scholarly journals An Optimal Iteration Method for Strongly Nonlinear Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Convergence Of Approximate Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

We introduce a new method, namely, the Optimal Iteration Perturbation Method (OIPM), to solve nonlinear differential equations of oscillators with cubic and harmonic restoring force. We illustrate that OIPM is very effective and convenient and does not require linearization or small perturbation. Contrary to conventional methods, in OIPM, only one iteration leads to high accuracy of the solutions. The main advantage of this approach consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solutions, which prove that OIPM is very efficient and accurate.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical 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.

scholarly journals Optimal Variational Method for Truly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Variational Method ◽  
Numerical Solution ◽  
Periodic Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Convergence Of Approximate Solutions ◽  
Good Agreement

The Optimal Variational Method (OVM) is introduced and applied for calculating approximate periodic solutions of “truly nonlinear oscillators”. The main advantage of this procedure consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. This approach does not depend upon any small or large parameters. A very good agreement was found between approximate and numerical solution, which proves that OVM is very efficient and accurate.

scholarly journals Energy Method to Obtain Approximate Solutions of Strongly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Potential Energy ◽  
Analytical Solutions ◽  
Energy Method ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Equivalent Representation ◽  
Strongly Nonlinear ◽  
Representation Form ◽  
Strongly Nonlinear Oscillators

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion.

A Global Residue Harmonic Balance Method for a Class of Nonlinear Jerk Equation

2013 ◽  
Vol 353-356 ◽  
pp. 3324-3327
Author(s):  
Xin Xue ◽  
Pei Jun Ju ◽  
Dan Sun
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
High Accuracy ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Accuracy Comparison ◽  
Comparison Of The Results ◽  
Strongly Nonlinear Oscillators

A new approach, namely the global residue harmonic balance, was developed to determine the accurately approximate periodic solution of a class of nonlinear Jerk equation containing velocity times acceleration-squared and velocity. Unlike other improved harmonic balance methods, all the forward harmonic residuals were considered in the present approximation to improve the accuracy. Comparison of the results obtained using this approach with the exact one and the existing results reveals that the high accuracy, simplicity and efficiency of the presented solution procedure. The method can be easily extended to other strongly nonlinear oscillators.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

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