The Approximate Analytical Solution Sequence for Fractional Oscillation Equations Based on the Fractional Variational Iteration Method

BAO Si-yuan; DENG Zi-chen

doi:10.3879/j.issn.1000-0887.2015.01.004

Volume 36 Issue 1

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BAO Si-yuan, DENG Zi-chen. The Approximate Analytical Solution Sequence for Fractional Oscillation Equations Based on the Fractional Variational Iteration Method[J]. Applied Mathematics and Mechanics, 2015, 36(1): 48-60. doi: 10.3879/j.issn.1000-0887.2015.01.004

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The Approximate Analytical Solution Sequence for Fractional Oscillation Equations Based on the Fractional Variational Iteration Method

doi: 10.3879/j.issn.1000-0887.2015.01.004

BAO Si-yuan¹,
DENG Zi-chen²

¹Department of Engineering Mechanics, School of Civil Engineering,Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R. China;²School of Mechanics and Civil. & Architecture,Northwestern Polytechnical University, Xi’an 710072, P.R.China

Funds: The National Natural Science Foundation of China(11202146)

Received Date: 2014-05-30
Rev Recd Date: 2014-11-27
Publish Date: 2015-01-15

Abstract

Abstract

The fractional calculus was introduced to describe the damped oscillator in viscoelastic medium and the Caputo-type fractional nonlinear oscillation equations were established. The fractional variational iteration method (FVIM) was modified with a small parameter and the Lagrange multiplier was derived. For the linear fractional oscillation equations, both the homogeneous equations and the sinusoidal force-excited nonhomogeneous equations were analyzed with the FVIM to obtain the approximate analytical solution sequence. The varying curves of the displacement for different values of the fractional order were given in the case of the Bagley-Torvik equation. The relationship between oscillator motion and fractional derivative was also studied according to the extent of memorability for different fractional orders. Compared with the ordinary variational iteration method, the proposed FVIM modified with a small parameter expands the interval of convergence significantly for the solution. In the end, the Van der Pol equation with fractional derivative as an example illustrates the method’s effectiveness and convenience to solve non-linear fractional differential problems.
- Caputo fractional derivative,
- nonlinear dynamics,
- fractional oscillation equation,
- fractional variational iteration method,
- approximate analytical solution

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References

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