Fractional Calculus, Probability and Non-local Operators ... · Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments A workshop on the occasion

Fractional Calculus, Probability and Non-local Operators:

Applications and Recent DevelopmentsNovember 6th - 8th 2013

BCAM - Alameda Mazarredo, 14 48009 Bilbao (Bizkaia), Basque Country, Spain

Scientific Committee

Michele Caputo (La Sapienza Rome University, IT, and Texas A&M University, US)Rudolf Gorenflo (Berlin Free University, DE) József Lörinczi (Loughborough University, UK) Yuri Luchko (Berlin Beuth University, DE) Francesco Mainardi (Bologna University, IT) Mark M. Meerschaert (Michigan State University, US) Gianni Pagnini (BCAM, ES)Enrico Scalas (East Piedmont University, IT BCAM, ES)

Invited Speakers

Francesco Mainardi (Bologna University, IT)Luisa Beghin (La Sapienza Rome University, IT)Michele Caputo (La Sapienza Rome University, IT, and Texas A&M University, US)Diego del Castillo Negrete (Oak Ridge National Laboratory, US)Rudolf Gorenflo (Berlin Free University, DE)George Karniadakis (Brown University, US)József Lörinczi (Loughborough University, UK)Yuri Luchko (Berlin Beuth University, DE)Mark M. Meerschaert (Michigan State University, US)Ralf Metzler (Potsdam University, DE)

Organizing Committee

Gianni Pagnini (BCAM) Enrico Scalas (East Piedmont University, and BCAM)

Mν(z) =∞�

n=0

(−z)n

n!Γ[−νn+ (1− ν)]

tDβ ∗f(t)=

1

Γ(m

−β)

�t

0

f(m

) (τ)

(t−

τ)β

+1−m

dτ

Co-funded by

A workshop on the occasion of the retirement of Professor

Francesco Mainardi

Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments

A workshop on the occasion of the retirement of Francesco Mainardi

Bilbao – Basque Country – Spain

6–8 November 2013

Program!"#$%"&

'(#%)(*&

+,--

+,.- &-/*01234(

+,5-

&-,&- &6/*!(78%9%"

&-,.- .:/*;%4($%

&-,5-

&&,&- !"##$%&'$() !"##$%&'$() !"##$%&'$()

&&,:- !"#$%"&

&6,&-

&6,.- 6-/*'"$%4"*<&= 6>/*?27(34%()

&6,5-

&.,&- &+/*'(8(@%3%

&.,.-

&5,.- '"%()*+#,%%-

&>,--

&>,6- 6&/*'"$%4"*<6= 65/*A1##28%

&>,:-

&B,-- *+, *+, *+,

C2@)23@(D*> EF183@(D*B !8%@(D*G

.-&"/0%- )*12#- 32456-

:/*H$1I2) B/*0(8%$$"

&>/*J2)482$$% &&/*K$L?F(2@ 6G/*?M%#$28

6+/*E2)82%8"*J(NF(@" .-/*ONF(%P%)*<&=

&./*Q28(8@"*Q%"8@( .&/*ONF(%P%)*<6=

&5/*R($%I28%3 6./*'"9" ./*H%NN(8%

7*&/,*8*6,+ !""&+45*"&#

.5/*S1342 6/*H%7% &/*TNF$2%4)28

&:/*Q8(N%(

5/*H"1(#(@( 6B/*?%P"83P%% 66/*'"U342)P"

&G/*V4(8"$( +/*0"8N128(

-.+/0 -.+/0 -.+/0

3-&,/$4, 9":5,/

../*;28242))%P"U( .6/*;%$23 6:/*A"#"3%)

>/*0(M2$(3

&B/*V892$ G/*0F2NFP%) !*,/*&8,

Social Events:

Tuesday 5, 19:00–21:00Get-together light dinner at Punto de Encuentro, Alamenda Mazarredo 20.

Saturday 9, 11:30–13:00Guided visit to Bilbao–Casco Viejo, meeting point at Arriaga Square/Arriaga Theater.

https://sites.google.com/site/fcpnlo/




6–8 November 2013

Invited Speakers

Francesco Mainardi, Round table moderator and conclusion speech

Luisa Beghin, On the fractional extensions of Gamma subordinator and Geometric Stable

processes

Michele Caputo, The homogeneity and evolution of economies

Diego del–Castillo–Negrete, Applications of fractional calculus to non-diffusive, non-

local transport in plasmas and fluids

Rudolf Gorenflo, Stochastic processes related to time-fractional diffusion-wave equation

Mohsen Zayernouri and George Em Karniadakis Fractional spectral and spectral ele-

ment methods

Jozsef Lorinczi, Ground state properties of a class of processes with jump discontinuities

Yuri Luchko, Multi-dimensional fractional wave equation and some properties of its fun-

damental solution

Mark. M. Meerschaert and Alla Sikorskii, Stochastic solutions for fractional wave equa-

tions

Ralf Metzler, Weak ergodicity breaking and ageing in anomalous diffusion




6–8 November 2013

Invited Talk:

On the fractional extensions of Gamma subordinator

and Geometric Stable processes

Luisa BeghinDipartimento di Scienze Statistiche, Sapienza Universita di Roma,

Piazzale Aldo Moro 5, I-00185 Roma, [email protected]

We define and study fractional versions of the well-known Gamma process, which are ob-tained by a time-change based upon an independent stable subordinator or its inverse. Asa preliminary result, we prove that the Gamma process is governed by a differential equa-tion expressed by means of the shift operator. Therefore the densities of the fractionalGamma processes introduced here are shown to satisfy differential equations involving thefractional shift operator (of order greater or less than one, in the two cases). As a con-sequence, the fractional generalization of some Gamma-subordinated processes (i.e. theVariance Gamma, the Geometric Stable and the Negative Binomial) are introduced andthe corresponding fractional differential equations are obtained.

References

[1] Beghin, L. 2013 Geometric stable processes and fractional differential equation relatedto them. Submitted, arXiv :1304.7915v1 [math-ph].

[2] Beghin, L. 2013 Fractional Gamma process and fractional Gamma-subordinated pro-cesses. Submitted, arXiv :1305.1753v1 [math-ph].

[3] Beghin, L., and Macci, C. 2014 Fractional discrete processes: compound and mixedPoisson representations. J. Appl. Probab., in press.

[4] Kozubowski, T.J. 1999 Univariate geometric stable laws. J. Comp. Anal. Appl. 1, 177–217.

[5] Kozubowski T.J., Meerschaert M.M., Podgorski K., Fractional Laplace motion, Adv.Appl. Probab., 38, (2006) 451-464.

[6] Kozubowski, T.J., and Podgorski, K. 2009 Distributional properties of the NegativeBinomial Levy process. Probab. Math. Statist. 29, 43–71.

[7] Kumar, A., Meerschaert, M.M., and Vellaisamy, V. 2011 Fractional normal inverseGaussian diffusion. Statist. Probab. Lett. 81, 146–152.

[8] Kumar A., Vellaisamy, V. 2012 Fractional normal inverse Gaussian process. Methodol.Comput. Appl. Probab. 14, 263–283.




6–8 November 2013

Invited Talk:The homogeneity and evolution of economies

Michele CaputoDipartimento di Fisica, Universita La Sapienza, Piazzale Aldo Moro 5, 00185 - Roma, Italy

Texas A&M University, 77843 College Station, Texas, [email protected]

We consider a model for the evolutions of m > 2 economies yi(t) where we assume thattheir interaction is based on the differences of the values of their evolution status. Since theeconomies have structures which cause delays, we introduce in the equations a mathematicalmemory formalism represented by a derivative of fractional order which leads to a system ofintegro-differential equations. The solution is given from a set of m linear equations in theLaplace Transform (LT) of the yi(t). Differently from a previous note, in the present oneeach economy is affected by a different memory. Is found that the asymptotic state of thestates of evolution of the economies are all equal to the initial value of the economy whichhas the memory represented by a fractional derivative of order smaller than the others andacts an attractor of the other economies. Two different methods: one probabilistic and onegeometric, for the estimate of the homogeneity and the evolution of the economies and theircomparisons are also presented and applied to study the evolution of the homogeneities of5 EU economies finding that they possibly drift towards the economy which has the longestmemory.




6–8 November 2013

Invited Talk:Applications of fractional calculus to

non-diffusive, non-local transport in plasmas and fluids

Diego del–Castillo–NegreteOak Ridge National Laboratory, Oak Ridge TN 37831-6169, USA

[email protected]

A review of the application of fractional calculus to non-diffusive transport modeling in

fluids and plasmas is presented. The main focus is on the study of transport of parti-

cles and heat. In the study of particle transport we consider chaotic transport by waves

in zonal flows in fluids, and test particle transport in plasma turbulence. In both sys-

tems the Lagrangian particle statistics is strongly non-Gaussian and the moments exhibit

super-diffusive anomalous scaling. Fractional diffusion models are proposed and tested in

the quantitative description of the observed non-diffusive phenomenology. Going beyond

particle transport, we review the application of fractional calculus to model non-local heat

transport. We first discuss experimental and numerical evidence of transport processes in

magnetically confined fusion plasmas in which the heat flux exhibits a non-local dependence

on the temperature gradient. Specific examples include fast propagation phenomena in per-

turbative experiments and numerical simulations of heat transport in stochastic magnetic

fields. We use fractional derivative operators to construct non-local transport models in one-

dimensional and two-dimensional bounded domains. Of particular interest the construction

of regularized fractional operators in finite-size domains leading to well-posed boundary

value problems. We conclude with a discussion of several applications to experiments in

controlled fusion plasmas including: up-hill transport, anomalous scaling of confinement

time, fast cold pulse propagation, heat wave propagation, and non-local transport in the

presence of transport barriers.

!"#$%&'()#*$&+#",-$*#!"#$%%"&'()%$*(+,"-."/%$0('!"1%$23',"

./0*,&#$&1,20(",3&4+-563*+(5"7$-5%**%*"$%03+%8"+-"+6%"+(2%9.$35+(-'30"8(..:*(-'9;3)%"%<:3+=-'"

4560"7203&83('3$8(!">:56?-"3'8"@3A'('(" ('" BCD"63)%"*6-;'"E32-'A"-+6%$" .35+*F" +63+"+6%"*-0:+(-'"+-"+6%"G3:56,"7$-H0%2"

"&""" 0,,0)0,(),()0,(),,(),( txxux

xxutxuRtxuD "

(*" 3" 7$-H3H(0(+," 8%'*(+," (." 21 I" J%$%" D (*" +6%" +%27-$30" .$35+(-'30" G37:+-" 8%$()3+()%" -."

