Fractional calculus: basic theory and applications (Part I)

Transcription

1 Frctionl clculus: bsic theory nd pplictions (Prt I) Diego del-cstillo-negrete Ok Ridge Ntionl Lbortory Fusion Energy Division P.O. Bo 28, MS 669 Ok Ridge, TN phone: (865) FAX: (865) e-mil: Lectures presented t the Institute of Mthemtics UNAM. August 25 Meico, City. Meico Outline. Introduction 2. Frctionl clculus 3. Frctionl diffusion nd rndom wlks 4. Numericl methods 5. Applictions: ) Turbulent trnsport b) Trnsport in fusion plsms c) Rection-diffusion systems 6. Some references

2 . Introduction Here we introduce the notion of frctionl integrl s strightforwrd generliztion of the stndrd, integer-order integrl, nd define the frctionl derivtive s the inverse opertion. To motivte the concept we discuss two emples: Abel s eqution nd het diffusion. The concepts discussed here re further elborted in the net section. Wht is frctionl derivtive? d n f d n L Hopitl (695): Wht if n=/2? Leibniz (695): This is n pprent prdo from which, one dy, useful consequences will be drwn It is usul prctice to etend mthemticl opertions, originlly defined for set of objects, to wider set of objects r r!r + " s s!r n! n!n + " #(s) s!r! n " n #N + $! " #C This mthemticl gmes hve, eventully, importnt physicl pplictions

3 D Frctionl integrls Integer order integrtion n" "n # = $ d $ d 2 L $ d n # n Interchnging the order of integrtion this cn be rewritten s D "n # = (n ")! $ ( " y) n" # ( y) dy Etending this epression for non-integer n=! we get Riemnn-Liouville frctionl integrl D "# $ = (#) ( " y) # " $ ( y) dy Emple: D "# µ = $(µ +) $(µ + # +) µ +# Frctionl derivtives Hving defined the frctionl integrl, define the frctionl derivtive s the inverse opertion D µ D "µ # =# D µ " = d N d N [ D #$ "]! = N " µ N = smllest integer > µ D m " = d N d N #(N#m [ D ) "] = d m d " m Riemnn-Liouville frctionl derivtive D " # = $(m ") m # ( y) ( y) " +m dy m "< # $ m Emples: D µ " = #(" +) #(" $ µ +) "$µ D µ "# e ik = ( ik) µ e ik

4 y v An emple: Abel s eqution g (,h) Let T(h) be the time it tkes prticle to go down sliding on the curve = "(y) Wht is the reltion between the function T(h) nd "(y)? Energy conservtion 2 v 2 = g h " y v = ds dt = + "2 dy dt h + " 2 T $ dy = $ dt = T 2g(h # y) f (y) = 2g + "2 T(h) = h f (y) # dy h " y #/ T = " D 2 h f f = " D / 2 y T S(t) T(,t) Het diffusion Consider br tht is heted on one end by time dependent source S(t). Wht is the reltion between S(t) nd the temperture T(,t) of the br? Het diffusion eqution " t T = # " 2 T Boundry conditions Initil conditions T( =,t) = S(t) T(,t = ) = Lplce trnsform T = 2 ˆ T (,s) = # $ e " st T(,t) dt st ˆ 2 = " # t e $ 4# (t$ ) S() d "(t) = T(,t) d 2"# t $ 3 / 2 # $ ˆ T "(t) = 2# D t $/ 2 S(t)

5 2. Frctionl clculus Here we present the precise definitions of the left nd right Riemnn- Liouville (RL) frctionl integrls nd derivtives, nd briefly discuss some of the min properties of these opertors. Of prticulr interest is the singulr behvior of the RL opertors t the boundries, nd the definition of the frctionl derivtives in the Cputo sense. Further detils on the mteril discussed here cn be found in [Smko-etl-993], [Podlubny-999] nd references therein. Riemnn-Liouville integrls Left integrl D "# $ = (#) b ( " y) #" $ ( y) dy " > > Right integrl D b "# $ = (#) b ( y " ) #" $ ( y) dy < b Given the reflection opertor Q " () = " ( + b # ) Q [ D "# $ ] = D "# b [ Q $ ] Q [ D "# b $ ] = D "# [ Q $ ]

