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Hilfer-Prabhakar derivatives and applications

International Conference on Fractional Calculus 9–10 June 2020 GhentAnalysis & PDE Center Ghent University, Belgium

Æivorad TomovskiUniversity of Ostrava

Ostrava June 9, 2020

OutlineIntroduction and preliminaries

Complete Monotonicity of the Mittag-Le✏er functionsHilfer-Prabhakar derivativeFractional Poisson Process

Introduction and preliminariesGosta Mittag-Le✏er, Mittag-Le✏er functionsFractional calculusFractional oscillation-relaxation di↵erential equations

Complete Monotonicity of the Mittag-Le✏er functionsComplete monotonicityFractional relaxation equationFractional oscillation equation

Hilfer-Prabhakar derivativeHilfer-Prabhakar derivativesCauchy problemsFree electron laser equation

Fractional Poisson ProcessFractional generalization of Poisson renewal processWaiting time densityFractional Poisson process - Hilfer-Prabhakar derivative

Zivorad Tomovski Hilfer-Prabhakar derivatives and applications



Gosta Mittag-Le✏er, Mittag-Le✏er functionsFractional calculusFractional oscillation-relaxation di↵erential equations

Gosta Mittag-Le✏er

I Gosta Mittag-Le✏er

(1846 - 1927)

- Swedish mathematician

- founder of the famous journalActa Mathematica (1882)





Mittag-Le✏er (M-L) functions

I One parameter M-L functionMittag-Le✏er, C.R. Acad. Sci. Paris 137 (1903) 554

E↵(z) =1X

n=0

zn

�(↵n + 1)(z 2 C,<(↵) > 0) (1)

I Two parameter M-L functionWiman, Acta Math. 29 (1905) 191

E↵,�(z) =1X

n=0

zn

�(↵n + �)(z ,� 2 C,<(↵) > 0) (2)

- entire functions of order ⇢ = 1/<(↵) and type 1





I Two parameter M-L function (cont.)- generalization of the exponential, hyperbolic and trigonometricfunctions

E1,1(z) = ez , E2,1(z2) = cosh(z)

E2,1(�z2) = cos(z), E2,2(�z2) = sin(z)/z

E1/2,1(z) = ez2

erfc(�z)

- Miller-Ross functionMiller, Ross, An Introduction to Fractional Calculus and FractionalDi↵erential Equations (1993)

Et(⌫, a) = t⌫1X

k=0

(at)k

�(⌫ + k + 1)= t⌫E1,⌫+1(at) (3)





Graphical representations





Integration of M-L functions

Z t

0⌧↵�1E↵,↵ (�a⌧↵) (t � ⌧)��1E↵,� (�b(t � ⌧)↵) d⌧

=E↵,� (�bt↵)� E↵,� (�at↵)

a� bt��1, a 6= b

Z t

0⌧↵�1E↵,↵ (�a⌧↵) (t � ⌧)��1E↵,� (�a(t � ⌧)↵) d⌧

= t↵+��1E↵,� (�at↵)

- Laplace transformPodlubny, Fractional Di↵erential Equations (1999)

L⇥t��1E↵,�(±at↵)

⇤(s) =

s↵��

s↵ ⌥ a, <(s) > |a|1/↵





I Three parameter M-L functionPrabhakar, Yokohama Math. J. 19 (1971) 7

E�↵,�(z) =

1X

n=0

(�)n�(↵n + �)

· zn

n!(z ,�, � 2 C,<(↵) > 0) (4)

- entire function of order ⇢ = 1/<(↵) and type 1

Lht��1E�

↵,� (!t↵)i(s) =

s↵��

(s↵ � !)�, |!/s↵| < 1

- Asymptotic expansionSaxena et al., Astrophys. Space Sci. 209 (2004) 299

E �↵,�(z) =

(�z)��

�(�)

1X

n=0

�(� + n)