-$8%$" ;6%$%3*" R 8%'-+%*" +6%" *73+(30"K(%*L" .$35+(-'30"7*%:8-98(..%$%'+(30"-7%$3+-$";(+6" #-:$(%$"

*,2H-0" || I" M5+:300," +6%," 63)%" 30*-" 5-'*(8%$%8" 3" *?%;%8" A%'%$30(L3+(-'" -." R " 3'8" +6%" 53*%"10 -." .$35+(-'30" 8(..:*(-'!" H:+" +-" 3)-(8" ('.03+(-'" -." '-+3+(-'*" ;%" $%*+$(5+" 3++%'+(-'" +-" +6%"

*73+(300," *,22%+$(5" *(+:3+(-'I"N%" 3*?O" G3'" ),( txu " ('" 3" '3+:$30";3," H%" ('+%$7$%+%8" 3*" +6%" *-P-:$'"7$-H3H(0(+,"8%'*(+,"E('"7-('+"!!"%)-0)('A";(+6"+(2%""F"-."3"$3'8-20,";3'8%$('A"73$+(50%""*+3$+('A"('"+6%"-$(A('""!#Q"3+"('*+3'+""#Q"R"N%"*6-;"+63+"+6(*"('8%%8"53'"H%"8-'%"('"+6%"%S+$%2%"53*%" 2 "E;6%$%"

2

2

xR F" I" T-$%-)%$!" (." 2;%" 53'" $%7035%" D H," 3'" -7%$3+-$" ;(+6" 8(*+$(H:+%8" -$8%$*O"

dDb ...)(]2,1(

;(+6"3"'-'9'%A3+()%"EA%'%$30(L%8F";%(A6+".:'5+(-'" (b FI"#-$"+6(*"*7%5(30"53*%"('"BUD""

),( txu ;3*" *6-;'" +-" H%" 3" 7$-H3H(0(+," 8%'*(+,I"N%"'%%8" +6%" (')%$*%9*+3H0%" *:H-$8('3+-$" " -."-$8%$"

2/ "-$"3'"377$-7$(3+%"A%'%$30(L3+(-'"-."(+"3'8"+6%"+6%-$,"-."/%$'*+%('".:'5+(-'*"E*%%"BVDFI"#-$"2-$%"H35?A$-:'8"-'"$%0%)3'+"*+-563*+(5*"3'8".:'5+(-'30"3'30,*(*";%"5(+%"BWD"3'8"BXDI""

#-$" +$%3+2%'+"-." +6%"2-$%"8(..(5:0+" " *(+:3+(-'" 21 " ";%" $%5:$" +-"3" *:H-$8('3+(-'" .-$2:03"8%$()%8"('"BYD"('";6(56" ),( txu " (*"$%03+%8"+-"+6%"(')%$*%9*+3H0%"*:H-$8('3+-$"-."-$8%$" / "3'8"+6%".:'832%'+30"*-0:+(-'"-."+6%"'%:+$30".$35+(-'30"8(..:*(-'"%<:3+(-'"('";6(56" O"

22

1

||)2/cos(||2)2/sin(||1

xtxttx

I"

Z6(*"'%:+$30".$35+(-'30"8(..:*(-'"%<:3+(-'"63*"H%%'"(')%*+(A3+%8"('"8%+3(0"H,">:56?-"('"B[DI"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

',$,",-2,6&3&

BUD"KI"1-$%'.0-!"\:I">:56?-!"TI"4+-P3'-)(5O"#:'832%'+30"*-0:+(-'"-."3"8(*+$(H:+%8"-$8%$".$35+(-'30"8(..:*(-'9;3)%"%<:3+(-'"3*"7$-H3H(0(+,"8%'*(+,I"$%&&"UX"EWQU[F!"W]V9[UXI""" """"""""""""""""""""""""""""""""""""""""BWD"MI^I"_-56:H%(O"1%'%$30".$35+(-'30"5305:0:*!"%)-0:+(-'"%<:3+(-'*!"3'8"$%'%;30"7$-5%**%*I"'(")*+,-./01,"23(4.,(5.67)+,"3+.89)3+:"9:&EWQUUF!"C`[9XQQI""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""B[D"\:I">:56?-O"#$35+(-'30";3)%"%<:3+(-'"3'8"8327%8";3)%*I"a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`F!"UXQX9UX[[I"" """"""""""""""""""""""""""""""""""""""""""""""BVD"KI>"" %?O">)+(4")2(.$1(<"23(4?.89)3+:.,(5.&77-2<,"23([email protected]%"1$:,+%$!"/%$0('"WQUQI""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" " " &




6–8 November 2013

Invited Talk:Stochastic processes related to

the time-fractional diffusion-wave equation

Rudolf GorenfloDepartment of Mathematics and Informatics, Free University of Berlin,

Arnimallee 3, D-14195 Berlin, [email protected]

!"#$%&'()#*$&+#",-$*#!"#$%%"&'()%$*(+,"-."/%$0('!"1%$23',"

./0*,&#$&1,20(",3&4+-563*+(5"7$-5%**%*"$%03+%8"+-"+6%"+(2%9.$35+(-'30"8(..:*(-'9;3)%"%<:3+=-'"

4560"7203&83('3$8(!">:56?-"3'8"@3A'('(" ('" BCD"63)%"*6-;'"E32-'A"-+6%$" .35+*F" +63+"+6%"*-0:+(-'"+-"+6%"G3:56,"7$-H0%2"

"&""" 0,,0)0,(),()0,(),,(),( txxux

xxutxuRtxuD "

(*" 3" 7$-H3H(0(+," 8%'*(+," (." 21 I" J%$%" D (*" +6%" +%27-$30" .$35+(-'30" G37:+-" 8%$()3+()%" -."

-$8%$" ;6%$%3*" R 8%'-+%*" +6%" *73+(30"K(%*L" .$35+(-'30"7*%:8-98(..%$%'+(30"-7%$3+-$";(+6" #-:$(%$"

*,2H-0" || I" M5+:300," +6%," 63)%" 30*-" 5-'*(8%$%8" 3" *?%;%8" A%'%$30(L3+(-'" -." R " 3'8" +6%" 53*%"10 -." .$35+(-'30" 8(..:*(-'!" H:+" +-" 3)-(8" ('.03+(-'" -." '-+3+(-'*" ;%" $%*+$(5+" 3++%'+(-'" +-" +6%"

*73+(300," *,22%+$(5" *(+:3+(-'I"N%" 3*?O" G3'" ),( txu " ('" 3" '3+:$30";3," H%" ('+%$7$%+%8" 3*" +6%" *-P-:$'"7$-H3H(0(+,"8%'*(+,"E('"7-('+"!!"%)-0)('A";(+6"+(2%""F"-."3"$3'8-20,";3'8%$('A"73$+(50%""*+3$+('A"('"+6%"-$(A('""!#Q"3+"('*+3'+""#Q"R"N%"*6-;"+63+"+6(*"('8%%8"53'"H%"8-'%"('"+6%"%S+$%2%"53*%" 2 "E;6%$%"

2

2

xR F" I" T-$%-)%$!" (." 2;%" 53'" $%7035%" D H," 3'" -7%$3+-$" ;(+6" 8(*+$(H:+%8" -$8%$*O"

dDb ...)(]2,1(

;(+6"3"'-'9'%A3+()%"EA%'%$30(L%8F";%(A6+".:'5+(-'" (b FI"#-$"+6(*"*7%5(30"53*%"('"BUD""

),( txu ;3*" *6-;'" +-" H%" 3" 7$-H3H(0(+," 8%'*(+,I"N%"'%%8" +6%" (')%$*%9*+3H0%" *:H-$8('3+-$" " -."-$8%$"

2/ "-$"3'"377$-7$(3+%"A%'%$30(L3+(-'"-."(+"3'8"+6%"+6%-$,"-."/%$'*+%('".:'5+(-'*"E*%%"BVDFI"#-$"2-$%"H35?A$-:'8"-'"$%0%)3'+"*+-563*+(5*"3'8".:'5+(-'30"3'30,*(*";%"5(+%"BWD"3'8"BXDI""

#-$" +$%3+2%'+"-." +6%"2-$%"8(..(5:0+" " *(+:3+(-'" 21 " ";%" $%5:$" +-"3" *:H-$8('3+(-'" .-$2:03"8%$()%8"('"BYD"('";6(56" ),( txu " (*"$%03+%8"+-"+6%"(')%$*%9*+3H0%"*:H-$8('3+-$"-."-$8%$" / "3'8"+6%".:'832%'+30"*-0:+(-'"-."+6%"'%:+$30".$35+(-'30"8(..:*(-'"%<:3+(-'"('";6(56" O"

22

1

||)2/cos(||2)2/sin(||1

xtxttx

I"

Z6(*"'%:+$30".$35+(-'30"8(..:*(-'"%<:3+(-'"63*"H%%'"(')%*+(A3+%8"('"8%+3(0"H,">:56?-"('"B[DI"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

',$,",-2,6&3&

BUD"KI"1-$%'.0-!"\:I">:56?-!"TI"4+-P3'-)(5O"#:'832%'+30"*-0:+(-'"-."3"8(*+$(H:+%8"-$8%$".$35+(-'30"8(..:*(-'9;3)%"%<:3+(-'"3*"7$-H3H(0(+,"8%'*(+,I"$%&&"UX"EWQU[F!"W]V9[UXI""" """"""""""""""""""""""""""""""""""""""""BWD"MI^I"_-56:H%(O"1%'%$30".$35+(-'30"5305:0:*!"%)-0:+(-'"%<:3+(-'*!"3'8"$%'%;30"7$-5%**%*I"'(")*+,-./01,"23(4.,(5.67)+,"3+.89)3+:"9:&EWQUUF!"C`[9XQQI""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""B[D"\:I">:56?-O"#$35+(-'30";3)%"%<:3+(-'"3'8"8327%8";3)%*I"aI"T3+6I@6,*I";<&EWQU[F!"Q[UQC!"U9UXI"""""""""""BYD"#I"T3('3$8(!"1I"@3A'('("3'8"KI"1-$%'.0-O"T%00('"+$3'*.-$2"3'8"*:H-$8('3+(-'"03;*"('".$35+(-'30"8(..:*(-'"7$-5%**%*I.$%&&"=&EWQQ[F!"YYU9YC]I"""""""""""""""""""""""""""""""""""""""""""""""""" " " """""""BCD"#I"T3('3$8(!"\:I">:56?-"3'8"1I"@3A'('(O"Z6%".:'832%'+30"*-0:+(-'"-."+6%"*735%9+(2%".$35+(-'30"8(..:*(-'"%<:3+(-'I"$%&&.<&EWQQUF!"UC[9U]WI""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""BXD"TI"TI"T%%$*563%$+"3'8"JI9@I"456%..0%$O"Z$(3'A:03$"3$$3,"0(2(+*".-$"5-'+(':-:*"+(2%"$3'8-2";30?*I";"3<9,4"2<.=+3<)44)4.,(5."9)2+.&77-2<,"23(4"::>& EWQQ`F!"UXQX9UX[[I"" """"""""""""""""""""""""""""""""""""""""""""""BVD"KI>"" %?O">)+(4")2(.$1(<"23(4?.89)3+:.,(5.&77-2<,"23([email protected]%"1$:,+%$!"/%$0('"WQUQI""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" " " &




