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A note on the definition of fractional derivatives applied in rheology

Acta Mechanica Sinica 2011年第6期27卷 866-876页

作者： Fan Yang Ke-Qin Zhu Department of Engineering Mechanics Tsinghua University 100084 Beijing China

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives-Riemann-Liouville （R-L） definition and caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model （Scott-Blair model）. We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R-L definition with lower terminal a →∞ and the caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal a →∞ （or, equivalently, the caputo derivatives with lower terminal a →∞） not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott-Blair model and give an exact solution of this equation under given initial conditions.

关键词： Fractional derivatives Rheology Riemann-Liouville definition caputo definition

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On the definition of fractional derivatives in rheology

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Theoretical & Applied Mechanics Letters 2011年第1期1卷 62-65页

作者： Fan Yang~(1,2,a)) and Ke-Qin Zhu~(1,b)) 1)Department of Engineering Mechanics,Tsinghua University,Beijing 100084,China 2)Department of Physics,the Chinese University of Hong Kong,Hong Kong,China Department of Physics the Chinese University of Hong Kong Hong Kong China

During the last two decades fractional calculus has been increasingly applied to physics, especially to *** is well known that there are obivious differences between Riemann-Liouville (R-L) definition and caputo definition,which are the two most commonly used definitions of fractional *** multiple definitions of fractional derivatives have hindered the application of fractional calculus in *** this paper,we clarify that the R-L definition and caputo definition are both Theologically unreasonable with the help of the mechanical analogues of the fractional element *** also find that to make them more reasonable Theologically,the lower terminals of both definitions should be put to—∞.We further prove that the R-L definition with lower terminal—∞and the caputo definition with lower terminal—∞are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular *** we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal—∞(or,equivalently,the caputo derivatives with lower terminal—∞) not only for steady-state processes,but also for transient processes.

关键词： fractional derivative caputo definition Riemann-Liouville definition Scott-Blair model

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