�(� � ↵(� + n))· (�z)�n

n!, |z | > 1

E �↵,�(z) ⇠ O

�|z |��

�, |z | > 1





I Four parameter M-L functionSrivastava, Tomovski, Appl. Math. Comput. 211 (2009) 198

E�,↵,� (z) =

1X

n=0

(�)n�(↵n + �)

· zn

n!(5)

(z ,�, � 2 C,<(↵) > max{0,<()� 1},<() > 0)

- entire function: order ⇢ = 1<(↵�)+1 , type � = 1

⇢

⇣<(↵)<()

<(↵)<(↵)

⌘⇢

Lht⇢�1E�,

↵,� (!t�)i(s) =

s�⇢

�(�)· 2 1

(⇢,�), (�,)

(�,↵)

��!

s�

�

! 2 C, min{<(�),<(),<(⇢),<(�)} > 0

p q

(ap,Ap)(bq,Bq)

�� z�=P1

k=0�(a1+A1k)···�(ap+Apk)�(b1+B1k)···�(bp+Bpk)

· zk

k! - Wright





Graphical representations





Fractional calculus

I Integral operators

- Riemann-Liouville (R-L) fractional integral

�Iµa+f

�(t) =

1

�(µ)

Z t

a

f (⌧)

(t � ⌧)1�µd⌧, t > a, <(µ) > 0.

(6)

I 0a+f (t) = f (t), (identity operator)

I �a+I�a+ = I �+�

a+ = I �a+I�a+, (semi-group property)

I �a+(t � a)s =�(s + 1)

�(s + 1 + �)(t � a)s+� , � � 0, s > �1





Generalized integral operator

Srivastava, Tomovski, Appl. Math. Comput. 211 (2009) 198

(E!;�,a+;↵,�')(t) =

Z t

a(t � ⌧)��1E �,

↵,� (!(t � ⌧)↵)'(⌧)d⌧ (7)

E�,↵,� (z) - four parameter M-L function (5)

- integral operator (7) appears in the solution of fractional di↵usion -

wave equations with a source term

Tomovski, Sandev, Appl. Math. Comput. (2012)

Tomovski, Sandev, Comput. Math. Appl. 62 (2011) 1554

Sandev, Metzler, Tomovski, J. Phys. A: Math. Theor. 44 (2011) 255203

Sandev, Tomovski, J. Phys. A: Math. Theor. 43 (2010) 055204

Sandev, Tomovski, Proc. Symp. Frac. Sig. Systems (Lisbon, 2009)





Generalized integral operator. Properties


kE!;�,a+;↵,�'k1 Mk'k1, 8' 2 L(a, b)

M = (b � a)<(�)1X

n=0

|(�)kn|(<(↵)n + �) |�(↵n + �)|

|!(b � a)<(↵)|n

n!





I Open problem


Lemma

The following inequality holds true⇣E!;�,k0+;↵,�E

!;�,k0+;↵,µ'

⌘(x)

⇣E!;�+�,k0+;↵,�+µ'

⌘(x) (8)

↵,�, �,!, �, µ 2 R+, 0 < k 1,↵ > k� 1

for any positive Lebesgue integrable function ' 2 L(a, b). In (8) equalityholds true when k = 1

- Open problem: Is it possible that relation in (8) holds true alsofor 0 < k < 1?

If it is, find necessary and su�cient conditions such that (8) holds

true for 0 < k < 1





Integral operators. Examples


⇣I�a+

h(t � a)��1E �,

↵,� (!(t � a)↵)i⌘

(x)

= (x � a)�+��1E �,↵,�+� (!(x � a)↵)

I�a+

⇣E!;�,a+;↵,�'

⌘= E!;�,

a+;↵,�+�' = E!;�,a+;↵,� I

�a+'

- Hardy-type inequality: Tomovski, Hilfer, Srivastava, Integral

Transform. Spec. Funct. 21 (2010) 797

�R10 x�↵p|

�I↵0+f

�(x)|pdx

�1/p �(1/p0)�(↵+1/p)

