some properties and computation of the mittag-leffler function · 2013. 12. 13. · of the...

Some properties and computation

of the Mittag-Leffler function

Renato Spigler

(with Moreno Concezzi)

Università “Roma Tre”, Rome, Italy

November , 2013 – Bilbao, FCPLNO Conference

to Francesco Mainardi, on his retirement

In the recent years there has been a wide interest in fractionalderivatives and in fractional differential equations.


This because, for instance, fractional time-derivatives may accountfor (time) delays, while fractional space-derivatives may explain anonlocal behavior, typically through a power law (rather thanexponential) decay.


This because, for instance, fractional time-derivatives may accountfor (time) delays, while fractional space-derivatives may explain anonlocal behavior, typically through a power law (rather thanexponential) decay.

Therefore, transport and diffusion in certain media might be bettermodeled by fractional partial differential equations.

On the other hand, at the time of the celebrated Bateman project,Special Functions such as the so-called Mittag-Leffler (M-L)functions and some of its generalizations were considered as nomore than curiosities, relegated among “miscellanea functions”, inthe corresponding volumes.

However, it turned out that the M-L functions play in the theory offractional ordinary differential equations (fODE) the same role ofthe exponential for classical ODEs.

However, it turned out that the M-L functions play in the theory offractional ordinary differential equations (fODE) the same role ofthe exponential for classical ODEs.

In fact, as u(t) := e−t is the unique solution of the ODE

du

dt= −u, t > 0, u(0) = 1,

so the M-L function

eα(t) := Eα(−tα) =

∞∑

n=0

(−1)ntαn

Γ(αn + 1),

with t > 0, 0 < α < 1, solves the fODE

∗Dαt v = −v , t > 0, v(0

+) = 1,

where

∗Dαt v(t) :=

1

Γ(1 − α)

∫

t

0

v ′(τ)

(t − τ)αdτ

denotes the so-called “Caputo fractional derivative” of order α.

We keep in mind also the generalized M-L functions

Eα,β(z) =∞∑

n=0

(−1)nzn

Γ(αn + β),

with z ∈ C, α, β > 0, noting that

Eα,1(z) ≡ Eα(z).

The previous property of the M-L functions,

eα(t) := Eα(−tα) =

∞∑

n=0

(−1)ntαn

Γ(αn + 1),

[that it solves the aforementioned Caputo fODE] confers themsome centrality within theory and applications of fODEs.

The previous property of the M-L functions,

eα(t) := Eα(−tα) =

∞∑

n=0

(−1)ntαn

Γ(αn + 1),

[that it solves the aforementioned Caputo fODE] confers themsome centrality within theory and applications of fODEs.

Consequently, their relevance in (pure and) applied mathematicsand sciences has increased considerably.

Among the numerous properties of the M-L functions eα(t) [orE (−tα)], we mention that it is an entire function of tα, for everyfixed value of α; α can be complex, with Re(α) > 0.

Among the numerous properties of the M-L functions eα(t) [orE (−tα)], we mention that it is an entire function of tα, for everyfixed value of α; α can be complex, with Re(α) > 0.

Note that for α = 1 we obtain e1(t) = e−t , so that eα(t) is a

generalization of e−t .

Figure: graphs of the M-L functions eα(t) ≡ E (−tα), for various α

(Mainardi).

In this talk we wish to discuss two issues:


• A conjecture, due to F. Mainardi concerning estimates of theM-L functions by certain simple rational functions.


• A conjecture, due to F. Mainardi concerning estimates of theM-L functions by certain simple rational functions.

• • A (new?) numerical approach to compute values of the M-Lfunctions.

• F. Mainardi has formulated the following conjecture:


For every t > 0, and for every fixed α, with 0 < α < 1, theestimates

gα(t) ≤ eα(t) ≤ fα(t)

hold, where

gα(t) :=1

1 + tαΓ(1 − α),

eα(t) :=

∞∑

n=0

(−1)ntαn

Γ(αn + 1),

fα(t) :=1

1 + tα

Γ(1+α)

.


For every t > 0, and for every fixed α, with 0 < α < 1, theestimates

gα(t) ≤ eα(t) ≤ fα(t)

hold, where

gα(t) :=1

1 + tαΓ(1 − α),

eα(t) :=

∞∑

n=0

(−1)ntαn

Γ(αn + 1),

fα(t) :=1

1 + tα

Γ(1+α)

.

