scholarly journals Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammad Heydari ◽  
Ghasem Brid Loghmani ◽  
Seyed Mohammad. Hosseini ◽  
Seyed Mehdi Karbassi
Keyword(s):  
Exact Solution ◽  
Harmonic Oscillator ◽  
Numerical Method ◽  
Nonlinear Oscillator ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Strongly Nonlinear ◽  
Hybrid Functions ◽  
Cardinal Functions ◽  
Strongly Nonlinear Oscillator

A numerical method for finding the solution of Duffing-harmonic oscillator is proposed. The approach is based on hybrid functions approximation. The properties of hybrid functions that consist of block-pulse and Chebyshev cardinal functions are discussed. The associated operational matrices of integration and product are then utilized to reduce the solution of a strongly nonlinear oscillator to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. The results are compared with the exact solution and results from several recently published methods, and the comparisons showed proper accuracy of this method.

scholarly journals Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. Mashayekhi ◽  
M. Razzaghi ◽  
O. Tripak
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Fredholm Integral Equations ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Hybrid Functions ◽  
Block Pulse Functions

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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.

scholarly journals A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Changqing Yang ◽  
Jianhua Hou
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions

A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

Motion Confinement in a Linear Chain of Coupled Oscillators With a Strongly Nonlinear End Attachment

2003 ◽  
Author(s):  
Leonid Manevitch ◽  
Oleg Gendelman ◽  
Andrey I. Musienko ◽  
Alexander F. Vakakis ◽  
Lawrence Bergman
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Oscillator ◽  
Linear Chain ◽  
Standing Waves ◽  
Strongly Nonlinear ◽  
Resonant Interactions ◽  
Energy Pumping ◽  
Weakly Coupled ◽  
Strongly Nonlinear Oscillator ◽  
Infinite Linear

We study the dynamics of a semi-infinite linear chain of particles that is weakly coupled to a strongly nonlinear oscillator at its free end. We analyze families of localized standing waves situated inside the lower or upper attenuation zones of the linear chain, corresponding to energy predominantly confined in the nonlinear oscillator. These families of standing waves are generated due to resonant interactions between the chain and the nonlinear attachment. A scenario for the realization of energy pumping phenomena in the system under consideration is discussed, and is confirmed by direct numerical simulations of the chain-attachment dynamic interaction.

Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Themistoklis P. Sapsis ◽  
Alexander F. Vakakis
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillator ◽  
Resonance Condition ◽  
Asymptotic Results ◽  
Strongly Nonlinear ◽  
Singular Perturbation Analysis ◽  
Local Bifurcation Analysis ◽  
The Family ◽  
Subharmonic Orbits ◽  
Strongly Nonlinear Oscillator

We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.

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