�R10 |f (x)|pdx

�1/p

1p + 1

p0 = 1





I Fractional derivatives

- R-L fractional derivative

�Dµ

a+f�(t) =

✓d

dt

◆n �I n�µa+ f

�(t), <(µ) > 0, n = [<(µ)] + 1

- Examples:Srivastava, Tomovski, Appl. Math. Comput. 211 (2009) 198Tomovski, Hilfer, Srivastava, Integral Transform. Spec. Funct. 21(2010) 797

⇣D�

a+

h(t � a)��1E�,

↵,� (!(t � a)↵)i⌘

(x)

= (x � a)��1E�,↵,�� (!(x � a)↵)

D�a+

⇣E!;�,a+;↵,�'

⌘= E!;�,

a+;↵,��' = E!;�,a+;↵,�D

�a+'





I Composite fractional derivative

- order 0 < µ < 1 and type 0 ⌫ 1Hilfer, Application of Fractional Calculus in Physics (Singapore:World Scientific, 2000)

�Dµ,⌫

a+ f�(t) =

✓I ⌫(1�µ)a+

d

dt

⇣I (1�⌫)(1�µ)a+ f

⌘◆(t) (9)

- The Hilfer-composite time derivative was used by Hilfer tosuccessfully describe the dynamics in glass formers over anextremely large frequency window

- From a practical point of view the description in terms ofcomposite-fractional operators increases the versatility of thesolution of the dynamic equation in the description of complexexperimental data over than ten orders of magnitude with lessparameters than traditional fit functions such as Havriliak-Negami





I Composite fractional derivative

Hilfer, Application of Fractional Calculus in Physics (Singapore:World Scientific, 2000)- Examples:

�Dµ,⌫

a+

⇥(t � a)��1

⇤�(x) =

�(�)

�(�� µ)(x � a)��µ�1

(x > a, 0 < µ < 1, 0 ⌫ 1,<(�) > 0)

⇣Dµ,⌫

a+

h(t � a)��1E�,

↵,� (!(t � a)↵)i⌘

(x)

= (x � a)��µ�1E�,↵,��µ (!(x � a)↵)

Dµ,⌫a+

⇣E!;�,a+;↵,�'

⌘= E!;�,

a+;↵,��µ'





Composite fractional derivative

- Hardy-type inequality: Tomovski, Hilfer, Srivastava, Integral Transform.Spec. Funct. 21 (2010) 797

✓Z 1

0x�↵p|

⇣D↵,�

0+ f⌘(x)|pdx

◆1/p

�(1/p0)

�(�(1� ↵) + 1/p)

✓Z 1

0|⇣D↵+��↵�

0+ f⌘(x)|pdx

◆1/p

1

p+

1

p0= 1

- Laplace transform: Hilfer, Application of Fractional Calculus in Physics(Singapore: World Scientific, 2000)

L⇥Dµ,⌫

0+ f (t)⇤= sµL [f (t)]� s⌫(µ�1)

⇣I (1�⌫)(1�µ)0+ f

⌘(0+) (10)





I Fractional derivatives

⌫ = 0, a = 0: classical R-L fractional derivative

�RLD

µ0+f�(t) =

d

dt

⇣I (1�µ)0+ f

⌘(t). (11)

- ⌫ = 1, a = 0: Caputo fractional derivativeCaputo, Elasticita e Dissipazione (Bologna: Zanichelli, 1969)

�CD

µ0+f�(t) =

✓I (1�µ)0+

d

dtf

◆(t). (12)

�CD

µ0+f�(t) =

�RLD

µ0+f�(t)� f (0+) · t�µ

�(1� µ), (13)

where 0 < µ < 1





Fractional oscillation-relaxation di↵erential equations

I Fractional relaxation-oscillation

Mainardi, Gorenflo, J. Comput. Appl. Math. 118 (2000) 175

d↵u(t;↵)

dt↵+ c↵u(t;↵) = 0, c > 0, 0 < ↵ 2

u(0;↵) = u0, 0 < ↵ 1, fractional relaxation

u(0+;↵) = u0, u(0+;↵) = 0, 1 < ↵ 2, fractional oscillation

- solution

u(t;↵) = u0E↵ (�(ct)↵)