This conjecture is supported by rather extensive numericalevaluations of the three functions, gα(t), eα(t), and fα(t).

To date, it seems that Thomas Simon proved Mainardi’s conjectureby probabilistic arguments, as a side-product of his work on certainstable laws.

To date, it seems that Thomas Simon proved Mainardi’s conjectureby probabilistic arguments, as a side-product of his work on certainstable laws.

Nevertheless, below we discuss such conjecture, providing analternative partial solution based on a more classical, directapproach.

Note that the prospective upper bound [i.e., fα(t)] is related to theconvergent power series approximation

eα(t) ∼ 1 −tα

Γ(1 + α),

valid for small values of t, while the prospective lower bound [i.e.,gα(t)] refers to the asymptotic approximation

eα(t) ∼t−α

Γ(1 − α),

valid for large values of t.

Both can be related to the [0/1] Padé approximants of eα(t).

More precisely, it was known that

eα(t) ∼∞

∑

n=1

(−1)n−1t−αn

Γ(1 − αn), t → +∞,

(see Erdeélyi et al., i.e., the Bateman project’s volumes, 1955),and that

eα(t) = 1 −tα

Γ(1 + α)+ . . . ∼ exp

[

−tα

Γ(1 + α)

]

, t → 0+.

F. Mainardi and his collaborators reported that their numericalcomputations, made to validate such conjecture, were accurate towithin 1% (relative errors).

Here are some results:

Figure: Mainardi’s graphs of gα(t), eα(t), and fα(t) for α = 0.25.

Our computations showed that the conjecture is valid within0.001%.

Here are our results:

10−5

100

105

10−2

10−1

100

t

f α(t

),e α

(t),

g α(t

)

eα(t)

fα(t)

gα(t)

10−5

100

105

10−4

10−3

10−2

10−1

t

ε f,ε g

α= 0.25

erroref

erroreg

Figure: Our graphs of gα(t), eα(t), and fα(t) for α = 0.25.

10−5

100

105

10−3

10−2

10−1

100

t

f α(t

),e α

(t),

g α(t

)

eα(t)

fα(t)

gα(t)

10−5

100

105

10−6

10−5

10−4

10−3

10−2

10−1

100

t

ε f,ε g

α= 0.5

erroref

erroreg


10−5

100

105

10−5

10−4

10−3

10−2

10−1

100

t

f α(t

),e α

(t),

g α(t

)

eα(t)

fα(t)

gα(t)

10−5

100

105

10−8

10−6

10−4

10−2

100

102

t

ε f,ε g

α= 0.75

erroref

erroreg


10−5

100

105

10−6

10−5

10−4

10−3

10−2

10−1

100

t

f α(t

),e α

(t),

g α(t

)

eα(t)

fα(t)

gα(t)

10−5

100

105

10−10

10−8

10−6

10−4

10−2

100

102

t

ε f,ε g

α= 0.9

erroref

erroreg


10−5

100

105

10−6

10−5

10−4

10−3

10−2

10−1

100

t

f α(t

),e α

(t),

g α(t

)

eα(t)

fα(t)

gα(t)

10−5

100

105

10−10

10−8

10−6

10−4

10−2

100

102

t

ε f,ε g

α= 0.95

erroref

erroreg


10−5

100

105

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

t

f α(t

),e α

(t),

g α(t

)

eα(t)

fα(t)

gα(t)

10−5

100

105

10−12

10−10

10−8

10−6

10−4

10−2

100

102

t

ε f,ε g

α= 0.99

erroref

erroreg


Ideas to prove the conjecture:


A) The three functions involved, i.e., gα(t), eα(t) and fα(t), are allknown to be completely monotone (CM) functions, therefore theycan be considered as the Laplace transforms of positive measures(in particular, Laplace transforms of nonnegative integrablefunctions).


A) The three functions involved, i.e., gα(t), eα(t) and fα(t), are allknown to be completely monotone (CM) functions, therefore theycan be considered as the Laplace transforms of positive measures(in particular, Laplace transforms of nonnegative integrablefunctions).

Moreover, it is known that

eα(t) =

∫ +∞

0e−rtKα(r) dr ,

where

Kα(r) =sinπα

π

rα−1

r2α + 2rα cos πα+ 1.

Finding the “kernels” for gα and fα corresponding to the functionKα for the M-L functions, is possible using certain known relations.

