Fractional Partial Differential Equation:

Transcription

1 Frctionl Prtil Differentil Eqution: Introduction nd Model Hong Wng Deprtment of Mthemtics, University of South Crolin IBM Visiting Fellow, Division of Applied Mthemtics, Brown University Division of Applied Mthemtics, Brown University April 12, 2018 Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 1 / 25 o

2 Acknowledgements Division of Applied Mthemtics, Brown University The OSD/ARO MURI Grnt W911NF nd the Ntionl Science Foundtion under Grnt DMS Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 2 / 25 o

3 Clssicl Fickin diffusion processes Diffusion describes the rndom movement of trcer prticles from high concentrtion to low concentrtion. Two fundmentl pproches were used to model diffusion. A deterministic/mcroscopic description vi second-order diffusion PDE for the PDF of prticle movements: Fick st up the diffusion eqution (1855) when studying how nutrients trvel through membrnes in living orgnisms, by mimicking the het conduction eqution of Fourier (1822). Einstein derived the diffusion eqution from first principles s prt of his work on Brownin motion (1905). A stochstic/microscopic description vi rndom wlk of prticles Brown observed nd investigted irregulr movement of smll pollen grin under microscope (1827). Person modeled diffusion process in terms of rndom wlk, when he studied on how mosquitoes spred mlri (1905). Bchelier used Brownin motion to model sset prices (1900). Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 3 / 25 o

4 The common ssumptions of Einstein nd Person the existence of men free pth, the existence of men witing time to perform jump. Under these pproximtions Person s pproches of rndom wlk yields Brownin motion, which then leds to stochstic differentil eqution nd is suited for microscopic description of diffusive trnsport. Einstein s derivtion yields Fickin diffusion eqution, which cn be viewed s Fokker-Plnck eqution of Brownin motion. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 4 / 25 o

5 Stochstic nd deterministic description of Fickin diffusion Let X be rndom vrible, F (x) = P[X x] nd p(x) = F (x) be its CDF nd PDF, respectively, nd µ q = E[X q ] be its qth moment. The common ssumptions of Einstein nd Person stte the vrince nd men witing time of rndomly selected prticle s motion re finite. If µ 1 = 0, µ 2 = σ 2 <, the FT ˆp hs the expnsion ˆp(k) = E[e ikx ] = e ikx p(x)dx = 1 ikµ 1k µ 2k 2 /2 + o(k 2 ) R = 1 σ 2 k 2 /2 + o(k 2 ), k 0. (1) { Let X 1, X 2,... be sequence of iid rndom vribles tht represent the rndom jumps of rndomly selected prticle with E[X i] = 0 nd E[X 2 i ] = σ 2. Lévy s continuity theorem = the prticle s loction S n := X X n stisfies E [ e ik(sn/ n) ] n = E [ e i(n 1/2 k)x j ] [ = 1 σ2 k 2 ( 1 )] n 2n + o n j=1 e σ2 k = e x2 2σ 2 e ikx dx = E [ e ikz] (2) {, 2πσ R S n/ n Z N(0, σ 2 ), s n. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 5 / 25 o

6 For ny fixed time t > 0 nd c >> 1, the rescled rndom wlk S ct := X X ct (3) { gives the prticle loction t time t > 0 fter ct jumps. As the jump size is reduced by 1/ c, the normlized prtition loction S ct / c stisfies E [ e ] ik(s ct / c) = [1 σ2 k 2 2c = E [ e ikz t ] =: ˆp(k, t) = ( 1 )] ct + o e tσ2 k 2 2 c 1 x2 e 2σ 2 t, 2πσ2 t (4) { S ct / c Z t by Lévy continuity theorem. Here Z t N(0, σ 2 t) is Brownin motion. Z t cn be written s n Ito type stochstic differentil eqution, which gives microscopic description of diffusion (Person s pproch) dz t = µdt + σdb t. (5) { where µ = 0 nd B t N(0, t) is the stndrd Brownin motion. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 6 / 25 o

7 Reltion to Fickin diffusion eqution Let ˆp(k, t) := e tσ2 k 2 2 be the FT of of the PDF p(x, t), which stisfies ˆp t = σ2 2 k2 ˆp = σ2 2 (ik)2 ˆp = σ2 2 p p 2 x 2 t = σ2 2 p 2 x. (6) { 2 This reltes the dispersivity K to the prticle jump vrince σ 2. The PDF stisfies Fickin diffusion eqution (s the Fokker-Plnck eqution of the SDE), which decys exponentilly. The equivlence between the PDE description nd the stochstic formultion lso hs mthemticl nd numericl impct One cn solve diffusion PDE (the Fokker-Plnck eqution) to find the PDF p(x, t) of the underlying stochstic process. One cn lso use prticle trcking method to numericlly solve diffusion PDE by simulting the underlying stochstic process. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 7 / 25 o