I Abel-Volterra integral equation of a second kind

u(t) +�

�(↵)

Z t

0

u(⌧)

(t � ⌧)1�↵d⌧ = f (t)

- solution represented via M-L function E↵ (��t↵)

- relaxation-oscillation phenomena (papers by Mainardi andGorenflo)

CD↵0+u(t) = D↵

u(t)�

m�1X

k=0

tk

k!u(k)(0+)

!= �u(t) + q(t)

m � 1 < ↵ m, uk(0+) = ck , k = 0, 1, 2, ...,m � 1u = u(t) - field variable; q(t) - given function





I Some fractional di↵erential and integral equations withcomposite fractional derivative


�Dµ,⌫

0+ y�(x) = �

⇣E!;�,0+;↵,�

⌘(x) + f (x)

⇣I (1�⌫)(1�µ)0+ y

⌘(0+) = c

- solution

y(x) = cxµ�⌫(1�µ)�1

�(µ� ⌫ + µ⌫)+ �xµ+�E�,

↵,�+µ+1 (!x↵)

+1

�(µ)

Z x

0(x � t)µ�1f (t)dt





I Some fractional di↵erential and integral equations withcomposite fractional derivative


x�Dµ,⌫

0+ y�(x) = �

⇣E!;�,0+;↵,�

⌘(x)

⇣I (1�⌫)(1�µ)0+ y

⌘(0+) = c1

- solution

y(x) = c2xµ�1

�(µ)+ c1

xµ�⌫(1�µ)�1

�(µ� ⌫ + µ⌫)

� �

�(µ)

Z x

0tµ�1(x � t)��1E�,

↵,�+1 (!(x � t)↵) dt

- c1 and c2 are arbitrary constants




Complete monotonicityFractional relaxation equationFractional oscillation equation

I e�↵,� (t,�) function

e�↵,� (t,�) = t��1E �↵,� (��t

↵)

e↵,� (t,�) = t��1E↵,� (��t↵)

e↵ (t,�) = E↵ (��t↵)

↵,�, � > 0,� 2 C

e�↵,� (t) ⌘ e�↵,� (t; 1)

e↵,� (t) ⌘ e↵,� (t; 1)

e↵ (t) ⌘ e↵ (t; 1)





Complete monotonicity

- A function f : [0,1) ! [0,1) is completely monotone (CM) iff 2 C [0,1), f is infinitely di↵erentiable on (0,1) and

(�1)m f (m) (x) � 0 (x > 0,m = 0, 1, 2, ..)

- Bernstein interpretation of CM function:

f (s) =

1Z

0

e�stK (t) dt





I Complete monotonicity

Pollard (1948): Bull. Am. Math. Soc. 54.

E↵ (�x) , x > 0 is CM if 0 < ↵ 1

E↵ (�x↵) , x > 0 is CM if 0 < ↵ 1

e↵ (x ;�) , x > 0 is CM if 0 < ↵ 1, � > 0

Schneider (1996): Exposition.Math. 14; Miller (1997/98): RealAnal. Exchange 23

E↵,� (�x) , x > 0 is CM if 0 < ↵ 1,� � ↵

e↵,� (x ;�) , x > 0 is CM if 0 < ↵ � 1, � > 0

Hanyga-Seredynska (2008): J. Stat. Phys. 131

E�↵,1 (�x↵) , x � 0 is CM i↵ ↵, � 2 (0, 1) .