6–8 November 2013

Invited Talk:Fractional spectral and spectral element methods

Mohsen Zayernouri and George Em KarniadakisDivision of Applied Mathematics, Brown University, Providence RI, 02912, USA

george [email protected]

We will present an overview of efficient high-order methods for time- and space-fractional

partial differential equations (FPDEs). To this end, we have developed a new theory for

suitable spectral basis functions, in terms of polyfractonomials, to employ them [1] in ap-

proximating the fractional differential operators. Specifically, we have developed a fractional

Petrov–Galerkin method [2], a fractional Tau method, and a fractional penalty method for

general linear FPDEs. We examine our schemes for fractional advection and fractional

diffusion equations. Numerical tests verify the theoretical exponential convergence. Other

fractional equations we consider include delay differential equations as well as stochastic

FPDEs. We are particularly interested in establishing the relative efficiency of fractional

spectral methods compared to the finite-difference approximations developed over the last

ten years.

References

[1] Zayernouri, M., and Karniadakis, G. E. 2013 Fractional Sturm–Liouville eigen-

problems: Theory and numerical approximation. J. Comput. Physics 252, 495–517.

[2] Zayernouri, M., and Karniadakis, G. E. 2013 Exponentially accurate spectral and apec-

tral element methods for fractional ODEs. J. Comput. Physics, to appear.




6–8 November 2013

Invited Talk:Ground state properties of a class of processes

with jump discontinuities

Jozsef LorincziSchool of Mathematics, Loughborough University Loughborough LE11 3TU, United Kingdom

[email protected]

I will consider a class of Levy processes subject to a regularity condition over the largejumps, recently introduced in joint work with K. Kaleta. This class includes a large numberof much studied processes as specific cases, in particular, it has a non-trivial overlap withsubordinate Brownian motions (though neither contains the other). Under a suitably chosenpotential the perturbed process has a stationary distribution (i.e., ground state). In thistalk I will discuss how should the potential be chosen so that the Feynman–Kac semigroupof the perturbed process is intrinsically ultracontractive (IUC). I will also explain what aresome interesting consequences of IUC. Another aspect I plan to discuss is the spatial decayproperties of the ground state and further eigenfunctions for given potentials.




6–8 November 2013

Invited Talk:Multi-dimensional fractional wave equation

and some properties of its fundamental solution

Yuri LuchkoBeuth Technical University of Applied Sciences, Luxemburger Str. 10, 13353 Berlin, Germany

[email protected]

Recently, a fractional generalization of the wave equation was introduced and analysed inthe case of one spatial variable (see [1, 2, 3, 4]). In contrast to the fractional diffusion-waveequation, the fractional wave equation contains fractional derivatives of the same orderα, 1 < α < 2, both in space (in the Riesz sense) and in time (in the Caputo sense).The solutions of the fractional wave equation were shown to exhibit properties of both thesolutions of the diffusion equation and those of the wave equation. In the one-dimensionalcase, the fundamental solution of the fractional wave equation can be interpreted as a spatialprobability density function evolving in time that possesses finite moments up to the order α[2, 3]. This property corresponds to a characteristic behaviour of the fundamental solutionof the diffusion equation. At the same time, the fundamental solution of the fractional waveequation can be treated as a damped wave whose amplitude maximum and the gravityand mass centres propagate with the constant velocities that depend just on the equationorder α. Moreover, the first, the second, and the Smith centro velocities of the dampedwaves described by the fractional wave equation are all constant in time [2]. In this talk,the problems mentioned above are considered for the multi-dimensional fractional waveequation. It turns out that most results proved in the one-dimensional case continue tohold true in the multi-dimensional case, too. To illustrate analytical findings, results ofnumerical calculations and plots are presented.

References

[1] Gorenflo, R., Iskenderov, A., and Luchko, Yu 2000 Mapping between solutions of frac-tional diffusion-wave equations. Fract. Calc. Appl. Anal. 3, 75–86.

[2] Luchko, Yu 2013 Fractional wave equation and damped waves. J. Math. Phys. 54,031505.

[3] Mainardi, F., Luchko, Yu., and Pagnini G. 2001 The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192.

[4] Metzler, R., and Nonnenmacher, T.F. 2002 Space- and time-fractional diffusion andwave equations, fractional Fokker–Planck equations, and physical motivation. ChemicalPhysics 284, 67–90.




6–8 November 2013

Invited Talk:Stochastic solutions for fractional wave equations

Mark. M. Meerschaert and Alla SikorskiiDepartment of Statistics and Probability, Michigan State University

619 Red Cedar Road, East Lansing MI 48823, USA

[email protected]

A fractional wave equation replaces the second time derivative by a fractional derivativeof order between one and two. In this paper, we show that the fractional wave equationgoverns a stochastic model for wave propagation, with deterministic time replaced by theinverse of a stable subordinator whose index is one half the order of the fractional timederivative.




6–8 November 2013

Invited Talk:Weak ergodicity breaking and ageing in anomalous diffusion

Ralf MetzlerInstitute for Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany

[email protected]

In 1905 Einstein formulated the laws of diffusion, and in 1908 Perrin published his Nobel-

prize winning studies determining Avogadro’s number from diffusion measurements. With

similar, more refined techniques the diffusion behaviour in complex systems such as the

motion of tracer particles in living biological cells or the tracking of animals and humans

is nowadays measured with high precision. Often the diffusion turns out to deviate from

Einstein’s laws. This talk will discuss the basic mechanisms leading to such anomalous

diffusion as well as point out its consequences. In particular the unconventional behaviour

of non-ergodic, ageing systems will be discussed within the framework of continuous time

random walks. Indeed, non-ergodic diffusion in the cytoplasm of living cells as well as in

membranes has recently been demonstrated experimentally.

References

[1] Barkai, E., Garini, Y., and Metzler, R. 2012 Strange kinetics of single molecules inliving cells. Phys Today 65, 29–35.




6–8 November 2013

Book of Abstracts

1. Franz Achleitner and Christian Kuehn, Traveling waves for reaction-diffusion equa-tions with bistable nonlinearity and nonlocal diffusion

2. Abdelouahab Bibi, On the covariance structure of a bilinear stochastic differentialequation

3. Umberto Biccari and Enrique Zuazua, Controllability of fractional evolution problemsvia the Hilbert Uniqueness Method

4. Alexander Blumen and Maxim Dolgushev, Exploring the applications of fractionalcalculus: hierarchically-built-polymers

5. Djillali Bouagada, Samuel Melchior and Paul Van Dooren, On computing H∞ normof an implicit fractional system

6. Felix S. Costa, Eliana C. Grigoletto, Jayme Vaz and Edmundo Capelas de Oliveira,Slowing-down of neutrons: fractional version

7. Sandra Carillo Singular kernel problems in materials with memory

8. Aleksei Chechkin, Natural and modified forms of distributed order fractional diffusionequations

9. Jose Manuel Corcuera, Asymptotics of weighted random sums

10. Carlota M. Cuesta, Franz Achleitner and Sabine Hittmeir, Travelling waves for anon-local Korteweg–de Vries–Burgers equation

11. Moustafa El-Shahed, Fractional calculus model of SVIR epidemic models with vacci-nation strategies

12. Mauro Fabrizio, Fractional thermo-mechanical systems. Dissipation and free energies

13. Luca Gerardo Giorda, Guido Germano and Enrico Scalas, Large-scale simulations ofsynthetic markets

14. Jose Luis Gracia and Martin Stynes, Analysis of a finite difference method for afractional-derivative two-point boundary value problem

15. Konstantinos Kalimeris, Attenuating models and reconstruction methods in photoa-coustic imaging

16. Andrea Mentrelli and Gianni Pagnini, Random front propagation in fractional diffu-sive systems

17. Sebastian Orze�l, Asymptotic behaviour of variance for subdiffusive processes withinCTRW scenario

18. Enrique Otarola, Abner Salgado and Ricardo Nochetto, A PDE approach to fractionaldiffusion: a priori and a posteriori error analyses

19. Paolo Paradisi, Linking fractional calculus and real data

20. Mirko D’Ovidio and Federico Polito, Diffusion-telegraph equations and related stochas-tic processes

21. Roberto Garra, Enzo Orsingher and Federico Polito, Fractional Klein-Gordon equa-tion and related processes

22. Yuriy Povstenko, Fractional heat conduction in a semi-infinite composed body

23. Liviu I. Ignat, Alejandro Pozo and Enrique Zuazua, Well-posedness and large timebehavior for a simplified model of the sonic boom propagation

24. Francesco Mainardi and Sergei Rogosin, George William Scott–Blair, the pioneer offractional calculus in rheology

25. Tommaso Ruggeri, Recent results in extended thermodynamics of dense and rarefiedpolyatomic gas: macroscopic approach and Maximum Entropy Principle

26. Nicy Sebastian and Rudolf Gorenflo, A generalized Laplacian model associated withthe fractional generalization of the Poisson processes

27. Nikolai Leonenko, Mark Meerschaert and Alla Sikorskii, Covariance structure of con-tinuous time random walk limit processes

28. Renato Spigler, Some properties and computation of the Mittag–Leffler functions

29. Antonio Lopes and Jose Tenreiro Machado, Fractional dynamics in forest fires

30. Vladimir Uchaikin and Renat Sibatov, Cosmic rays propagation in galaxy: a fractionalapproach

31. Vladimir Uchaikin and Renat Sibatov, Fractional differential model of anisotropicdiffusion of cosmic rays in random magnetic fields

32. Maria Veretennikova and Vassili Kolokoltsov, Controlled fractional dynamics with ap-plications

33. Enrico Scalas and Noelia Viles, A Functional Limit Theorem for stochastic integralsdriven by a time-changed symmetric α-stable Levy Process

34. Giuliano Vitali, Soil hydrology: a fractional physical model

35. Santos B Yuste and Joaquın Quintana-Murillo, Adaptive finite difference methods forfractional diffusion equations




6–8 November 2013

Traveling waves for reaction-diffusion equations with bistablenonlinearity and nonlocal diffusion

Franz AchleitnerVienna University of Technology

Wiedner Hauptstraße 8-10, 1040 Vienna, Austria

[email protected]

Christian KuehnVienna University of Technology

Wiedner Hauptstraße 8-10, 1040 Vienna, Austria

[email protected]

Oral presentation

We consider a single component reaction-diffusion equation in one spatial dimension withbistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller (RF)type. Our main result shows the existence, uniqueness and stability of a traveling wavesolution connecting the two stable homogeneous steady states. This is a first step to studythe nonlocal complex Ginzburg-Landau equation (NCGLE) which has been derived recentlyas a model for electrochemical systems with migration coupling.

