On the numerical computation of the Mittag-Leffler function

doi:10.1016/j.cnsns.2014.03.014

Communications in Nonlinear Science and Numerical Simulation

Volume 19, Issue 10, October 2014, Pages 3419-3424

https://doi.org/10.1016/j.cnsns.2014.03.014 Get rights and content

Highlights

•
An algorithm is developed to approximate $E_{α, 1} (- t^{α})$ .
•
Errors are clearly lower than those of usual asymptotic approximations.
•
The algorithm runs in average 2277 times faster than computing function $E_{α, 1}$ .

Abstract

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

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 $e_{α} (t) = E_{α} (- t^{α})$ where $α$ denotes the fractional order, t stands for time and $E_{α}$ represents the one parameter MLf to be recalled in the sequel.

The paper is organized as follows. Section 2 introduces the fundamental aspects of $E_{α} (t)$ and $e_{α} (t)$ . Section 3 develops the approximation for the numerical calculation of $e_{α} (t)$ and analyses its computational load. Finally, Section 4 outlines the main conclusions.

Section snippets

The Mittag-Leffler function

The MLf, defined as $E_{α} (t) = \sum_{n = 0}^{+ \infty} \frac{t^{α n}}{Γ (α n + 1)}$ is a special function, first studied and discussed in [16], [15], [17], which generalises the standard exponential $e^{t} = \sum_{n = 0}^{+ \infty} \frac{t^{n}}{Γ (n + 1)}$ . It can in its turn be generalised [25], [26] as $E_{α, β} (t) = \sum_{n = 0}^{+ \infty} \frac{t^{α n}}{Γ (α n + β)}$ which 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 $ϕ_{α} (t)$

Function $e_{α} (t)$ was calculated using Matlab and the routine in [20] for reference purposes. Weights $ϕ_{α} (t)$ were determined for $α \in] 0, 1 [$ with a step of $Δ α = 0.01$ , and for $t \in [10^{- 5}, 10^{5}]$ with 20 logarithmically-spaced points per decade. They are shown in Fig. 3: those for $α < 0.5 \land t > 1$ were not further considered because numerical instability arises spoiling the results.

Cuts of this surface for constant values of $α$ can be approximated by functions of the type $ϕ_{α} (t) = \frac{1}{1 + e^{- x_{1} (α) \log_{10} t + x_{2} (α)}}$ The 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.

References (26)

Dumitru Baleanu et al.
Fractional calculus: models and numerical methods
(2012)
Kai Diethelm
The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type
(2010)
Lars Garding
Mathematics and mathematicians: mathematics in Sweden before 1950
(1994)
Harry Bateman
(1955)
H.J. Haubold et al.
Mittag-Leffler functions and their applications
J Appl Math
(2011)
R. Hilfer
Application of fractional calculus in physics
(2000)
C.M. Ionescu
The human respiratory system: an analysis of the interplay between anatomy, structure, breathing and fractal dynamics
(2013)
Mumtaz Ahmad Khan et al.
On some properties of the generalized Mittag-Leffler function
Springer-Plus
(2013)
A.A. Kilbas et al.
Theory and applications of fractional differential equations
(2006)
J.A. Tenreiro Machado et al.
Benchmarking computer systems for robot control
IEEE Trans Edu
(1995)

F. Mainardi

Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models

(2010)

Francesco Mainardi

On some properties of the Mittag-Leffler function $e_{α} (- t^{α})$ , completely monotone for $t > 0$ with $0 < α < 1$

Discrete Contin Dyn Syst Ser B

(2013)

Agnieszka B. Malinowska et al.

Introduction to the fractional calculus of variations

(2012)

Cited by (26)