Gorenflo-Mainardi (1996): PreprintA-14/96, FachbereichMathematik and Informatik,Free University, Berlinhttp://www.math.fu-berlin.de/publ/index.html(1997) Fractals and Fractional Calculus in Continuum Mechanics,Springer-Verlag, Wien, Berlin, New York

e↵ (t) =

8>><

>>:

1R

0e�rtK↵ (r) dr ,↵ 2 (0, 1]

1R

0e�rtK↵ (r) dr + 2

↵et cos( ⇡

↵ ) cos⇥t sin

�⇡↵

�⇤,↵ 2 (1, 2]

K↵ (r) =r↵�1

⇡

sin (⇡↵)

r2↵ + 2r↵ cos (⇡↵) + 1> 0,↵ 2 (0, 1]

limt!0+

e↵ (t) = 1

1Z

0

K↵ (r) dr =

⇢1, 0 < ↵ 1

1� 2↵ , 1 < ↵ 2






e↵,� (t) =

1Z

0

e�rtK↵,� (r) dr , (0 < ↵ < � 1)

K↵,� (r) =r↵��

⇡

r↵ sin (⇡↵) + sin [⇡ (� � ↵)]

r2↵ + 2r↵ cos (⇡↵) + 1> 0

e↵,� (t) =

1Z

0

e�rtK↵,� (r) dr+2

↵et cos(

⇡↵ ) cos

ht sin

⇣⇡↵

⌘� ⇡

↵(� � 1)

i,

↵ 2 (1, 2] , � > 0

Tomovski, Pogany, Srivastava (2014), J. Franklin Institute

e↵,� (t) , t > 0 is CM if ↵ 23

2, 2

◆,� 2

1,↵+

3

2

◆





Fractional relaxation equation (Gorenflo-Mainardi)

du

dt+ a

d↵u

dt↵+ u (t) = q (t) , u (0+) = c0, 0 < ↵ < 1

u (t) = c0u0 (t) +

tZ

0

q (t � ⌧) u� (⌧) d⌧, u� (t) = �u00 (t)

u0 (t) =

1Z

0

e�rtK↵ (r ; a) dr

K↵ (r ; a) =1

⇡

ar↵�1 sin (↵⇡)

(1� r)2 + a2r2↵ + 2 (1� r) ar↵ cos (↵⇡)> 0, u0 (t) is CM.

u� (t) =

1Z

0

e�rtrK↵ (r ; a) dr , u� (t) is CM.





Fractional oscillation equation (Gorenflo-Mainardi)

d2v

dt2+ a

d↵v

dt↵+ v (t) = q (t) , 0 < ↵ < 2

v (0+) = c0, v 0 (0+) = c1.

(a) v (t) = c0v0 (t)� c1v00 (t)�

tZ

0

q (t � ⌧) v 00 (⌧) d⌧ (0 < ↵ < 1)

(b) v (t) = c0v0 (t) + c1

tZ

0

v0 (⌧) d⌧ �tZ

0

q (t � ⌧) v 00 (⌧) d⌧, (1 < ↵ < 2)

v0 (t) = f↵ (t; a) + g↵,� (t; a)





f↵ (t; a) =

1Z

0

e�rtH↵ (r , a) dr

H↵ (r , a) =1

⇡

ar↵�1 sin (↵⇡)

(r2 + 1)2 + a2r2↵ + 2 (r2 + 1) ar↵ cos (↵⇡)

H↵ (r , a) > 0 (0 < ↵ < 1) , r > 0

H↵ (r , a) < 0 (1 < ↵ < 2) , r > 0

In case (a) : f↵ (t) is CM; In case (b) : � f↵ (t) is CM

- Oscillatory character

g↵,� (t) = 2<(

⇢e i� + a�⇢e i�

�↵�1

2⇢e i� + a↵ (⇢e i�)↵�1 e(⇢ei�)t

),⇣⇢ > 0,

⇡

2< � < ⇡

⌘

g↵,� (t) > 0 (0 < ↵ < 1)

g↵,� (t) < 0 (1 < ↵ < 2)





I Tomovski, Pogany, Srivastava (2014): J. Franklin Inst.