In particular, we consider a scalar quantity u : R+ × R → U ⊂ R, (t, x) �→ u(t, x), which isgoverned by the equation

∂u

∂t= Da

θu+ f(u) , x ∈ R , t ∈ R+ , (1)

for some function f : R → R and fixed parameters 1 < a ≤ 2 and |θ| ≤ min{a, 2− a}.Equation (1) models nonlocal diffusion and reaction of a quantity u(t, x) in a one-dimensionalspatial domain over time. The Riesz-Feller operator Da

θ models nonlocal diffusion. It is de-fined as a Fourier multiplier operator

F [Daθf ](k) = −ψθ

a(k)F [f ](k) , k ∈ R , (2)

for sufficiently well-behaved functions f(x) and a two-parameter symbol

ψθa(k) = |k|ae(i sgn(k)θπ/2) , 0 < a ≤ 2 , |θ| ≤ min{a, 2− a} . (3)

The function f , which models the reaction, is assumed to satisfy

f ∈ C1(R) , f(−1) = f(1) = 0 , f �(−1) < 0 , f �(1) < 0 . (4)

A special case is the real Ginzburg-Landau equation

∂u

∂t=

∂2u

∂x2+ u(1− u2) , x ∈ R , t ∈ R+ . (5)




6–8 November 2013

On the covariance structure of a bilinear stochastic

differential equation

Abdelouahab BibiUniversity of Constantine

[email protected]

Poster presentation

This talk is concerned with a study of general bilinear stochastic differential equations.We provide conditions for second-order and strict-sense stationarities of the state process.Explicit formulas for the mean and covariance functions for the state process are given. Alinear representation is obtained and the optimal linear filter and its asymptotic behaviorare investigated. The problem of parameters estimation for some particular case is alsoconsidered.

References

[1] Arnold, L. 1974 Stochastic differential equations, theory and applications. New–York,J. Wiley.

[2] Lebreton, A., M. Musiela 1983. A look at bilinear model for multidimensional stochasticsystems in continuous time. Statistics & Decisions 1, 285–303.

[3] Liptser, R.S., and A.N., Shirayev 1978 Statistics for random processes. I, II. Springer.




6–8 November 2013

Controllability of fractional evolution problems

via the Hilbert Uniqueness Method

Umberto BiccariBCAM - Basque Center for Applied Mathematics

Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country – Spain

[email protected]

Enrique ZuazuaBCAM - Basque Center for Applied Mathematics & Ikerbasque Basque Foundation for Science

Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country - Spain

[email protected]

Oral presentation

In some recent works, Ros–Oton and Serra [1, 2, 3] have analysed the Dirichlet problemfor the fractional Laplacian, proving some regularity properties and deriving a Pohozaevidentity for this operator. Starting from their results, we study a fractional wave andSchrodinger equation, both with Dirichlet boundary conditions on a bounded domain ofRn. Our analysis, in particular, will focus on the problem of the controllability of theseequations via the application of the Hilbert Uniqueness Method.

References

[1] Ros–Oton, X. and Serra, J. 2013 The Dirichlet problem for the fractional Laplacian:

regularity up to the boundary. To appear in J. Math. Pure Appl., arXiv: 1207.5985.

[2] Ros–Oton, X. and Serra, J. 2012 The Pohozaev identity for the fractional Laplacian.Preprint arXiv: 1207.5986.

[3] Ros–Oton, X. and Serra, J. 2013 The extremal solution for the fractional Laplacian.Preprint arXiv: 1305.2489.




6–8 November 2013

Exploring the applications of fractional calculus:hierarchically-built-polymers

Alexander Blumen and Maxim DolgushevTheoretical Polymer Physics, University of Freiburg

Hermann-Herder-Straße 3, D-79104 Freiburg, Germany

[email protected] and [email protected]

Oral presentation

As discussed by Mainardi [5], expressions involving fractional derivatives have a long history

in the theory and modeling of viscoelastic materials, going back to the nineteen thirties [3].

Already then it was realized that dynamical aspects of polymeric materials may often be

modeled using fractional operators. In this respect polymers are very challenging [8], since,

through changes in their chemical structure, in the medium surrounding them, and in their

density in solution they can display vast dynamical domains in which “anomalous” behaviors

occur [4, 7]. The classical approaches to model such behaviors are connected to the theory

of generalized Gaussian structures (GGS) [4].

Exemplarily, we take now the stiffness of polymers into account and extend the GGS for-

malism to semiflexible tree-like structures; among them are dendrimers and regular hyper-

branched structures [2]. Semiflexibility leads to restrictions on the bonds’ orientations, thus

fixing the form of the generalized harmonic potential [1, 2], from which different mechan-

ical relaxation forms for distinct semiflexible architectures follow. Furthermore, the GGS

theory is naturally related to fractional generalized Langevin equations of non-Markovian

nature [6, 7].

References

[1] Dolgushev, M. and Blumen, A. 2009 J. Chem. Phys. 131, 044905; 2013 J. Chem. Phys.

138, 204902.

[2] Furstenberg, F., Dolgushev, M., and Blumen, A. 2012 J. Chem. Phys. 136, 154904;

2013 J. Chem. Phys. 138, 034904.

[3] Gemant, A. 1936 Physics - J. Gen. Appl. Physics 7, 311; 1938 Phil. Mag. 25, 540.

[4] Gurtovenko, A.A. and Blumen, A. 2005 Adv. Polym. Sci. 182, 171.

[5] Mainardi, F. 2012 Fract. Calc. Appl. Anal. 15, 712.

[6] Mura, A., Taqqu, M.S., and Mainardi, F. 2008 Physica A 387, 5033.

[7] Panja, D. 2010, J. Stat. Mech. P06011.

[8] Sokolov, I.M., Klafter, J., and Blumen, A. 2002 Physics Today, 55 (11), 48.

Fractional Calculus, Probability and Non-local Operators: Applications and Recent DevelopmentsA workshop on the occasion of the retirement of Francesco Mainardi

Bilbao – Basque Country – Spain6–8 November 2013

On Computing H! norm of an Implicit Fractional System

Djillali BOUAGADA 1;SamuelMELCHIOR2andPaulV anDOOREN2

1LMPA,DepartmentofMathematicsandComputerScience" UniversityofMostaganem

BP. 227, Mostaganem-27000 - Algeria2ICTEAM,UniversitecatholiquedeLouvain "Belgium

Djillali BOUAGADALMPA, Department of Mathematics and Computer Science-University of Mostaganem

BP. 227, Mostaganem-27000 - Algeria

[email protected]

Oral presentation

The last few years many e!orts have been made to develop fractional systems in di!erentfields of research. Some applications of fractional order systems can be found in [5] and [4].The H! norm of a stable transfer function arises often in control theory. Recently a methodfor the computation of the L2-gain and H!-norm for fractional systems has been proposedin [1]. In this paper we give another formulation for computing the H!-norm, which usesthe concept of parahermitian transfer functions [2]. An algorithm is then derived for thee"cient calculation of the H!-norm of a fractional systems. It is based on the computationof level sets [3] and yields several di!erent approaches for the calculation of the maximumsingular value of the transfer function, as a function of the frequency !!, where 0 < " # 1.

References

[1] L. Fadiga, C. Farges, J. Sabatier, M. Moze, On computation of H!-norm forcommensurate fractional order systems, 50th IEEE Conference on decision and control,

and European control conference 2011, Orlando, FL, USA, December 12-15, 2011.

[2] Y. Genin, Y. Hachez, Y. Nesterov, R. Stefan, P. Van Dooren, S. Xu, Positivityand linear matrix inequalities, European Journal of Control, vol.8(3), pp. 275-298, 2002.

[3] Y. Genin, P. Van Dooren, V. Vermaut, On stability radii of generalized eigenvalueproblems, Proceedings European Control Conf, 1997.

[4] T. Kaczorek, Selected problems of fractional systems theory, Springer-Verlag, Berlin(2011).

[5] J. Sabatier, O. Agrawal, J.T. Machado, Advances in Functional Calculus: The-

oritical developments and Applications in physics and Engineering, Springer, London(2007).




6–8 November 2013

Slowing-down of neutrons: fractional version

F. Silva Costa

Departament of Mathematics, DEMATI - UEMA65054-970 Sao Luis, MA, Brazil

[email protected]

E. Contharteze Grigoletto, J. Vaz Jr. and E. Capelas de Oliveira

Department of Applied Mathematics, Imecc Unicamp13083-859 Campinas, SP, Brazil

[email protected]; [email protected]; [email protected]

Oral presentation

The fractional version for the diffusion of neutrons in a material is studied. The concept offractional derivative is presented, in the Caputo and Riesz senses, only. Using this concept,we discuss a fractional partial differential equation associated with the slowing-down ofneutrons, whose analytical solution is presented in terms of the Fox’s H-function. As aconvenient limit case, the classical solution is recovered.



Singular Kernel Problems in Materials with Memory

Sandra CarilloDipartimento di Scienze di Base e Applicate per l’Ingegneria - Sez. MATEMATICA,

SAPIENZA Universita‘ di Roma, Via A. Scarpa 16, ROME, [email protected], [email protected]

Oral presentation

Recent results concerning evolution problems in materials with memory obtained in [1, 2,3, 4] are reconsidered. In particular, the case of singular kernel problems studied in [3, 4]is analyzed in the cases of viscoelasticity, on one side, and thermodynamics with memory,on the other one. Notably, an important role, in establishing the a priori estimates neededto prove the solution existence, is played by the free energy associated to the material withmemory itself. Here, in turn, the papers by Fabrizio, Gentili, Reynold [6] and by Fabrizio,Gentili, Golden [5], are referred to as far as the description of the models is concerned.Perspectives and opens problems are also mentioned.