In fact, from

1

sα + λ=

∫ +∞

0e−rttα−1Eα,α(−λt

α) dt.

(known in the literature), it follows

1

Γ(1 − α)Eα,α

(

−rα

Γ(1 − α)

)

≤sinπα

π

rα−1

r2α + 2rα cos πα+ 1

≤ Γ(1 + α)Eα,α(−rαΓ(1 + α)).

However, such result seems to be useless.

Here are some computed “kernels” (to within MATLAB errors).Results are not at all conclusive since Kα(r) is singular at r = 1when α = 1 (note that sinπα tends to vanish, however).

10−5

100

105

10−10

10−8

10−6

10−4

10−2

100

102

104

r

H1(

s),K

α(s)

,H2(

s)

Kα(s)

H1(s)

H2(s)

10−5

100

105

10−3

10−2

10−1

100

101

102

103

r

ε H1,

ε H2

α= 0.25

erroreH1

erroreH2

Figure: Comparing the graphs of the three “kernels” for α = 0.25.

10−5

100

105

10−8

10−6

10−4

10−2

100

102

104

r

H1(

s),K

α(s)

,H2(

s)

Kα(s)

H1(s)

H2(s)

10−5

100

105

10−5

10−4

10−3

10−2

10−1

100

r

ε H1,

ε H2

α= 0.5

erroreH1

erroreH2


10−5

100

105

10−10

10−8

10−6

10−4

10−2

100

102

r

H1(

s),K

α(s)

,H2(

s)

Kα(s)

H1(s)

H2(s)

10−5

100

105

10−5

10−4

10−3

10−2

10−1

100

101

r

ε H1,

ε H2

α= 0.75

erroreH1

erroreH2


10−5

100

105

10−12

10−10

10−8

10−6

10−4

10−2

100

102

r

H1(

s),K

α(s)

,H2(

s)

Kα(s)

H1(s)

H2(s)

10−5

100

105

10−5

10−4

10−3

10−2

10−1

100

101

102

r

ε H1,

ε H2

α= 0.9

erroreH1

erroreH2


10−5

100

105

10−12

10−10

10−8

10−6

10−4

10−2

100

102

r

H1(

s),K

α(s)

,H2(

s)

Kα(s)

H1(s)

H2(s)

10−5

100

105

10−5

10−4

10−3

10−2

10−1

100

101

102

103

r

ε H1,

ε H2

α= 0.95

erroreH1

erroreH2


10−5

100

105

10−12

10−10

10−8

10−6

10−4

10−2

100

102

r

H1(

s),K

α(s)

,H2(

s)

Kα(s)

H1(s)

H2(s)

10−5

100

105

10−6

10−4

10−2

100

102

104

r

ε H1,

ε H2

α= 0.99

erroreH1

erroreH2


B) Another approach might perhaps be based on properties of CMfunctions and their Padé approximants.

C) A direct approach might consist in comparing power series,considering that the rational functions gα and fα can be expandedin geometric series, in suitable intervals. Recall that

1

1 + x=

∞∑

n=0

(−1)nxn,

for |x | < 1, but we can write

1

1 + x=

1

x

1

1 + 1x

=

∞∑

n=0

(−1)nx−n−1

when |x | > 1. The special case x = 1 (for us only x > 0 matters)should be treated separately.

C) A direct approach might consist in comparing power series,considering that the rational functions gα and fα can be expandedin geometric series, in suitable intervals. Recall that

1

1 + x=

∞∑

n=0

(−1)nxn,

for |x | < 1, but we can write

1

1 + x=

1

x

1

1 + 1x

=

∞∑

n=0

(−1)nx−n−1

when |x | > 1. The special case x = 1 (for us only x > 0 matters)should be treated separately.

In our problem, either x := tαΓ(1 − α), or x := tα/Γ(1 + α).

It seems natural divide the problem into three cases according to


(I) 0 < tα < 1/Γ(1 − α) (< Γ(1 + α) < 1);


(I) 0 < tα < 1/Γ(1 − α) (< Γ(1 + α) < 1);

(II) 1/Γ(1 − α) < tα < Γ(1 + α) (< 1)


(I) 0 < tα < 1/Γ(1 − α) (< Γ(1 + α) < 1);

(II) 1/Γ(1 − α) < tα < Γ(1 + α) (< 1)

(III) tα > Γ(1 + α)) (> 1/Γ(1 − α)).