e�↵,� (t) =

1Z

0

e�rtK�↵,� (r) dr , (↵ 2 (0, 1] ,� > 0, � > 0)

K�↵,� (r) =

r↵��

⇡

sinh�arctg r↵ sin(⇡↵)

r↵ cos(⇡↵)+1 + ⇡ (� � ↵�)i

[r2↵ + 2r↵ cos (⇡↵) + 1]�/2> 0

✓↵ 2

✓0,

1

2

�,� 2 (0, 1) , � > 0,� � ↵�

◆

en↵,� (t) =

1Z

0

e�rtK n↵,� (r) dr +

2 (�1)n�1

↵n (n � 1)!et cos(⇡/↵)

cosht sin

⇣⇡↵

⌘� ⇡

↵(� � 1)

i n�1X

l=0

(1� n)l cl(↵n � � � n + 2)l

(↵ 2 (1, 2] ,� > 0, � = n 2 N)Zivorad Tomovski Hilfer-Prabhakar derivatives and applications




e↵,� (t) t > 0 is CM if ↵ 23

2, 2

◆,� 2

1,↵+

3

2

◆.

e�↵,� (t) t > 0 is CM if ↵ 2✓0,

1

2

�, � 2 (0, 1) , � > 0, ↵� �.

Wright function:

� (⇢,�↵; z) =1X

n=0

1

� (⇢� n↵)

zn

n!

↵ 2 (0, 1) , ⇢ 2 C





Stankovic (1970): Publ. Inst. Math. (Beograd)

(↵,�, �; u) = u��↵��1�� ↵�,�↵;�u�↵

�> 0

u > 0,↵ 2 (0, 1) ,� � ↵�

e�↵,� (t;�) =1

� (�)

1Z

0

e��xK (↵,�, �; x) dx

K (↵,�, �; x) = x (��1)/↵�1 ⇣↵,�, �; tx�1/↵

⌘> 0

↵,� 2 (0, 1) , �,� > 0, � � ↵�


e�↵,� (t;�) is CM if ↵,� 2 (0, 1) , �,� > 0,� � ↵�.





Stankovic (1970): Publ. Inst. Math. (Beograd)

��x�p�1��p,�↵;�x�↵

�� 1

↵⇡

�� p+1

↵

�

�cos ↵⇡

2

� p+1↵

(p > �1)


��e�↵,� (t;�)��

�⇣� � ��1

↵

⌘�⇣

��1↵

⌘

⇡↵��1↵ � (�)

⇥cos�⇡↵2

�⇤��((��1)/↵)(t > 0)

↵ 2 (0, 1) ,� > 1, �,� > 0, ↵� > � � 1




Hilfer-Prabhakar derivativesCauchy problemsFree electron laser equation

Hilfer-Prabhakar derivatives

I Prabhakar integral operator

Prabhakar (1971): Yokohama Math. J. 19

⇣E�↵,�,�,a+'

⌘(t) =

tZ

a

(t � ⌧)��1 E�↵,� (� (t � ⌧)↵)' (⌧) d⌧

⇣E�↵,�,�,0+'

⌘(t) =

⇣' ⇤ e�↵,�

⌘(t)

E�↵,�,�,0+E

�0

↵,�0,�,0+ = E�+�0

↵,�+�0,�,0+

Sobolev space

Wm,1 [a, b] =

⇢f 2 L1 [a, b] :

dm

dtmf 2 L1 [a, b]

�, m = 1, 2, ..