References

[1] Carillo S., Valente V. and Vergara Ca!arelli G. 2011 A result of existence and unique-ness for an integro-di!erential system in magneto-viscoelasticity, Applicable Anal-isys: An International Journal, (90) n. 12, (2011) 1791–1802, ISSN: 0003-6811, doi:10.1080/00036811003735832.

[2] Carillo S., Valente V. and Vergara Ca!arelli G. 2011 An existence theorem for themagneto-viscoelastic problem, Discrete and Continuous Dynamical Systems Series S. ,(5) n. 3, (2012), 435 – 447 doi: 10.3934/dcdss.2012.5.435;

[3] Carillo S., Valente V. and Vergara Ca!arelli G. A Singular Viscoelasticity Problem: anExistence and Uniqueness Result, Di!erential and Integral Equations, (2013), 1–10, inpress.

[4] Carillo S., Valente V. and Vergara Ca!arelli G. Heat Conduction With Memory: ASingular Kernel Problem. (2013), Evolution Equations and Control Theory, submitted.

[5] M. Fabrizio, G. Gentili, and J.M. Golden, Non-isothermal Free Energies for LinearTheories with Memory, Mathematical and Computer Modelling, 39 (2004), 219-253.

[6] M. Fabrizio, G. Gentili, D.W.Reynolds On rigid heat conductors with memory Int. J.Eng. Sci. 36 (1998), pp. 765–782.




6–8 November 2013

Natural and modified forms ofdistributed order fractional diffusion equations

Aleksei ChechkinInstitute for Physics and Astronomy,

University of Potsdam, 14476 Potsdam-Golm, Germany

Akhiezer Institute for Theoretical Physics, NSC Kharkov Institute of Physics and Technology,

Akademicheskaya st 1, 61108 Kharkov, Ukraine

[email protected]

Oral presentation

We consider diffusion-like equations with time and space fractional derivatives of distributed-

order for the kinetic description of anomalous diffusion and relaxation phenomena, whose

mean squared displacement does not changes as a power law in time. Correspondingly, the

underlying processes cannot be viewed as self-affine random processes possessing a unique

Hurst exponent. We show that different forms of distributed-order equations, which we call

”natural” and ”modified” ones, serve as a useful tool to describe the processes which become

more anomalous with time (retarding subdiffusion and accelerated superdiffusion) or less

anomalous demonstrating the transition from anomalous to normal diffusion (accelerated

subdiffusion and truncated Levy flights). Fractional diffusion equation with the distributed-

order time derivative also accounts for the logarithmic diffusion (strong anomaly).




6–8 November 2013

Asymptotics of weighted random sums

Jose Manuel CorcueraFacultat de Matematiques, Universitat de Barcelona,

Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain

[email protected]

Oral presentation

In this talk we show how fractional calculus appears as a natural tool to get limit theoremsof weighted random sums. More specifically we consider the limit of Riemannian sumswhen the integrator converges stably to a Brownian motion, then with the help of fractionalcalculus we show that under mild conditions on the integrand the stably limit is the integralwith respect to the above mentioned Brownian motion.




6–8 November 2013

Travelling waves for anon-local Korteweg–de Vries–Burgers equation

Carlota M. Cuesta

Department of Mathematics, University of the Basque Country,

Aptdo. 644, E-48080 Bilbao, Basque Country – Spain

[email protected]

Franz Achleitner and Sabine Hittmeir

Vienna University of Technology, Institute for Analysis and Scientific Computing,

Wiedner Hauptstr. 8-10, AUT-1040 Wien, Austria

Oral presentation

A Korteweg–de Vries–Burgers (KdV–Burgers) equation with non-local diffusion has been

derived in the analysis of a shallow water flow by performing formal asymptotic expansions

associated to the triple-deck regularisation (i.e. using an extension of classical boundary

layer theory) used in fluid mechanics. The non-local operator is of fractional type of order

between 1 and 2. Travelling wave solutions has are analysed in order to study the possibility

of shock formation and its nature in the full problem. We show rigorously the existence of

these waves. In absence of the dispersive term, the existence of travelling waves and their

monotonicity has been established previously by two of the authors in a previous work.

In contrast, travelling waves of the non-local KdV–Burgers equation are not in general

monotone, as is the case for the corresponding classical (or local) KdV–Burgers equation.

This requires a more complicated existence proof compared to our previous work. Moreover,

the travelling wave problem for the classical KdV–Burgers equation is usually analysed via

a phase-plane analysis, which is not possible due to the non-local diffusion operator and

we apply instead approximation that make use of fractional calculus results available in

the literature. In addition we discuss the monotonicity of the waves in terms of a control

parameter and prove their stability in case they are monotone.




6–8 November 2013

Frcational calculus model of SVIR epidemic models withvaccination strategies

Moustafa El-ShahedDepartment of Mathematics, College of Education,

P.O. Box 3771, Unizah-Qasssim, Qasssim University, Saudi Arabia

[email protected]

Oral presentation

This paper deals with the fractional order SVIR model with vaccination strategies. We

give a detailed analysis for the asymptotic stability of disease free and positive fixed points.

Adams–Bashforth–Moulton algorithm have been used to solve and simulate the system of

differential equations.




6–8 November 2013

Fractional thermo-mechanical systems.Dissipation and free energies

Mauro FabrizioDipartimento di Matematica, Universita di Bologna,Piazza di Porta San Donato 5, I-40126 Bologna, Italy

[email protected]

Oral presentation

Within the fractional derivative framework, we study thermo-mechanical models with mem-ory and compare them with the classical Volterra theory. This comparison shows significantdifferences in the type of kernels and predicts important changes in the behavior of fluidsand solids. Moreover, an analysis of the thermodynamic restrictions provides compatibilityconditions on the kernels and allows us to determine certain free energies, which in turnenables the definition of a topology on the history space. Finally, an analogous analysis iscarried out for the phenomenon of heat propagation with memory.




6–8 November 2013

Large-scale simulations of synthetic markets

Luca Gerardo GiordaBCAM - Basque Center for Applied Mathematics


[email protected]

Guido GermanoScuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I-56126 Pisa, Italy

[email protected]

Enrico ScalasDip. Scienze e Tecnologie Avanzate, Univ. Piemonte Orientale, & BCAM - Basque C. Appl. Math.

Viale T. Michel 11, I-15121 Alessandria, Italy

[email protected]

Oral presentation

High-frequency trading has been experiencing an increase of interest both for practical pur-poses within financial institutions and within academic research; recently Furse et al. [1]reviewed the state of the art and gave an outlook analysis. Therefore, models for tick-by-tick financial time series [2] are becoming more and more important. Together withhigh-frequency trading comes the need for fast simulations of full synthetic markets forseveral purposes including (but not limited to) scenario analyses for risk evaluation. Thesesimulations are very suitable to be run on massively parallel architectures. Aside more tra-ditional large-scale parallel computers, high-end personal computers equipped with severalmulti-core CPUs and general-purpose GPU programming are gaining importance as cheapand easily available alternatives. A further option are FPGAs. In all cases, developmentcan be done in a unified framework with standard C or C++ code and calls to appropriatelibraries like MPI (for CPUs) or CUDA for (GPGPUs). Here we present such a prototypesimulation of a synthetic regulated equity market. The basic ingredients to build a syn-thetic share are two sequences of random variables, one for the inter-trade durations andone for the tick-by-tick logarithmic returns. Our results of extensive simulations are basedon several distributional choices for the above random variables, including Mittag–Lefflerdistributed inter-trade durations and alpha-stable tick-by-tick logarithmic returns.

References

[1] Furse, C., Haldane, A., Goodhart, C., Cliff, D., Zigrand, J.-P., Houstoun, K., Linton,O., and Bond, P. 2011 The Future of Computer Trading in Financial Markets. Workingpaper, Foresight, Government Office for Science, London, UK.http://www.bis.gov.uk/foresight/our-work/projects/current-projects/computer-trading

[2] Engle, R. F., and Russell, J. R. 1998 Autoregressive conditional duration: a new model

for irregularly spaced transaction data. Econometrica 66, 1127–1162.



Analysis of a finite di!erence method for afractional-derivative two-point boundary value problem

Jose Luis GraciaIUMA and Department of Applied Mathematics, University of Zaragoza

Pedro Cerbuna, 12 50009 - Zaragoza, Spain

[email protected]

Martin StynesDepartment of Mathematics, National University of Ireland

Western Road, Cork, Ireland

[email protected]

Oral presentation

A linear two-point boundary value problem is considered, whose leading term is a Caputo-type fractional derivative of order !, where 1 < ! < 2. A standard finite di!erence methodis used to discretize this problem on a uniform mesh. The convergence of this numericalmethod in the discrete maximum norm is proved under realistic hypotheses on the derivativesof the solution—in particular, the second-order derivative of the solution may be unboundednear one endpoint of the interval. Some numerical results will be presented.




6–8 November 2013

Attenuating models and reconstruction methods inphotoacoustic imaging

Konstantinos KalimerisRadon Institute of Computational and Applied Mathematics,

Altenberger Str. 69, AUT-4040 Linz, Austria

[email protected]

Oral presentation

The common underlying mathematical equation of Photoacoustic Imaging is the wave equa-tion for the pressure. In this talk some of the existing models are presented, taking intoaccount acoustic attenuation under the different physical properties of the biological tissueand a strong causality property. That is, the solutions of these equations are zero beforethe initialization and satisfy a finite front wave propagation speed. In some of the differ-ent models presented here, the relevant ”attenuated” wave equation incorporate fractionalderivative terms. A family of time reversal imaging functionals for strongly causal acous-tic attenuation models is presented. The time reversal techniques are based on recentlyproposed ideas of Ammari et al for the thermo-viscous wave equation. In particular, anasymptotic analysis provides reconstruction functionals from first order corrections for theattenuating effect. Finally, a novel approach for higher order corrections is described.




6–8 November 2013

Random front propagation in fractional diffusive systems

Andrea MentrelliDepartment of Mathematics & CIRAM, University of Bologna

via Saragozza 8, I-40123 Bologna, Italy

[email protected]

Gianni PagniniBCAM - Basque Center for Applied Mathematics & Ikerbasque Basque Foundation for Science


[email protected]

Oral presentation

Modelling the propagation of interfaces is of interest for several applied problems, espe-

cially for those related to chemical reactions where the reacting interface separates two

different compounds. The level-set method [1] is a successfully front tracking technique

widely studied and used. When the front propagation occurs in systems characterized by

an underlying random motion, the front gets a random character and a tracking method

for fronts with a random motion is desired. The level-set method is randomized assuming

the motion of the interface particles a random diffusive motion [2]. In particular, it is here

considered the case when such motion is governed by the time-fractional diffusion equation.

