  
Space Method ◽  
Good Agreement

A new algorithm called multistep reproducing kernel Hilbert space method is represented to solve nonlinear oscillator’s models. The proposed scheme is a modification of the reproducing kernel Hilbert space method, which will increase the intervals of convergence for the series solution. The numerical results demonstrate the validity and the applicability of the new technique. A very good agreement was found between the results obtained using the presented algorithm and the Runge-Kutta method, which shows that the multistep reproducing kernel Hilbert space method is very efficient and convenient for solving nonlinear oscillator’s models.

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