I Prabhakar derivativeGarra, Gorenflo, Polito, Tomovski (AMC, 2014)

D�↵,�,�,0+f (x) =

dm

dxmE��↵,m��,�,0+f (x)

f 2 L1 [0, b] , f ⇤ e��↵,m�� 2 Wm,1 [0, b] , m = [�]

↵,�, �,� 2 C, < (↵) ,< (�) > 0

I Hilfer-Prabhakar derivativeGarra, Gorenflo, Polito, Tomovski (AMC, 2014)

⇣D�,µ,⌫

↵,�,0+f⌘(t) =

✓E��⌫↵,⌫(1�µ),�,0+

d

dt

⇣E��⌫↵,(1�⌫)(1�µ),�,0+f

⌘◆(t)

µ 2 (0, 1) , ⌫ 2 [0, 1] , f 2 L1 [a, b] , f ⇤ e��(1�⌫)↵,(1�⌫)(1�µ) 2 AC 1 [0, b]





I Regularized version of Hilfer-Prabhakar derivative⇣CD�,µ

↵,�,0+f⌘(t) =

✓E��⌫↵,⌫(1�µ),�,0+E

��(1�⌫)↵,(1�⌫)(1�µ),�,0+

d

dtf

◆(t)

=

✓E��↵,1�µ,�,0+

d

dtf

◆(t)

I Laplace transform of the Hilfer-Prabhakar derivative

LhD�,µ,⌫

↵,�,0+fi(s) = sµ

�1� �s�↵

�� L [f ] (s)

�s�⌫(1�µ)�1� �s�↵

��⌫ hE��(1�⌫)↵,(1�⌫)(1�µ),�,0+f (t)

i

t=0+

I Laplace transform of the regularized version ofHilfer-Prabhakar derivative

LhCD�,µ

↵,�,0+fi(s) = sµ

�1� �s�↵

�� L [f ] (s)� sµ�1�1� �s�↵

��f (0+)





I Caputo-Fabrizio, Progr. Fract. Di↵. Appl. (2015)

CFD↵a+f (t) =

M(↵)

1� ↵

Z t

aexp

✓� ↵

1� ↵(t � ⌧)

◆f 0(⌧) d⌧

I Attangana-Baleanu, Thermal Science (2016)

ABCD↵a+f (t) =

M(↵)

1� ↵

Z t

aE↵

✓� ↵

1� ↵(t � ⌧)↵

◆f 0(⌧) d⌧





Cauchy problems

8>>>><

>>>>:

D�,µ,⌫↵,�,0+u (x , t) = K @2u(x,t)

@x2 , t > 0, x 2 RE��(1�⌫)↵,(1�⌫)(1�µ),�,0+u (x , 0+) = g (x)

limx!±1

u (x , t) = 0

µ 2 (0, 1) , ⌫ 2 [0, 1] ,� 2 R, K ,↵ > 0, � � 0

u (x , t) =1

2⇡

+1Z

�1

^g (k)

1X

n=0

��Kk2

�ntµ(n+1)�⌫(µ�1)�1E�(n+1�⌫)

↵,µ(n+1)�⌫(µ�1) (�t↵)

!e�ikxdk





I Cauchy problems8>>><

>>>:

CD�,µ↵,�,0+u (x , t) = K @2u(x,t)

@x2 , t > 0, x 2 Ru (x , 0+) = g (x)lim

x!±1u (x , t) = 0

µ 2 (0, 1) , � 2 R, K ,↵ > 0, � � 0

u (x , t) =1

2⇡

+1Z

�1

^g (k)

1X

n=0

��Kk2tµ

�nE�n↵,µn+1 (�t

↵)

!e�ikxdk





Free electron laser equation

8<

:

dydx = �i⇡g

xR

0(x � t) e i⌘(x�t)y (t) dt, g , ⌘ 2 R, x 2 (0, 1]

y (0) = 1(

D�,µ,⌫↵,!,0+y (x) = �E$

↵,µ,�,0+y (x) + f (x) , x > 0, f (x) 2 L1 (0,1)

E��(1�⌫)↵,(1�⌫)(1�µ),�,0+y (0+) = k

y (x) = k1X

n=0

�nx⌫(1�µ)+µ+2µn�1E�+n($+�)��⌫↵,⌫(1�µ)+2µn+µ (!x

↵)

+1X

n=0

�nE�+n($+�)↵,µ(2n+1),!,0+f (x)