6 Q[ D "# $ ] = (#) +b" $ ( u) ( + b " " u) "# du z = "u + + b = "(#) b dz= ( z ) # D b $ + b z [ ] # Q $ Semi-group property The frctionl integrls stisfy the following importnt semigroup property "# D "# D b D "$ = D "#"$ " > D b "$ = D b "#"$ " > D "# D "$ = (#) ($) dw ( " w) "# w (u) du (w " u) "$ w u=w Fubini s theorem u = "(#) " dw " dul = " du " dwl "($) w du (u) u u dw ( w) # (w u) $ Bet function $(") $(#) B (",#) = $(" + #) B (",#) ( $ u) $"$# = "(# + $) (u) (# +$ du = ( u) #$ D )

7 Riemnn-Liouville derivtives n = ["] + b [#]=integrl prt of # Left derivtive D " # = $(n ") n n # ( u) du ( u) "n + Right derivtive D " b # = ( $) n (n $") n n b # ( u) du ( u $ ) "$n + Reciprocity D " D #" $ = $ D " D #" $ = (")(n #") n n dz du z $ ( u) ( # z) "#n + ( z # u) #" Echnging the order of the integrls nd using the definition of the Bet function we get: = "(#)"(n $#) n B (#,n $# )* - du ( u) n ) ( ( $ u) $n, = + "(n) # n # n du $ ( u) ( u) n using (n ")! = " n " D #n n $ = $ # u "n $ du = ( " u) "n D

8 However D "# D #" [ j $ ] # $ = $() " D k j= = ( " ) #" j # " j + j "# $ < j The previous formule cn be generlized s D " D #$ = D "#$ " # $ # D "# D $ " [ j ] $ = D $ "# () " D k j= = ( " ) #" j # " j + Composition of frctionl derivtives " n " D # n [ n $ ]= D +# $ " # " # n $ ( " D # n * = D +n $() + ) n+ - j= j+"+n,( + j +" + n) $ ( j ) () + n "# $ < n D " D # " [ $ ] = D +# # $() D j $ m j= j" [ ] = " j m " # $ < m In prticulr, frctionl derivtives commute D " D # # [ $ ] = D [ D " $ ] if nd only if D " # [ j $ ] = = D "# [ j $ ] = = j =,Ln j =,Lm

9 Behvior ner the lower terminl Let # " (k) () "() = $ k! k= k D " # = $ # (k) () k! D " k= k" using k = $ ( k + ) $ ( k +#" ) D " # ( # ) k#" D k$" $ " # = # (k) () k +$" k= m $ (k ) () lim " D # $ = lim " k +# k= ( ) #k m " # $ < m lim " D # $ = unless " (k ) () = k =,Lm " Fourier Fourier-Lplce trnsforms $ ˆ "(k) = e ik "() d "() = 2# #$ $ e $ik ˆ " (k) d F [ "# D $ ] = ("i k) $ ˆ (k) F D +" [ # $ ] = i k # ˆ $ (k) Lplce $ ˆ "(s) = e # st "(t) dt "(t) = c +i$ c#i$ e st ˆ "(s) ds L[ D " t #] = s " ˆ n$ [ ] t= # (s) $ s k D t "$k$ # k= n " # $ < n These re nturl generliztions of the Fourier-Lplce trnsform of regulr derivtives Note tht we hve tken = "# b = #

