scholarly journals Solving the Oscillation Equation With Fractional Order Damping Term Using a New Fourier Transform Method

2017 ◽  
Vol 13 (5) ◽  
pp. 7393-7397
Author(s):  
OZLEM OZTURK MIZRAK
Keyword(s):  
Fourier Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fourier Transformation ◽  
Cosmic Time ◽  
Damping Term ◽  
Fractional Damping ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Oscillation Equation

We propose an adapted Fourier transform method that gives the solution of an oscillation equation with a fractional damping term in ordinary domain. After we mention a transformation of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Fourier transformation, via an example for 1 < alpha < 2; where alpha is the order of fractional derivative. Then, we give an application of the results.

scholarly journals Dynamics of the helmholtz oscillator with fractional order damping

2013 ◽  
Vol 23 ◽  
pp. 12-15
Author(s):  
Adolfo Ortiz ◽  
Jesús Seoane ◽  
J. Yang ◽  
Miguel Sanjuán
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Critical Value ◽  
Periodic Motions ◽  
Damping Term ◽  
Fractional Damping ◽  
Runge Kutta Method ◽  
History Of

The dynamics of the nonlinear Helmholtz Oscillator with fractional order damping are studied in detail. The discretization of differential equations according to the Grünwald-Letnikov fractional derivative definition in order to get numerical simulations is reported. Comparison between solutions obtained through a fourth-order Runge-Kutta method and the fractional damping system are comparable when the fractional derivative of the damping term a is fixed at 1. That proves the good performance of the numerical scheme. The effect of taking the fractional derivative on the system dynamics is investigated using phase diagrams varying a from 0.5 to 1.75 with zero initial conditions. Periodic motions of the system are obtained at certain ranges of the damping term. On the other hand, escape of the trajectories from a potential well result at a certain critical value of the fractional derivative. The history of the displacement as a function of time is shown also for every a selected.

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