Hence, the probability density function for interface particle displacement emerges to be

the M-Wright function, originally obtained by Mainardi [3] and generally referred to as the

Mainardi function. Some analytical and numerical results are shown and discussed.

References

[1] Sethian, J. A., and Smereka, P. 2003 Level set methods for fluid interfaces. Ann. Rev.

Fluid Mech. 35, 341–372.

[2] Pagnini, G., and Bonomi, E. 2011 Lagrangian formulation of turbulent premixed com-

bustion. Phys. Rev. Lett. 107, 044503.

[3] Mainardi, F. 1996 Fractional relaxation-oscillation and fractional diffusion-wave phe-

nomena. Chaos, Solitons & Fractals 7, 1461–1477.




6–8 November 2013

Asymptotic behaviour of variance for subdiffusive processeswithin CTRW scenario

Sebastian Orze�lHugo Steinhaus Center

Institute of Mathematics and Computer ScienceWroc�law University of Technology,

Wyb. Wyspianskiego 27, 50-370 Wroc�law, Polande-mail address: [email protected]

Oral presentation

We study asymtotic properties of variance for an Ito diffusion X(τ) subordinated by an in-verse subordinator S(t), i.e. the inverse process to some Levy subordinator (process startingfrom zero and having almost all non-negative and increasing trajectories). Considered pro-cess Y (t) = X(S(t)) is directly related to Continous-Time Random Walk (CTRW) processand it can be used as a model of subdiffusion. For that reason, the model has many ap-plications in finance [1, 2, 5], physics [3, 4] and other fields. We focus on subordinatorsdefined by Levy exponents that behaves asymptotically (i.e. in 0+ or\and +∞) like powerfunction tα, where 0 < α < 1 and we consider Ito diffusions X(τ) with bounded and lineardrift coefficients. We study also asymptotic behaviour of time-domain relaxation functionφp(t) = 1− EeipB(S(t)), p ≥ 0, where B(τ) is a standard Brownian motion.

References

[1] Janczura J., Orze�l S., Wy�lomanska A., Subordinated alpha-stable Ornstein-Uhlenbeckprocess as a tool for financial data description, Physica A 390, 4379-4387 (2011).

[2] Magdziarz M., Orze�l S., Weron A., Option pricing in subdiffusive Bachelier model,Journal of Statistical Physics 145, 187-203 (2011).

[3] Orze�l S., Mydlarczyk W., Jurlewicz A., Accelerating subdiffusions governed by multipleorder time-fractional diffusion equations: stochastic representation by a subordinatedBrownian motion and computer simulations, Physical Peview E 87, 032110 (2013).

[4] Orze�l S., Weron A., Fractional Klein-Kramers dynamics for subdiffusion and Ito for-mula, Journal of Statistical Mechanics, P01006 (2011).

[5] Orze�l S., Wy�lomanska A., Calibration of the subdiffusive arithmetic Brownian motionwith tempered stable waiting-time, Journal of Statistical Physics 143, 447-454 (2011).




6–8 November 2013

A PDE approach to fractional diffusion:a priori and a posteriori error analyses

Enrique Otarola and Abner Salgado

Department of Mathematics, University of Maryland,College Park, MD 20742-4015, USA

[email protected]; [email protected]

Ricardo Nochetto

Dep. of Mathematics and Inst. for Physical Science and Technology, University of Maryland,College Park, MD 20742, USA

[email protected]

Oral presentation

The purpose of this work is the study of solution techniques for problems involving frac-tional powers of [the Dirichlet Laplace] symmetric coercive elliptic operators in a boundeddomain with Dirichlet boundary conditions. This operator can be realized as the Dirichletto Neumann map of a degenerate/singular elliptic problem posed on a semi-infinite cylinder,which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapiddecay of the solution of this problem, we propose a truncation that is suitable for numericalapproximation. We discretize this truncation using first degree tensor product finite ele-ments. We derive suboptimal a priori error estimates for shape regular discretizations andoptimal error estimates for anisotropic discretizations, both estimates in weighted Sobolevspaces. Next, as a first step to design an adaptive algorithm, we present a computable aposteriori error estimator, which relies on the solution of small discrete problems on stars.It exhibits built-in flux equilibration and is equivalent to the energy error up to data os-cillation. A simple adaptive strategy is designed, which reduces error and data oscillation.We present numerical experiments to illustrate the a priori and a posteriori error estimatesas well as the adaptive method’s performance.




6–8 November 2013

Linking fractional calculus to real data

Paolo ParadisiIstituto di Scienza e Tecnologie dell’Informazione (ISTI-CNR),

Via Moruzzi 1, I-56124 Pisa, Italy

[email protected]

Oral presentation

I will review some well-known theoretical findings about fractional calculus and, in partic-ular, the links between fractal intermittency, the Continuous Time Random Walk (CTRW)model and the emergence of Fractional Diffusion Equations (FDE) for anomalous diffusion.In this framework, I will show how fractional operators are associated with the existence ofrenewal events, a typical feature of complex systems. I will also discuss the possibile connec-tions with critical phenomena. Then, I will introduce some statistical methods allowing tounderstand when a real system could be described by means of fractional models. Finally,I will show some applications to real data from nano-crystal fluorescence intermittency,human brain dynamics and atmospheric turbulence.




6–8 November 2013

Diffusion-telegraph equations andrelated stochastic processes

Mirko D’Ovidio

Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Universita ”La Sapienza” di RomaVia A. Scarpa 16, I-00161 Roma, Italy

[email protected]

Federico Polito

Department of Mathematics, University of TorinoVia Carlo Alberto 10, I-10123, Torino, Italy

[email protected]

Oral presentation

In this talk we present the explicit stochastic solution to a diffusion-telegraph equation inwhich the time-operator is related to the so-called Prabhakar operator, that is an integraloperator with a generalized Mittag–Leffler function in the kernel. The stochastic solutionis given as a Levy process process time-changed with the inverse process to a linear combi-nation of stable subordinators. Note that the considered framework interpolates and thusgeneralizes directly fractional diffusion equations and fractional telegraph equations.




6–8 November 2013

Fractional Klein–Gordon equation and related processes

Roberto Garra

Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Universita La Sapienza di Roma

Via A. Scarpa 16, I-00161 Roma, Italy

[email protected];

Enzo Orsingher

Dipartimento di Statistica, Universita ”La Sapienza” di Roma

P.le Aldo Moro 5, I-00185 Roma, Italy

[email protected]

Federico Polito

Department of Mathematics, University of Torino

Via Carlo Alberto 10, I-10123, Torino, Italy

[email protected]

Oral presentation

A fractional formulation of the one-dimensional Klein–Gordon equation is studied by means

of generalized fractional integrals in the sense of McBride (SIAM J.Math. Anal., 1975). In-

deed, by using a series of simple transformations of variables, this equation can be reduced

to a fractional hyper-Bessel-type equation that can be explicitly solved by using the the-

ory of fractional power of operators. Exact solutions are found in terms of multi-index

Mittag–Leffler functions. A telegraph-type process related to the solution of the fractional

Klein–Gordon equation is studied. We also treat the two-dimensional case and related

processes. We then discuss other mathematical approaches to fractional hyper-Bessel-type

equations by means of fractional Hadamard derivatives and sequential operators involving

Caputo derivatives. Also in this case exact solutions are found, that can be of interest for

applications in mathematical physics.



Fractional heat conduction in a semi-infinite composed body

Yuriy PovstenkoJan D!lugosz University in Czestochowa

Armii Krajowej 13/15, 42-200 Czestochowa Poland

[email protected]

Oral presentation

Starting from the pioneering papers [1], [2], [3], considerable interest has been shown insolutions to fractional di"usion-wave equation. In this paper, we consider the fractionalheat conduction equations with the Caputo derivatives of di"erent orders ! and " in atwo-layer medium composed of a region 0 < x < L and a region L < x < !:

# !T1

#t!= a1

#2T1

#x2, 0 < x < L; (1)

# "T2

#t"= a2

#2T2

#x2, L < x < !, (2)

under initial conditions

t = 0 : T1(x, t) = f(x), 0 < ! " 2,#T1(x, t)

#t= F (x), 1 < ! " 2, 0 < x < L, (3)

t = 0 : T2(x, t) = 0, 0 < " " 2,#T2(x, t)

#t= 0, 1 < " " 2, L < x < !, (4)

and conditions of perfect thermal contact

x = L : T1(x, t) = T2(x, t), (5)

x = L : k1D1!!RL

#T1(x, t)

#x= k1D

1!"RL

#T2(x, t)

#x, 0 < ! " 2, 0 < " " 2. (6)

The boundary surafce at x = 0 is kept insulated.The Laplace transform with respect to time is used, and the approximate solution for smallvalues of time is obtained in terms of the Mittag-Le#er function Eµ,#(z) and the Mainardifunction M(µ, z) [1], [2].

References

[1] Mainardi, F. 1996 The fundamental solutions for the fractional di!usion-wave equation.Appl. Math. Lett. 9(6), 23–28.

[2] Mainardi, F. 1996 Fractional relaxation-oscillation and fractional di!usion-wave phe-

nomena. Chaos, Solitons Fractals 7, 1461–1477.

[3] Wyss, W. 1986 The fractional di!usion equation. J. Math. Phys. 27, 2782–2785.




6–8 November 2013

Well-posedness and Large Time Behavior for a SimplifiedModel of the Sonic Boom Propagation

Liviu I. IgnatInstitute of Mathematics “Simion Stoilow” of the Romanian Academy

21 Calea Grivitei Street, 010702 Bucharest - Romania

[email protected]

Alejandro PozoBCAM - Basque Center for Applied Mathematics

Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country - Spain.

[email protected]

Enrique ZuazuaBCAM - Basque Center for Applied Mathematics & Ikerbasque Basque Foundation for Science

Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country - Spain.

[email protected]

Oral presentation

One of the main concerns in current aeronautics research is related to the control and

reduction of the noise generated by aircrafts and, in particular, to handling the sonic boom

phenomenon produced by supersonic flights [1]. In this presentation we will focus on the

so-called Augmented Burgers Equation, which models the propagation of the sonic boom

from the near field of aircrafts down to the ground level [2]. This equation, which includes

nonlinear phenomena as well as a non-local operator, is given by�ut = uux + νuxx +

Cθ Kθ ∗ uxx, (x, t) ∈ R× (0,∞),

u(x, 0) = u0(x), x ∈ R,

where ν, C, θ > 0 are constant parameters and Kθ is a non-symmetric kernel defined as:

Kθ(z) =

�e−z/θ, z > 0,

0, elsewhere.

