Technical Brief

Equivalence of Initialized Fractional Integrals and the Diffusive Model

[+] Author and Article Information
Jian Yuan

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: yuanjianscar@gmail.com

Youan Zhang

Institute of Technology,
Yantai Nanshan University,
Yantai 265713, China
e-mail: zhangya63@sina.com

Jingmao Liu

Shandong Nanshan International Flight Co., Ltd.,
Yantai 265713, China
e-mail: liujingmao@nanshan.com.cn

Bao Shi

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: baoshi781@sohu.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 5, 2017; final manuscript received December 10, 2017; published online January 10, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(3), 034501 (Jan 10, 2018) (4 pages) Paper No: CND-17-1251; doi: 10.1115/1.4038777 History: Received June 05, 2017; Revised December 10, 2017

Fractional calculus is viewed as a novel and powerful tool to describe the stress and strain relations in viscoelastic materials. Consequently, the motions of engineering structures incorporated with viscoelastic dampers can be described by fractional-order differential equations. To deal with the fractional differential equations, initialization for fractional derivatives and integrals is considered to be a fundamental and unavoidable problem. However, this issue has been an open problem for a long time and controversy persists. The initialization function approach and the infinite state approach are two effective ways in initialization for fractional derivatives and integrals. By comparing the above two methods, this technical brief presents equivalence and unification of the Riemann–Liouville fractional integrals and the diffusive representation. First, the equivalence is proved in zero initialization case where both of the initialization function and the distributed initial condition are zero. Then, by means of initialized fractional integration, equivalence and unification in the case of arbitrary initialization are addressed. Connections between the initialization function and the distributed initial condition are derived. Besides, the infinite dimensional distributed initial condition is determined by means of input function during historic period.

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