� = 0, ⌫ = 0, µ ! 1, f ⌘ 0, � = �i⇡g , ! = i⌘, ↵ = $ = k = 1




Fractional generalization of Poisson renewal processWaiting time densityFractional Poisson process - Hilfer-Prabhakar derivative

Fractional Poisson process

- Poisson process - renewal process with waiting time pdf of exp. type

� (t) = �e��t , � > 0, t � 0

with moments: hT i = 1

�,⌦T 2↵=

1

�2, ...., hT ni = 1

�n

- Survival probability:

(t) = P (T > t) = e��t , t � 0

d

dt (t) = �� (t) , t � 0, (0+) = 1

Khintchine (1960): Mathematical Methods in the Theory of Queuing

p0 (t) = e��t ,d

dtpk (t) = � (pk�1 (t)� pk (t)) , pk (0) = 0, k = 1, 2, 3, ..

pk (t) = P (N (t) = k) =(�t)k

k!e��t , t � 0, k = 0, 1, 2, ..





I Fractional generalization of Poisson renewal processBeghin-Orsinher (2009): Electronic Journ. Prob. 14

⇤D�t (t) = � (t) , t > 0, 0 < � 1; (0+) = 1

(t) = P (T > t) = E�

��t�

�, 0 < � 1.

p0 (t) = E�

��t�

�, ⇤D

�t pk (t) = pk�1 (t)� pk (t) , k = 1, 2, 3, ..

pk (t) = P (N (t) = k) =tk�

k!E (k)�

��t�

�, k = 0, 1, 2, ..





Waiting time density

I Mittag-Le✏er waiting time density (CTRW)Hilfer, Anton (1995), Phys. Rev. E 51.

fµ (t) = � d

dtEµ (�tµ) = tµ�1Eµ,µ (�tµ) , µ > 0

1Z

0

e�st fµ (t) dt =1

1 + sµ(|sµ| < 1)

I Prabhakar-Mathai waiting time density Mathai, 2006: FCAA

gµ,⌫ (t) = tµ⌫�1E⌫µ⌫,µ (�tµ) , µ, ⌫ > 0

1Z

0

e�stgµ,⌫ (t) dt =1

(1 + sµ)⌫(|sµ| < 1)





I Fractional Poison Process associated with regularizedversion of Hilfer-Prabhakar derivativeGarra, Gorenflo, Polito, Tomovski (2014): Appl. Math. Comput.

8<

:

cD�,µ⇢,��,0+pk (t) = ��pk (t) + �pk�1 (t) , k � 0, t,� > 0

pk (0) =

⇢1 : k = 00 : k � 1

(� > 0, � � 0, 0 < ⇢ 1, 0 < µ 1)

- Define probability generating function of the counting numberN (t) , t � 0 :

G (v , t) =1X

k=0

vkpk (t)





I (cont.)⇢

cD�,µ⇢,��,0+G (v , t) = �� (1� v)G (v , t) , |v | 1

G (v , 0) = 1

G (v , t) =1X

k=0

(��tµ)k (1� v)k E�k⇢,µk+1 (��t

⇢) , |v | 1

G (v , t) =1X

k=0

vk1X

r=k

(�1)r�k✓r

k

◆(�tµ)r E�r

⇢,µr+1 (��t⇢)

pk (t) =1X

r=k

(�1)r�k✓r

k

◆(�tµ)r E�r

⇢,µr+1 (��t⇢) , k � 0, t � 0

� = 0, pk (t) =1X

r=k

(�1)r�k✓r

k

◆(�tµ)r

� (µr + 1)= (�tµ)k E k+1

µ,µk+1 (��tµ)

v = 1,1X

k=0

pk (t) = 1.





I How to evaluate the mean value of N (t)?⇢

cD�,µ⇢,��,0+EN (t) = �, t > 0EN (t) |t=0 = 0, t > 0

EN (t) = �tµE�⇢,1+µ (��t⇢) , t � 0.


university of ostrava · hilfer, application of fractional calculus in physics (singapore: world...

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