We will prove the existence and uniqueness of solution for any initial data u0 ∈ L1(R) ∩L∞(R), using a fixed point argument. Then, by means of a scaling method and some time-

decay estimates of the Lp-norms of the solutions, we will obtain their large-time behaviour

and give a explicit form of the developed asymptotic profile. Numerical tests will be also

provide in order to illustrate the results.

References

[1] Alonso, J. J. and Colonno, M. R. 2012Multidisciplinary Optimization with Applications

to Sonic-Boom Minimization. Annu. Rev. Fluid Mech. 44, 505–526.

[2] Rallabhandi, S. K. 2011 Advanced sonic boom prediction using Augmented Burgers

Equation. Journal of Aircraft 48(4), 1245–1253.




6–8 November 2013

George William Scott-Blair, the pioneerof fractional calculus in rheology

Francesco MainardiUniversita degli Studi di Bologna

via Irnerio, 46, I-40126 - Bologna, Italy

[email protected]

Sergei RogosinIMAPS, Aberystwyth University

Penglais, SY23 3BZ - Aberystwyth, Ceredegin, UK

[email protected]

Oral presentation

The report is devoted to the description of the role of the british chemist GeorgeWilliam Scott-

Blair as a pioneer of fractional modelling at the study of the rheological problems. He pro-

posed the simplest fractional model of viscoelastic material, namely, the fractional element

model in the form ([1], [2])

τij = Eλα−∞Dα

t γij , (0 ≤ α ≤ 1)

where E is the elastic modulus, λ is the relaxation time, τij is the ij-component of the stress

tensor τ based on the cartesian coordinates and γij is the ij-component of the strain tensor

γ, the fractional derivative is understood in Liouville sense. From phenomenological point

of view (see, e.g., [3]) the fractional element model can be naturally generalized from springs

anddashpots by simply replacing the ordinary derivatives in their constitutive equations by

the fractional ones.

The role of G.W.Scott-Blair in rheology was recognized by rheological society (see [4], [5]).

Recently, a special role of G.W.Scott-Blair as a pioneer of creation of fractional models in

rheology was discovered and recognized too (see, [6]).

References

[1] Scott-Blair, G.W. 1947 The role of psychophysics in rheology. J. Colloid Sci. 2, 21-32.

[2] Scott-Blair, G.W. 1974 An Introduction to Biorheology, Elsevier, New York.

[3] Pan Yang, Yee Cheong Lam, Ke-Qin Zhu. 2010 Constitutive equation with fractionalderivatives for generalized UCM model. J. of Non-Newtonian Fluid Mech. 165, 88–97.

[4] Doraiswamy, D. The Origins of Rheology; A Short Historical Excursion. In: RheologyBulletin, 71, No. 1 (Jan. 2002).

[5] The Scott Blair Collection. http://www.aber.ac.uk/en/is/collections/scottblair/

[6] Mainardi, F. 2012 An historical perspective on Fractional Calculus in Linear Viscoelas-ticity. Frac. Calc. Appl. Anal. 15, No. 4, 712-717.




6–8 November 2013

Recent results in extended thermodynamics of dense andrarefied polyatomic gas: macroscopic approach and

Maximum Entropy Principle

Tommaso RuggeriDepartment of Mathematics & CIRAM, University of Bologna

via Saragozza 8, I-40123 Bologna, Italy

[email protected]

Oral presentation

After a brief survey on the principles of Rational Extended Thermodynamics of monatomic

gas (entropy principle, constitutive equations of local type, symmetric hyperbolic systems,

main field, principal sub-system) we present in this talk a recent new approach to deduce

hyperbolic system for dense gases not necessarily monatomic. In the first part of the talk we

study extended thermodynamics of dense gases by adopting the system of field equations

with a different hierarchy structure to that adopted in the previous works. It is the theory

of 14 fields of mass density, velocity, temperature, viscous stress, dynamic pressure and heat

flux. As a result, all the constitutive equations can be determined explicitly by the caloric

and thermal equations of state as in the case of monatomic gases. It is shown that the

rarefied-gas limit of the theory is consistent with the kinetic theory of gases. In the second

part, we limit the result to the physically interesting case of rarefied polyatomic gases and

we show a perfect coincidence between extended thermodynamics and the procedure of

Maximum Entropy Principle. The main difference with respect to usual procedure is the

existence of two hierarchies of macroscopic equations for moments of suitable distribution

function, in which the internal energy of a molecule is taken into account.

References

[1] Arima, T., Taniguchi, S., Ruggeri, T., and Sugiyama, M. 2012. Continuum Mech.

Thermodyn. 24, 271–292.

[2] Arima, T., Taniguchi, S., Ruggeri, T., and Sugiyama, M. 2012. Phys. Lett. A 376,

2799–2803.

[3] Arima, T., Taniguchi, S., Ruggeri, T., and Sugiyama, M. 2012. Continuum Mech.

Thermodyn. (2012) DOI 10.1007/s00161-012-0271-8.

[4] Boillat, G., and Ruggeri, T. 1997. Continuum Mech. Thermodyn. 9, 205–212.

[5] Muller, I., and Ruggeri, T. 1998 Rational Extended Thermodynamics. 2nd ed., Springer

Tracts in Natural Philosophy 37. New–York, Springer–Verlag.

[6] Pavic, M., Ruggeri, T., and Simic, S. 2013. Physica A 392, 1302–1317.

[7] Ruggeri, T. 2012. Quart. of Appl. Math. 70, 597–611.




6–8 November 2013

A generalized Laplacian model associated with the fractionalgeneralization of the Poisson processes

Nicy SebastianIndian Statistical Institute, Chennai Centre,

Taramani, 600113 Chennai, India

[email protected]

Rudolf GorenfloDepartment of Mathematics and Informatics, Free University of Berlin,

Arnimallee 3, D-14195 Berlin, Germany

Oral presentation

We have provided a fractional generalization of the Poisson renewal processes by replacingthe first time derivative in the relaxation equation of the survival probability by a frac-tional derivative of order α (0 < α ≤ 1). A generalized Laplacian model associated withthe Mittag–Leffler distribution is examined. We also discuss some properties of this newmodel and its relevance to time series. Distribution of gliding sums, regression behaviorsand sample path properties are studied. Finally we introduce the q-Mittag–Leffler processassociated with the q-Mittag–Leffler distribution.




6–8 November 2013

Covariance structure of continuous time random walk limitprocesses

Nikolai N. LeonenkoCardiff University

Sengennydd Road, Cardiff CF24, 4YU, UK

[email protected]

Mark M. MeerschaertMichigan State University

619 Red Cedar Road, East Lansing, MI 48824, USA

[email protected]

Alla SikorskiiMichigan State University

619 Red Cedar Road, East Lansing, MI 48824, USA

[email protected]

Oral presentation

A time changed Levy process can arise as the limit of contonuous time random walk, wherethe random walk jumps are separated by random waiting times [4]. We consider randomtime changes to the inverse or hitting time processes of a Levy subordinator L. When theouter process is a homogeneous Poisson process, and L is the standard stable subordinator,the time changed process is fractional Poisson process [1, 2, 3, 5]. The use of time-changedprocesses in modeling often requires the knowledge of their second order properties suchas the covariance function. This paper provides the explicit expression for the covariancefunction for time changed Levy processes. Several examples useful in applications arediscussed.

References

[1] L. Beghin and E. Orsinger (2009) Fractional Poisson processes and related randommotions. Electron. J. Probab., 14, 1790–1826.

[2] F. Mainardi, R. Gorenflo and E. Scalas (2004) A fractional generalization of the Poissonprocesses. Vietnam Journ. Math., 32, 5364.

[3] F. Mainardi, R. Gorenflo, A. Vivoli (2007) Beyond the Poisson renewal process: Atutorial survey. J. Comput. Appl. Math., 205, 725735.

[4] M. M. Meerschaert and H.-P. Scheffler (2008) Triangular array limits for continuoustime random walks. S tochastic Processes and Their Applications, 118, 1606–1633.

[5] M. M. Meerschaert, E. Nane and P. Vellaisamy (2011) The fractional Poisson processand the inverse stable subordinator, Electronic Journal of Probability, 16, Paper no.59, pp. 16001620.




6–8 November 2013

Some properties and computation ofthe Mittag–Leffler functions

Renato SpiglerDepartment of Mathematics, University ”Roma Tre”,Largo San Leonardo Murialdo 1, I-00146 Rome, Italy

[email protected]

Oral presentation

A conjecture concerning estimates of the so-called Mittag–Leffler functions, important inthe framwork of fractional differential equations, was proposed recently by F. Mainardi.We discuss such conjecture and propose a possible new numerical aproach to compute theMittag–Leffler functions.



Fractional Dynamics in Forest Fires

Antonio M. LopesInstitute of Mechanical Engineering, Faculty of Engineering, University of Porto

Rua Dr. Roberto Frias, 4200-465 - Porto Portugal

[email protected]

J. A. Tenreiro MachadoInstitute of Engineering, Polytechnic of Porto

Rua Dr. Antonio Bernardino de Almeida - 431 Porto Portugal

[email protected]

Oral presentation

Every year forest fires consume large areas, being a major concern in many countries likeAustralia, United States and Mediterranean Basin European Countries (e.g., Portugal,Spain, Italy and Greece). Understanding patterns of such events, in terms of size andspatiotemporal distributions, may help to take measures beforehand in view of possiblehazards and decide strategies of fire prevention, detection and suppression. Traditionalstatistical tools have been used to study forest fires. Nevertheless, those tools might not beable to capture the main features of fires complex dynamics and to model fire behaviour[1]. Forest fires size-frequency distributions unveil long range correlations and long memorycharacteristics, which are typical of fractional order systems [2]. Those complex correla-tions are characterized by self-similarity and absence of characteristic length-scale, meaningthat forest fires exhibit power-law (PL) behaviour. Forest fires have also been proved toexhibit time-clustering phenomena, with timescales of the order of few days [3]. In thispaper, we study forest fires in the perspective of dynamical systems and fractional calculus(FC). Public domain forest fires catalogues, containing data of events occurred in Portugal,in the period 1980 up to 2011, are considered. The data is analysed in an annual basis,modelling the occurrences as sequences of Dirac impulses. The frequency spectra of suchsignals are determined using Fourier transforms, and approximated through PL trendlines.The PL parameters are then used to unveil the fractional-order dynamics characteristicsof the data. To complement the analysis, correlation indices are used to compare and findpossible relationships among the data. It is shown that the used approach can be useful toexpose hidden patterns not captured by traditional tools.