10 Some limittions of the RL definition Although well-defined mthemticl object, the Riemmn-Liouville definition of the frctionl derivtive hs some problems when it comes to pply it in physicl problems. In prticulr: The derivtive of constnt is not zero The RL derivtive is in generl singulr t the lower limit D " A = A # $" ( $ ) " lim " D # $ = unless " (k ) () = This might be n issue when using RL opertors for writing evolution equtions This might be n issue when pplying boundry conditions The Lplce trnsform of the RL derivtive depends on the frctionl derivtive t zero L[ D " t #] = s " ˆ n$ [ ] t= # (s) $ s k D t"$k$ # k= This might be n Issue when solving Initil vlue problems Cputo frctionl derivtive These problems cn be resolved by defining the frctionl opertors in the Cputo sense. Consider the cse < " < 2 k +2$" D " # $ #() ($") ( $") $ #() " (2 $") ( $") = # (k +2) () $ "$ k + 3 $" k= singulr terms D " [#() $ #() $ # () ()] = k= # (k +2) () $ regulr terms k +2$" ( k + 3$" ) Define the Cputo derivtive by subtrcting the singulr terms using C D " # = D " [#() $ #() $ # () ()] #" (u) # (k +2) () $ $ du = ( du $ u k! $ u k= C D " # = $(2 ") # (u) ( u) k $ " du

11 In generl C D " # = $(n ") n u # du ( u) "n + C D " b # = ($)n (n $") n u # du ( $ u) "$n + nd s epected: C D " A = lim " D # $ = L[ D " t #] = s " ˆ Note tht in n infinite domin n$ [ ] t= # (s) $ s k # ("$k$) k= C "# D $ = "# D $ C D # " $ = D # " $ 3. Frctionl diffusion nd rndom wlks Here we discuss the close connection between frctionl clculus (in prticulr frctionl diffusion equtions) nd the theory of continuous time rndom wlks (CTRW). We discuss the spce-time scling properties of the solution of the frctionl diffusion eqution nd the reltions with the Levy stble distributions. We present two emples of Levy distributions one in the contet of chotic dvection [del-cstillo- Negrete-998, 2] nd nother in the contet of finncil mthemtics [Mntegn-Stnley-994]. CTRWs were originlly introduced in [Montroll-Weiss-965]. Further detils cn be found in the review ppers [Montroll-Shlesinger-984], [Metzler-Klfter-2], nd references therein. For discussion in the contet of turbulent trnsport in plsms see [del-cstillo-negrete-etl-24]. For redble introduction to stochstic processes nd Levy distributions see [Pul-Bschngel-999], [Bouchrd- Georges-99] nd references therein. For discussion on the Green s function of the frctionl diffusion eqution see [Minrdi et l. Frctionl Clculus nd Applied Anlysis, 4, (2)].

12 The continuous time rndom wlk [Montroll-Weiss-965]! n!! n = witing time! (" ) = witing time pdf n! n = jump!(" ) = jump size pdf Mster eqution t * P(,t ) =! ()" (t )dt + " (t # t ) $( # ) P(,t )d dt ( ) +, t # Contribution from prticles tht hve not moved during (,t) Contribution from prticles locted t nd jumping to during (,t) The Montroll-Weiss mster eqution cn be solved using Lplce-Fourier trnsforms ˆ P (k,s) =! " (s) s!" (s) # ˆ (k) We wnt to tke the long time, nd lrge scle limit. To do this, introduce the smll prmeters epsilon nd delt nd consider " * (t) = # " $ t #( ) " * () = # " $ # ) ( Then, the limit of interest is " # $ # In terms of Lplce trnsforms then L [" * (t)](s) = " ˆ (# s) F " * () [ ](k) = ˆ " # k rescled MW eqution P ˆ *(k,s) = " # ($ s) s " # ($ s) ˆ ( k)

13 Trpping pdf Gussin Mrkovin cse "(t) = µ e #µ $ Jumps pdf "() = " (# s) = $# s < t > +L + # s < t > ˆ " (# k) = e $ 2 k 2 # 2 / 2 $# 2 2 k 2 /2 +L 2 2# $ e / 2$ /µ nd $ trnsport scles Continuum distinguished limit " = 2 # 2 " # $ # finite 2 t $ Lplce trnsform Fourier trnsform L [! t P ] = s P "# () F [! 2 ˆ P ] = "k 2 P ˆ sp ˆ "= "# k 2 P ˆ! t P = "! 2 P Non-Gussin, non-mrkovin cse Trpping pdf with memory Jumps pdf "(t) ~ t #( $ +) "() ~ #(+$ ) " (# s) $c (# s) +L ˆ "(# k) $c 2 (# k ) +L Continuum distinguished limit " # $ # " = 2 2 t # $ s! ˆ P " s! " = "# k $ ˆ P Lplce trnsform Fourier trnsform " L [! t P ] = s " P # s " # $ () " [ ˆ ] = # k " P ˆ F! P! " t P = #! $ P Frctionl diffusion eqution

