On some properties of the Mittag-Leffler function $E_\alpha(-t^\alpha)$, completely monotone for $t > 0$ with $0 < \alpha < 1$

Francesco Mainardi

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On some properties of the Mittag-Leffler function $E_\alpha(-t^\alpha)$, completely monotone for $t > 0$ with $0 < \alpha < 1$

Francesco Mainardi

arXiv:1305.0161v4May 1, 2013math-phcs.NAmath.CAmath.CVmath.MPmath.NA

TL;DR

The paper examines the fractional-relaxation function Eα(−t^α) and the need for simpler completely monotone approximations at small and large times. It relates its frequency and relaxation-time spectra and compares asymptotic approximants, including rational functions that perform well over much of the time range.

Problem
The paper examines how to characterize the spectra and approximate Eα(−t^α) with simpler completely monotone functions at small and large times.
Source: §4 · p. 12
Method
The authors compare two sets of completely monotone asymptotic approximants and relate the function's frequency and relaxation-time spectra.
Source: §4 · p. 12
Results
The frequency and relaxation-time spectra coincide, and the large-time rational approximation also approximates short times well but fails in an intermediate range.
Source: §4 · p. 12
Takeaways & Limitations
The rational approximants provide upper and lower bounds in the reported numerical examples over a large time range.
Source: §4 · p. 12
Whether the rational functions bound Eα(−t^α) for every t > 0 and 0 < α < 1 remains an open problem.
Source: §3.2 · p. 11, §4 · p. 12

Abstract

from arXiv · show

We analyse some peculiar properties of the function of the Mittag-Leffler (M-L) type, $e_\alpha(t):= E_\alpha(-t^\alpha)$ for $0 <\alpha < 1$ and $t > 0$, which is known to be completely monotone (CM) with a non negative spectrum of frequencies and times, suitable to model fractional relaxation processes. We first note that these two spectra coincide so providing a universal scaling property of this function. Furthermore, we consider the problem of approximating our M-L function with simpler CM functions for small and large times. We provide two different sets of elementary CM functions that are asymptotically equivalent to $e_\alpha(t)$ as $t \to 0$ and $t \to \infty$.

1 Introduction

The Mittag-Leffler function has become central to fractional calculus, while its broad generalizations have sometimes obscured its relevance to applied science. The introduction emphasizes that many models use only a few parameters and may need only a modestly generalized Mittag-Leffler function.

The Mittag-Leffler function plays a fundamental role in fractional calculus and in treating integral and differential equations of fractional order.Source: §1 · pp. 1–2
Earlier applications used the function to represent responses in dielectric and viscoelastic media, and it was also discussed in connection with nerve conduction.Source: §1 · p. 2
The function was treated as a special case of Fox H functions from the 1960s onward, within a framework allowing an arbitrary number of parameters.Source: §1 · p. 2
General Fox H-function classification can obscure applied relevance, while many mathematical models use only one to three parameters.The authors suggest that generalized Mittag-Leffler functions with two or three indices may suffice for some applications.Source: §1 · p. 2

2 The Mittag-Leffler function in fractional relaxation processes

For 0 < α < 1, the paper treats Eα(−t^α) as a completely monotone fractional-relaxation function. Its frequency and relaxation-time spectra coincide under a reciprocal change of variable, yielding a universal scaling property.

The Mittag-Leffler function is defined by a power series and is an entire-function generalization of the exponential, which it recovers at α = 1.Source: §2 · p. 3
The paper studies the negative real axis for 0 < α < 1, where the function supplies the solution to a fractional relaxation equation.The two fractional relaxation formulations are equivalent by Laplace transformation.Source: §2 · pp. 3–4
Since Kα(r) is non-negative for all r in the integral, the formula proves that eα(t) is a completely monotone (CM) function, by the Bernstein theorem.The Bernstein theorem provides a necessary and sufficient condition for a CM function to be a real Laplace transform of a non-negative measure.Source: §2 · p. 5
Changing variables by τ = 1/r converts the frequency representation into a relaxation-time representation, and the two spectral distributions are identical.The distribution describes a continuous mixture of elementary exponential relaxation processes.Source: §2 · p. 5
For α = 1, the function reduces to exp(−t), whose spectral distribution is the Dirac delta centered at r = 1.Source: §2 · p. 6

3 Asymptotic approximations to the Mittag-Lefler function

The section contrasts the conventional stretched-exponential and power-law limits with two rational completely monotone approximants. It reports that the long-time rational approximant also works numerically at short times, while its proposed bounds on the target remain conjectural.

3.1 The two common asymptotic approximations

The common short- and long-time behaviors are a stretched exponential and a negative power law, respectively, with eα(t) interpolating between them.The stretched exponential describes rapid small-time decay, and the power law describes slow large-time decay.Source: §3.1 · p. 7
The short-time stretched-exponential and long-time power-law curves can cross eα(t), changing the sign of their relative errors.The short-time curve stays below eα(t); the long-time curve starts above it, and an intersection may occur.Source: §3.1 · p. 9

3.2 The two rational asymptotic approximations

The proposed upper and lower bounds on eα(t) remain conjectural, although the authors report them in numerical examples over a large time range.They leave the bound conjecture open for proof or disproof.Source: §3.2 · p. 11
The rational long-time approximation gα(t) is also numerically good at short times, failing mainly in an intermediate range.The authors report this behavior for their numerical examples.Source: §3.2 · p. 12
The proposed first [0/1] Padé approximants are rational completely monotone functions derived from the short-time power series and the long-time asymptotic series.The authors prove gα(t) ≤ fα(t) for t ≥ 0 and 0 < α < 1 using the gamma-function reflection formula.Source: §3.2 · pp. 9, 10, 12

4 Conclusions

For 0 < α < 1 and t > 0, the paper discusses the completely monotone Mittag-Leffler function Eα(−t^α), whose frequency and time spectra coincide. It compares asymptotic approximations and reports numerical bounds and behavior of the new rational approximants.

4 Conclusions

As a completely monotone function, it admits non-negative frequency and time spectra; these spectra are equal, providing a universal scaling property.Source: §4 · p. 12
Two sets of completely monotone approximants are compared for small and large times; the latter set is described as probably new.Both sets are asymptotically equivalent to Eα(−t^α) in their respective time regimes.Source: §4 · p. 12
The new rational pair provides numerical upper and lower bounds over a large time range, while validity for every t > 0 and 0 < α < 1 is posed as a conjecture.The reported bounding behavior is based on numerical examples.Source: §4 · p. 12
The long-time rational approximation also works well at short times, failing only in an intermediate range.The paper notes this behavior as surprising.Source: §4 · p. 12

Final remark

The final remark reports a proof of the bounding conjecture by Thomas Simon and notes Renato Spigler’s earlier partial proof near x = 0.

Final remark

Thomas Simon provided a proof of the conjecture using probability arguments.The paper identifies Simon with the University of Lille.Source: Final remark · p. 13
Renato Spigler had earlier proved part of the conjecture for x = t^α in a right neighborhood of x = 0.The paper describes this neighborhood as the most critical set for establishing the conjecture.Source: Final remark · p. 13
The paper says Spigler’s partial proof preceded Simon’s proof and concerned the conjecture in equation (3.11).The partial result was presented at a July 2013 SIAM Annual Meeting.Source: Final remark · p. 13