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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access November 26, 2015

A detailed analysis for the fundamental solution of fractional vibration equation

  • Li-Li Liu and Jun-Sheng Duan
From the journal Open Mathematics

Abstract

In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.

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Received: 2015-9-13
Accepted: 2015-10-21
Published Online: 2015-11-26

©2015 Li-Li Liu, Jun-Sheng Duan

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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