14 Solution of frctionl diffusion eqution Consider the initil vlue problem of the symmetric frctionl diffusion eqution in n infinite domin " t # = l $ D [ + r D ] # "(,t = ) = " () l=" ("#) 2cos $ /2 r=" (+ #) 2cos $ /2 "#$ # Using F [ "# D $ ](k) = ("i k) $ (k) F[ D # " $ ](k) = ( i k) # $(k) ˆ "(k) = ˆ " (k) ep t l #ik { [ $ + r( ik) $ ]} Then, using ( mi k) " = k " ep #i "$ 2 k k ( * ) We cn write the solution s ˆ "(k) = ˆ " (k) L ˆ # $ (k) where 2 L ˆ (k) = ep 4 $ t k, " + i# k "# k tn 3. ( " 54-2 )/ * 84 And using the convolution theorem "(,t)= ( L $ ( )" () d 2#

15 Some emples Gussin Levy L " = 2 # = $ = L " =.5 # = $ = Levy Levy L " =.5 # = $ =.35 L " =.5 # = $ =.5 Probbilistic interprettion "(,t)= Probbility of being t t time t "(,t)= 2# ( L $ ( )" () d Trnsition probbility In the cse lph=2, thet= the solution reduces to L ˆ "# (k) = ep { $ t k 2 L } 2 () = 2 " t ep $ # 2 4 t ( ) Which s epected corresponds to normlized Gussin with vrince " 2 =2 t Consistent with the Brownin rndom wlk for " = 2 probbility is Gussin. the trnsition Wht is probbilistic interprettion for " # 2?

16 Levy distributions Consider set of independent, identiclly distributed rndom vribles { l n } n=,2ln Prob( l < l n < l + dl)= p(l) Given this, we would like to know wht is the probbility distribution of the normlized sum S N = B N N # l n " A N Prob( < S N < + d)=p() =?? n= In the contet of rndom wlk the nswer to this question gives the probbility of finding prticle t given point t given time provided we know the probbility distribution of the individul steps An nswer to this question is provided by the Centrl Limit Theorem ccording to which: # P() " Gussin provided l 2 = $ l 2 p(l) dl< # N#$ However, wht hppens if l 2 = "????? "# A probbility density is stble if there re constnts nd b such tht $ p( l+b )" p( l+b ) = dl p[ (z # l)+b ] p( 2 l + b 2 ) = p(z + b) #$ The importnce of the stble distributions is given by the following Theorem (Levy-Khintchine) A probbility density L() cn be limiting distribution S N = B N only if it is stble N # l n " A N Prob < S N < + d n= " N #$ L() The Gussin is stble, nd ccording to the CLT we know tht is the ttrctor of of ll processes by finite vrince. However, the fmily of stble distributions is much bigger.

17 Levy-Khintchine cnonicl representtion: A probbility density L " # (k) = e ik = d L # () e ik - Is stble if nd only if L " # (k) = ep i$ k + c k " + i# k k tn / ) ) "./ ( ( 2 < " # 2 "< # < $ **,, But ccording to our previous clcultion this is the Green s fuunction of the spce frctionl diffusion eqution. So we hve gone full circle: from stochstic process to frctionl diffusion equtions nd bck to stochstic processes. We first derived the frctionl diffusion eqution s continuum limit for non-gussin rndom wlks. Then, we solved the frctionl eqution, nd showed tht the solution is given in terms of Levy stble distributions. Spce-time frctionl diffusion eqution C D " t # = l $ D [ + r D ] #!(, t = ) = "() Symmetric cse l=r = " 2cos #$ /2 Solution in terms of Fourier-Lplce trnsforms! (k, s) = s " # s " + $ k! (k, t = ) [ E! ("c t! )]( s) = s! " s! + c E! ( z) = # n z n "! n + Mittg-Leffler function! (, t) = 2" $ e # ik E #$ (# k ( t )dk