(I) 0 < tα < 1/Γ(1 − α) (< Γ(1 + α) < 1);

(II) 1/Γ(1 − α) < tα < Γ(1 + α) (< 1)

(III) tα > Γ(1 + α)) (> 1/Γ(1 − α)).

besides the two cases for tα = 1/Γ(1 − α), and tα = Γ(1 + α).

In fact, observe that, being 0 < α < 1, we have

0 <1

Γ(1 − α)< Γ(1 + α) < 1.

Recall also that

Γ(1 − α)Γ(1 + α) =πα

sinπα≥ 1;

and1 < 1 + α < 2 ⇒ 0 < Γ(1 + α) < 1.

All series involved are alternating series, i.e., their terms havealternating sign.

Recall that, when, in∞∑

n=0

(−1)nan = a0 − a1 + a2 − a3 + . . . ,

with an ≥ 0 for every n, and an → 0 as n → ∞ monotonically, notonly convergence is guaranteed, but estimating the sum, say S , ispossible according to:

S ≥ 0, S ≤ a0, S ≥ a0 − a1, S ≤ a0 − a1 + a2, etc.

All series involved are alternating series, i.e., their terms havealternating sign.

Recall that, when, in∞∑

n=0

(−1)nan = a0 − a1 + a2 − a3 + . . . ,

with an ≥ 0 for every n, and an → 0 as n → ∞ monotonically, notonly convergence is guaranteed, but estimating the sum, say S , ispossible according to:

S ≥ 0, S ≤ a0, S ≥ a0 − a1, S ≤ a0 − a1 + a2, etc.

This property holds for all the geometric series involved, but itdoes not hold for the M-L functions:

xn

Γ(αn + 1)

does not tend to zero monotonically, as n → ∞, for every fixedx > 0.

To date, we were only able to prove part of the conjecture,proceeding as follows:


1) we first show that, for every 0 < x := tα < 1/Γ(1 − α) (case(I)), and for every fixed 0 < α < 1, the M-L function eα(t) is aseries with alternating signs and indeed with the general termdecaying to 0 as n → ∞ monotonically.


1) we first show that, for every 0 < x := tα < 1/Γ(1 − α) (case(I)), and for every fixed 0 < α < 1, the M-L function eα(t) is aseries with alternating signs and indeed with the general termdecaying to 0 as n → ∞ monotonically.

Since the two geometric series corresponding to the rationalfunctions gα(t) and fα(t) enjoy the same property in such interval,estimates can be established in the form:

2)

gα(t) ≤ 1 − xΓ(1 − α) + x2Γ(1 − α)2 ≤ 1 −

x

Γ(1 + α)≤ eα(t)

for the lower bound, and

3)

eα(t) ≤ 1 −x

Γ(1 + α)+

x2

Γ(2α + 1)

≤ 1 −x

Γ(1 + α)+

x2

Γ(1 + α)2−

x3

Γ(1 + α)3≤ fα(t)

for the upper bound,

As for (1), setting

F (x , y) :=xy

Γ(αy + 1)=: ϕ(y),

with 0 < x := tα < 1/Γ(1 − α), 0 < α < 1, y ≥ 0, we obtainϕ(0) = 1 and

ϕ′(y) < 0 iff ψ(αy + 1) >1

αln x ≡ ln t, (1)

where ψ denotes the logarithmic derivative of Γ.

Now, being merely t > 0, the ln t on r.h.s. can be either positiveor negative.

However, confining t to being 0 < tα < 1/Γ(1 − α) (case (I)),and being

ψ(αy + 1) ≥ −γ

where γ = 0.57721... is the Euler-Mascheroni constant, sinceαy + 1 ≥ 1, we infer that (1) holds true provided that0 < tα ≤ e−αγ (approximately, 0 < t < 0.56).

0 1 2 3 4 5−20

−15

−10

−5

0

5

x

ψ(x

)

DefiningG (α) := e−αγΓ(1 − α)

for 0 < α < 1, we have G (0+) = 1, and

G ′(α) = −e−αγΓ(1 − α)[γ + ψ(1 − α)],

wherefrom G ′(α) ≥ 0 since ψ(1 − α) ≤ −γ. Therefore G (α) > 1for all y > 0, and this amounts to the validity of our claim.

