A detailed analysis for the fundamental solution of fractional vibration equation

Li-Li Liu; Jun-Sheng Duan

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Li-Li Liu, and Jun-Sheng Duan. "A detailed analysis for the fundamental solution of fractional vibration equation." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275840>.

@article{Li2015,
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.},
author = {Li-Li Liu, Jun-Sheng Duan},
journal = {Open Mathematics},
keywords = {Fractional derivatives and integrals; Fractional differential equations; Laplace transform; Fractional vibration},
language = {eng},
number = {1},
pages = {null},
title = {A detailed analysis for the fundamental solution of fractional vibration equation},
url = {http://eudml.org/doc/275840},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Li-Li Liu
AU - Jun-Sheng Duan
TI - A detailed analysis for the fundamental solution of fractional vibration equation
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - 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.
LA - eng
KW - Fractional derivatives and integrals; Fractional differential equations; Laplace transform; Fractional vibration
UR - http://eudml.org/doc/275840
ER -

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