Communications in Nonlinear Science and Numerical Simulation
Short communicationOn the numerical computation of the Mittag-Leffler function
Introduction
During the last decades Fractional Calculus (FC) became a major area of research and development and we can mention its application in many scientific areas ranging from mathematics and physics, up to biology, engineering, and earth sciences [18], [22], [14], [19], [6], [9], [21], [11], [2], [1], [13], [7]. The Mittag-Leffler function (MLf) plays an important role in FC, being often called by scholars the “queen” of the FC functions. Nine decades after its first formulation by the Swedish Mathematician [3] Magnus Gösta Mittag-Leffler (1846–1927), the MLf became a relevant topic, not only from the pure mathematical point of view, but also from the perspective of its applications.
Bearing these ideas in mind, this short communication addresses the application of the MLf and real-time calculation in control systems of the expression where denotes the fractional order, t stands for time and represents the one parameter MLf to be recalled in the sequel.
The paper is organized as follows. Section 2 introduces the fundamental aspects of and . Section 3 develops the approximation for the numerical calculation of and analyses its computational load. Finally, Section 4 outlines the main conclusions.
Section snippets
The Mittag-Leffler function
The MLf, defined asis a special function, first studied and discussed in [16], [15], [17], which generalises the standard exponential . It can in its turn be generalised [25], [26] aswhich is the two-parameter MLf. Its main properties and applications can be found in [5] and in chapter 18 of [4]; it is of great importance in Fractional Calculus (and thus in the study of dynamic systems of fractional order) [24]. It also appears when
Determining the weight function
Function was calculated using Matlab and the routine in [20] for reference purposes. Weights were determined for with a step of , and for with 20 logarithmically-spaced points per decade. They are shown in Fig. 3: those for were not further considered because numerical instability arises spoiling the results.
Cuts of this surface for constant values of can be approximated by functions of the typeThe curve fitting was
Conclusions
The MLf is an important function in mathematics, numerical calculus, engineering and applied sciences that are studied with the formalism of FC. Currently new phenomena are discovered and analysed adopting the FC perspective and requiring efficient computational schemes for expressions involving the MLf. This paper joined two recently proposed asymptotic expressions for deriving a fast numerical procedure useful in expressions common in fractional order control algorithms.
Acknowledgment
This work was partially supported by Fundação para a Ciência e a Tecnologia, through IDMEC under LAETA.
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