As for (2),

gα(t) ≤ 1 − xΓ(1 − α) + x2Γ(1 − α)2 ≤ 1 −

x

Γ(1 + α)≤ eα(t)

is equivalent (after a little algebra) to

tαΓ(1 − α) ≤ 1 −1

Γ(1 − α)Γ(1 + α)= 1 −

sinπα

πα,

and the r.h.s. is a number less than 1, hence we have an effectiverestriction to the values of t in case (I).

As for (3),

eα(t) ≤ 1 −x

Γ(1 + α)+

x2

Γ(2α + 1)

≤ 1 −x

Γ(1 + α)+

x2

Γ(1 + α)2−

x3

Γ(1 + α)3≤ fα(t),

it is equivalent to (after a little algebra)

0 < tα < Γ(1 + α)

(

1 −Γ(1 + α)2

Γ(2α+ 1)

)

.

The fraction on the r.h.s. can be shown to be positive (by Alsinaand Tomàs’ inequality), hence this is an effective restriction to thevalues of t in case (I).

There are several useful inequalities involving the gamma function,established over the years:

• W. Gautschi

1

(k + 1)1−λ<

Γ(k + λ)

Γ(k + 1)<

1

k1−λ, 0 < λ < 1, k = 1, 2, . . .

• D. Kershaw

1(

x − 12 +√

λ+ 14

)1−λ<

Γ(x + λ)

Γ(x + 1)<

1(

x + 12)1−λ

,

0 < λ < 1, x > 0.

• L. Lorch

1

(k + λ)1−λ<

Γ(k + λ)

Γ(k + 1)<

1(

k + 12)1−λ

, 0 < λ < 1, k = 1, 2, . . .

Lorch also proved that his upper bound holds even for λ > 2, whilethe opposite inequality holds when 1 < λ < 2. A. Laforgia provedthat Lorch’s inequalities also hold for real positive values of k.

• Alsina and M.S. Tomás

1

n!≤

Γ(1 + x)n

Γ(1 + nx)≤ 1, 0 ≤ x ≤ 1, n ∈ N.

• J. Sándor (generalized Alsina-Tomás inequalities as)

1

Γ(1 + a)≤

Γ(1 + x)a

Γ(1 + ax)≤ 1, 0 ≤ x ≤ 1, a ≥ 1.

etc.However, they seem to be not enough to prove the conjecture.

• • Computing the M-L functions eα(t) solving numerically acertain fODE.

Several algorithms have been developed over the years to computethe M-L functions. As often, computing a variety of SpecialFunctions, a few different methods have been adopted on differentintervals: power series for small arguments, asymptoticapproximations for large arguments, etc.

• • Computing the M-L functions eα(t) solving numerically acertain fODE.

Several algorithms have been developed over the years to computethe M-L functions. As often, computing a variety of SpecialFunctions, a few different methods have been adopted on differentintervals: power series for small arguments, asymptoticapproximations for large arguments, etc.

A code is also available in MATLAB, based on such works(MATLAB code due to I. Podlubny (2006), following contributionsgiven by R. Gorenflo, J. Loutchko, and Yu. Luchko (2002), (2006);(2012).

Since the M-L function plays for fODEs the same role of theexponential for classical ODEs, and in particular, eα(t) solves thesimple fODE

∗Dαt v = −v , t > 0, v(0

+) = 1,

where ∗Dαt v(t) is the Caputo derivative, one could evaluate eα(t)

solving numerically such an fODE.

A Predictor-Corrector method, based on the so-calledGrünwald-Letnikov (G-L) approximation of the Caputo derivativewas used with h = 0.05 as integration stepsize.(The number of terms in the G-L approximation, as well as thestepsize, were actually chosen according to the value of α.)

0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

y(t)

α= 0.25

u(t)eα(t)

Figure: u(t) is the numerical solution of the fODE for the M-L function,while eα(t) is computed by MATLAB, with α = 0.25.

0 5 10 15 200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

y(t)

α= 0.5

u(t)eα(t)

Figure: u(t) is the numerical solution of the fODE for the M-L function,while eα(t) is computed by MATLAB, for α = 0.50.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

y(t)

α= 0.75

u(t)eα(t)


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

y(t)

α= 0.9

u(t)eα(t)


0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

y(t)

α= 0.95

u(t)eα(t)


Happy retirement, Francesco! . . .

hoping that you will keep going, as an active researcher.

some properties and computation of the mittag-leffler function · 2013. 12. 13. · of the...

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