A new population model for urban infestations
2023, Chaos, Solitons and Fractals
The spread of rodents and insects in cities, in particular in summer periods, poses significant health, economic, social and environmental threats. The analysis of incidence and identification of seasonal and weather determinants are crucial for addressing intervention strategies. In this context, we investigate the occurrence of rats and cockroaches infestations in the city of Madrid, Spain, with differential equation models. Birth–death fluxes, sine–cosine waves and temperature conform the mechanistic nature of the models. Available data on citizen-reported sightings from 2010 until 2013 are fitted by parameter calibration and uncertain-error measurement. Numerical simulations show that the time series are adequately explained by the proposed models. This strongly suggests that the models can be used to predict future infestation dynamics, which can guide health policy measures.
Multi-field formulations for solving plane problems involving viscoelastic constitutive relations
2023, Applications in Engineering Science
This article reports a multi-field numerical formulation for solving plane problems involving viscoelastic materials. Stress fields satisfying equilibrium equations are constructed using Airy’s potentials which are expressed as a linear combination of $C^{2}$ basis functions. The strain field is derived from a continuous displacement field obtained from a linear combination of $C^{0}$ basis functions. An appropriate linear combination of these stress and displacement basis functions is determined such that the resulting stress and strain fields satisfy the constitutive relation subjected to the satisfaction of the constraints arising from the boundary conditions. Since a viscoelastic constitutive relation involves stress, strain, and their rates, stress and displacement degrees of freedom or their rates can be considered as optimization variables for minimizing the error in satisfying the constitutive relation. Two Algorithms are proposed based on this choice of optimization variable. Accuracy and efficiency of the proposed algorithms are studied through five different boundary value problems involving four forms of the viscoelastic constitutive relations and for two loading histories. Using the developed rectangular element, viscoelastic beam bending problem is solved for the different constitutive relations studied.
Law of two-sided exit by a spectrally positive strictly stable process
2020, Stochastic Processes and their Applications
For a spectrally positive strictly stable process with index in (1, 2), we obtain (i) the sub-probability density of its first exit time from an interval by hitting the interval’s lower end before jumping over its upper end, and (ii) the joint distribution of the time, undershoot, and jump of the process when it makes the first exit the other way around. The density of the exit time is expressed in terms of the roots of a Mittag-Leffler function. Some theoretical applications of the results are given.
Numerical techniques for solving truss problems involving viscoelastic materials
2020, International Journal of Non-Linear Mechanics
We develop a methodology for solving truss problems involving viscoelastic materials where, of all the member forces that satisfy the nodal force equilibrium equation and nodal displacements that satisfy the displacement boundary conditions, those member forces and nodal displacements that satisfy the constitutive relation are sought. Since a rate type viscoelastic constitutive relation involves the rate of the stress or strain, this study explores the use of member forces, nodal displacements, and support reactions or their rates as independent variables. Assuming small deformations, the nodal force equilibrium and the displacement boundary condition results in a linear equality constraint between the independent variables. Then we find the unknown independent variables such that the root mean squared error in the constitutive relation of the members of the truss is minimized subject to the satisfaction of the linear constraint at selected times. The objective function is evaluated at selected times or integrated over subintervals of time. We explore six possible solution methods and benchmark them for their accuracy and efficiency. We study statically determinate and indeterminate truss whose members are modeled using rate and integral type viscoelastic constitutive relations for creep and oscillatory loading. For the standard linear solid model, we find that the proposed methods are more accurate than ABAQUS and, at times, require lesser computational wall time. We also demonstrate the applicability of the proposed methodology to fractional order and nonlinear viscoelastic constitutive relations.
Tsallis–Mittag-Leffler distribution and its applications in gas prices
2020, Physica A: Statistical Mechanics and its Applications
The theory of Mittag-Leffler process is a fundamental concept in probability and stochastic process. This paper introduces the class of Tsallis–Mittag-Leffler distributions. In special case, our results include a new Tsallis $q$ -Weibull distribution, one-parametric Mittag-Leffler distribution and some other well-known distributions. Finally, to illustrate the applicability of our distribution, a real data set on gas prices in time series data based on 100 days moving average is analyzed.
Mittag-Leffler-Gaussian distribution: Theory and application to real data
2019, Mathematics and Computers in Simulation

View all citing articles on Scopus

View full text

Short communicationOn the numerical computation of the Mittag-Leffler function

Highlights

Abstract

Introduction

Section snippets

The Mittag-Leffler function

Determining the weight function ϕα(t)

Conclusions

Acknowledgment

Fractional calculus: models and numerical methods

The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type

Mathematics and mathematicians: mathematics in Sweden before 1950

Mittag-Leffler functions and their applications

J Appl Math

Application of fractional calculus in physics

The human respiratory system: an analysis of the interplay between anatomy, structure, breathing and fractal dynamics

On some properties of the generalized Mittag-Leffler function

Springer-Plus

Theory and applications of fractional differential equations

Benchmarking computer systems for robot control

IEEE Trans Edu

Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models

On some properties of the Mittag-Leffler function eα(-tα), completely monotone for t>0 with 0<α<1

Discrete Contin Dyn Syst Ser B

Introduction to the fractional calculus of variations

Short communication
On the numerical computation of the Mittag-Leffler function

Determining the weight function $ϕ_{α} (t)$

On some properties of the Mittag-Leffler function $e_{α} (- t^{α})$ , completely monotone for $t > 0$ with $0 < α < 1$