References

[1] Alvarado, E., Sandberg, D. and Pickford, S. 1988 Modeling Large Forest Fires as Ex-treme Events. Northwest Science 72, 66-75.

[2] Mainardi, M. 2010 Fractional Calculus and Waves in Linear Viscoelasticity. ImperialCollege Press, London.

[3] Telesca, L., Amatulli, G., Lasaponara, R., Lovallo, M. and Santulli, A. 2005 Time-scaling properties in forest-fire sequences observed in Gargano area (southern Italy).Ecological Modelling 185, 531-544.




6–8 November 2013

Cosmic rays propagation in galaxy: a fractional approach

Vladimir Uchaikin and Renat SibatovUlyanovsk State University, Russian Federation

[email protected]; ren [email protected]

Oral presentation

In framework of so-called compound diffusion model, the cosmic rays propagation process

is separated in two processes: longitudinal (random motion along a magnetic force line)

and transversal (motion with the force line, performing random walk in space). Originally,

both these motions were considered as normal Gaussian processes. Nowadays, there exist

some reasons to refuse the simple models and pass to more realistic models including finite

velocity of free motion and multiscale (fractal) character of interstellar magnetic fields.

A fractional approach to description of cosmic rays anomalous walking longitudinal pro-

cess in the compound diffusion model is considered. The process is constructed as a one-

dimensional asymmetric walk of a particle with a finite constant speed and α-type asymp-

totic of the free path distribution. The anomalous diffusion equation following from the

relevant integral equation includes the fractional material derivative operator. It is solved

by means of the Fourier–Laplace transform method and the solutions are represented for

0 < α < 1 in terms of elementary function, for 1 < α < 2 in terms of Levy-stable distribu-

tions and for α = 2 in terms of the Gaussian distribution.

This model is modified by involving exponential truncation of free path distribution and

coupling collision points with perpendicular displacement events. Analytic investigation

shows that this model reveals different behavior in various time domains: we observe the

superdiffusion in the parallel direction and subdiffusion in the perpendicular one in inter-

mediate time region, and normal diffusion in both directions at asymptotically large time.

The results are compared with those of other authors and the reasons of discrepancy are

discussed.




6–8 November 2013

Fractional differential model of anisotropic diffusion ofcosmic rays in random magnetic fields

Vladimir Uchaikin and Renat SibatovUlyanovsk State University, Russian Federation

[email protected]; ren [email protected]

Oral presentation

The new results on the development of the fractional differential model of cosmic raystransport in quasi-regular magnetic fields are presented. The combination of the compounddiffusion model proposed by Getmantsev in the 60s with the modern theory based on thegeneralized kinetic equations of fractional order enriches the overall picture of transportallowing to take into account non-exponential nature of particle path lengths along magneticfield lines, the finite value of particles velocity, a qualitative difference between longitudinaland transverse diffusion modes. The problems of particle escape from the galaxy, anisotropyand chemical composition of the primary cosmic radiation are considered. The prioritycomputational problems within the framework of this model are discussed.




6–8 November 2013

Controlled fractional dynamics with applications

Maria VeretennikovaUniversity of Warwick

Mathematics Institute, Zeeman building, Coventry CV4 7AL

[email protected]

Prof. V. KolokoltsovUniversity of Warwick

Statistics Department, Zeeman building, Coventry CV4 7AL

[email protected]

Oral presentation

Recently fractional calculus has drawn attention of many scientists. For instance it comes up

in mathematical description of anomalous diffusion, which has been observed in experiments

and is useful in modeling a vast range of natural phenomena. We study the continuous

time random walk model for anomalous diffusion. In this talk we start by explaining the

fundamentals of fractional calculus and then describe how it arises in particle evolution

models.

Let X1, X2, ...XNt be i.i.d. r.v., and T1, . . . , Tn be i.i.d. positive r.v. Define

Z(t) = inf(n : T1 + . . .+ Tn > t). (1)

Then, under certain assumptions on the distributions of Xi and Ti, more precisely they

should belong to normal domains of attraction of certain stable laws, the scaled averages

Ef(X1 + . . .+XZ(t)) are known to converge to solutions of fractional equations, such as

D∗β0,tf = Lf, (2)

For example L is of the form L = −(−∆)α/2

,α ∈ (0, 2], and D∗β0,t is the Caputo time

derivative, with β ∈ (0, 1).

We extend this convergence result to the case when either Xi or Ti can be controlled and

look at scaling effects for different variations of this process. We obtain the corresponding

general limiting equation for the optimal payoff function S(t, y) of the controlled continuous

time random walk. It is of the following form:

D∗β0,tS(t, y) = LS(t, y) +H(t, y,DyS(t, y))

When the control is linear quadratic, the limiting payoff equation turns into a fractional

in time Riccati equation. Extensions of the model include for instance studying the effectof time and position dependence on the optimal payoff equation and other generalisations

related to Feller processes. We analyse well-posedness of simplified versions of the limiting

equations and derive existence and uniqueness of the classical solution to a simplified Caputo

fractional Hamilton-Jacobi-Bellman equation under certain conditions. Finally we consider

applications of the theory to finances and game theory. In particular, in collaboration with

A. Chertok, we use this to model the net flow of orders in a limit order book and give a

probabilistic description of the limiting process via fractional distributions. As for game

theory, we extend this to an N -particle system and to interacting N -particle systems.




6–8 November 2013

A Functional Limit Theorem for stochastic integrals drivenby a time-changed symmetric α-stable Levy Process

Enrico ScalasUniversita del Piemonte Orientale and BCAM-Basque Center for Applied Mathematics

viale Michel 11, I15121, Italy and Mazarredo, 14, E48009 Bilbao, Basque Country-Spain

[email protected]

Noelia VilesUniversitat de Barcelona

Gran via de les Corts Catalanes, 585 08007 - Barcelona Catalonia-Spain

[email protected]

Oral presentation

A continuos time random walk (CTRW) can be formally defined as a random walk subor-

dinated to a counting renewal process. CTRWs became a widely used tool for describing

random processes that appear in a large variety of physical models and also in finance.

The main motivation of our work comes from the physical model given by a damped har-

monic oscillator subject to a random force (Levy process) studied in the paper of Sokolov.

We study the convergence of a class of stochastic integrals with respect to the compound

fractional Poisson process.

Under proper scaling and distributional hypotheses, we establish a functional limit theorem

for the integral of a deterministic function driven by a time-changed symmetric α-stableLevy process with respect to a properly rescaled continuous time random walk in the Sko-

rokhod space equipped with the Skorokhod M1-topology. The limiting process is the cor-

responding integral but with respect to a time-changed α-stable Levy process where the

time-change is given by the functional inverse of a β-stable subordinator.

References

[1] Scalas, E., and Viles, N. 2013 A functional limit theorem for stochastic integrals drivenby a time-changed symmetric α-stable Levy process. Submitted.

[2] Sokolov, I. M. 2011 Harmonic oscillator under Levy noise: Unexpected properties inthe phase space. Phys. Rev. E. 83, 041118.




6–8 November 2013

Soil hydrology: a fractional physical model

Giuliano VitaliDipartimento di Scienze Agrarie, Universita di Bologna

Viale Fanin 44, I-40127 Bologna, Italy

[email protected]

Oral presentation

Soil hydrogy is a main branch of soil physics, whose aim is understanding hydrology of

cropped land, both in the surface and underground. On the surface a main problem is

understanding runoff and its relation to erosion and infiltration process, whose interest

is recently grown in connection to climate change and the related increased frequency of

extreme events, related to flood and loss of fertile soils. Infiltration is the process by which

water enters the vadose zone, the unsatured layer where water redistributes and finally feeds

water-table. Both surface and underground complexity can be ascribed to the multiple

viewpoint soil can be interpreted (e.g. granular vs porous medium, cracked vs viscoelastic

system) whose aspect and behaviour changes with climate (pedological layers), ecology

(organic content) and water (mechanical properties), as much as its intrinsic heterogeneity.

The major dynamical processes, runoff and water redistribution, are still largely represented

by lumped models which make use of empirical parameters hardly portable from one soil

to soil. Even the fractal approach of the past decades have been used to represent soil

properties (e.g. particle/pore size distribution) without a clear connection to a physical

model. In the present study fractional PDE models are proposed to reinterpretate both

surface and porous system water dynamics. The models have been identified making use

of the analogy with the ladder model already used to coin physical models of viscoelastic

systems. The approach, based on the continued fraction discrete interpretation of fractional

derivative, allows to get for runoff process and water redistribution a new perspective model.

Simulations are developed using a lattice approach where node occupation (by water) at

steady state under different flux intensity, show how dimensionality of flow stay in the range

2 : 1 for runoff and 3 : 1 for redisribution, where the 1-dimensional case reflects a generation

of rills on the surfaces and of preferential paths in the underground.



Adaptive Finite Di!erence Methods for Fractional Di!usionEquations

Santos B. Yuste! and J. Quintana-Murillo†

Departamento de Fısica, Universidad de ExtremaduraAvda. Elvas s/n, 06071 Badajoz, Spain

[email protected], †[email protected]

Oral presentation

Standard finite di!erence methods when applied to fractional di!erential are increasinglyslow and memory demanding due to the fact that, typically, the number of operationsrequired to calculate the numerical solution scale as the square of the number of timesteps.To reduce this number of timesteps is, then, of paramount importance. A way of doingthis is by resorting to algorithms with variable timesteps. These algorithms can be used tobuild adaptive methods in which the size of the timesteps is chosen according to the behaviorof the solution: large timesteps are used when the solution changes slowly whereas smalltimesteps are employed when the solution changes fast. Although in general the way inwhich the timesteps are chosen depends on the problem one wants to solve, here we presenta relatively general and e"cient procedure in which the size of the timesteps are dynamicallyobtained by means of a method that is an adaptation, to fractional di!usion equations, ofthe well-known step doubling method employed in ordinary di!erential equations. A keyissue is to what extent the variability of the size of the timesteps a!ects to the stability ofthe method. We address this problem by obtaining the stability regions for several di!erentways of choosing the size of the timesteps. These results are checked numerically.

Documents

Fractional Calculus, Probability and Non-local Operators ... · Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments A workshop on the occasion