On the definition of fractional derivatives in rheology

AbstractDuring the last two decades fractional calculus has been increasingly applied to physics, especially to rheology. It 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 derivatives. The multiple definitions of fractional derivatives 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 unreasonable with the help of the mechanical analogues of the fractional element model. We also find that to make them more reasonable rheologically, 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 points. Thus 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