18 Self-similrity nd non-diffusive scling Similrity vrible!= t " # / $ Fundmentl solution "(, t) =t #$ / K ( ) = " K! # $ cos (! z) E ( z ( )dz " (,# t) = # " ( $ / # $ /,t ) + Self-similr scling * Moments n = # n "(,t) d =t n$ / # n K() d n ~ t n " /# 2 " # > < Super-diffusion Sub-diffusion # determines the symptotic scling in spce! (, t) =t "# / $ K ( )!= t " # / $ Spce similrity vrible = " K! # $ cos (! z) E ( z ( )dz! " # "( + #) K (! ) ~! For fied t nd lrge "( +# )! (, t ) ~! ~! (+" )

19 determines the symptotic scling in time! (, t) = " # "$ / K (# "$ / )! = t "# / $ Time similrity vrible = " K! # $ cos (! z) E ( z ( )dz! " # "( + #) K (! ) ~!! " For fied : K (! ) ~ +! "("# )!! (, t) ~ t " for t ~ ( ) ( t # " for t $ ~ t! t ~ t!" An emple from finnce The simplest models of the fluctutions of ssets prices ssume n underlying geometric Brownin motion, nd led to Gussin distribution of price vritions. In 963 by Mndelbrot pointed out the eistence of ft tils due to lrge events. Bsed on this he proposed to use Levy stble distributions to model prices. Mtegn nd Stnley showed tht the probbility distribution of price vritions of the SP5 strongly deprts from Gussin nd is well-fitted ner the center by Levy stble distribution of inde #=.4. Levy Gussin [Mtegn-Stnley, Nture 995]

20 r (t) Pssive sclr diffusion! r (t) = r (t) " r () = ensemble verge M(t ) =! r = men " 2 (t) =! r #! r [ ] 2 = vrince P(! r,t) = probbility distribution homogenous, isotropic turbulence Brownin rndom wlk lim t! " M(t ) = V t! 2 (t) = Dt P(" r,t) = Gussin V= trnsport velocity D=diffusion coefficient Coherent structures cn give rise to nomlous diffusion trnsport region trpping region trpping event echnge region flight event t Coherent structures correltions lim t! "! 2 (t) ~ t " P(,t ) = non - Gussin

21 An emple from chotic dvection Qusigeostrophic flow 2! " v The rotting nnulus eperiment Solomon, Weeks, Swinney, Phys. Rev. Lett. 7, 3975 (993). Test prticle trnsport Eperiment Solomon, Weeks, Swinney, PRL, 7, 3975 (993) Model [del-cstillo-negrete-998]

22 Non-Gussin distribution functions Probbility distribution functions of prticle displcements. P P Ft tils Gussin [del-cstillo-negrete-998] Solomon, Weeks, Swinney, PRL, 7, 3975 (993) Hmiltonin chotic dvection model d "# =! dt "y dy "# = dt " 2! = " ln [cosh ] + j = # j $ j () cosk j (y " c j t) tnh() zonl flow regulr neutrl modes u [del-cstillo-negrete-998]

23 integrble flow! = "ln[ cosh ]+#! () cos k y +c sher flow trnsport trpping region seprtri chotic trnsport! =! +" 2! 2 () cos( k 2 y #$ t) nonintegrble perturbtion chotic orbit echnge region stochstic lyer Self-similr probbility distributions P t! / 2 P! X =! "!! = (" # " )t #$ / 2 collpse of curves! P * (X, t) = t "# / 2 F( X t "# / 2 ) X n ~ t n! / 2 P * (X,t ) =! " / 2 P * (! " / 2 X,! t)