8 Stochstic derivtion of n dvection-diffusion PDE For µ 0, the stochstic process µt + Z t stisfies the Ito SDE (5). Moreover, it hs FT E [ e ik(µt+zt)] = e ikµt tσ2 k 2 2 =: ˆp(k, t), (7) { which solves ˆp ( t = iµk σ2 2 k2)ˆp = µ p x + σ2 2 p 2 x 2, (8) { which inverts to n dvection-diffusion eqution s the Fokker-Plnk PDE of the SDE (5). p t + µ p x σ2 2 p 2 x 2 = 0 (9) { Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 8 / 25 o

9 Derivtion of Fickin diffusion PDE with vrible diffusivity Derivtion of the clssicl conservtion lw Let c(x, t) be the concentrtion of solute nd q(x, t) be the flux. In smll cube of side δx with the cross-sectionl re A = (δx) 2, the mss chnge δm nd δc over time δt re δm(x, t) = q(x δx/2, t)aδt q(x + δx/2, t)aδt = δq(x, t)aδt, δc(x, t) = δm(x, t) Aδx = δq(x, t)δt. δx Tking the limit s δx, δt 0 + yields mss conservltion lw δc(x, t) δt δq(x, t) = = c δx t = q x (10) { (11) { Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBMApril Visiting 12, 2018 Fellow, Division 9 / 25 o

10 The clssicl Fick s lw c(x, t) q(x, t) = K x (12) { ssumes the prticles jump loclly to the left nd right neighboring cells with equl probbility δc(x, t) c(x + δx/2, t) c(x δx/2, t) q(x, t) K = K. (13) { δx δx The form (12) is cler for constnt K nd is ssumed for vrible K. Inserting Fick s lw into (11) yields the clssicl Fickin diffusion PDE c(x, t) t = ( c(x, t) ) K = K 2 c(x, t). (14) { x x x 2 The second equl sign holds for constnt K. (14) is self-djoint. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 10 / 25 o

11 Derivtion of the Fokker-Plnck PDE from the SDE Consider the Ito SDE with vrible drift µ nd voltility σ dz t = µ(t, Z t)dt + σ(t, Z t)db t. (15) { For ny smooth nd rpidly decying f(x), Ito s Lemm sttes Y t = f(z t ) stisfies dy t = f (Z t)dz t f (Z t)dz 2 t = ( f (Z t)µ+f (Z t)σ 2 /2 ) dt+f (Z t)σdb t (16) { Integrting (16) on ny time intervl [, b] gives Y b Y = f(z b ) f(z ) = b ( f (Z t)µ+f (Z t)σ 2 /2 ) b dt+ f (Z t)σdb t. (17) { Tking the expecttion of (17) (recll E(B t ) = 0) yields E[f(Z b ) f(z )] = f(x) [ p(x, b) p(, t) ] b p(x, t) dt = f(x) dxdt R R t b ( = f (x)µ(x, t) + f (x)σ 2 (x, t)/2 ) p(x, t)dxdt. R (18) { Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 11 / 25 o

12 Integrting the terms on the right-hnd side by prts nd using the fct tht f is rbitrty to get the Fokker-Plnck PDE p t + ( ) ( µ(x, t)p(x, t) 2 σ 2 ) x x 2 2 p(x, t) = 0. (19) { For vrible drift nd voltility, the governing PDE is in conservtive form. Retining conservtion is of crucil importnce in mny pplictions (e.g., subsurfce porous medium flow nd trnsport, especilly when the problem hs high uncertinty). The vrible σ is in the different plce from tht in the Fickin diffusion PDE. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 12 / 25 o

13 Frctionl clculus hs history lmost s long s its integer cousin Prehistoricl development Frctionl clculus stemmed from question by L Hopitl (1695) to Leibniz on the mening of dn y dx for n = 1/2. Leibniz s reply n (Sept ):... This is n pprent prdox from which, one dy, useful consequences will be drwn.... Euler observed tht the differentition formul d n x α dx = α(α 1) (α n + Γ(α + 1) n 1)xα n = Γ(α n + 1) xα n hs mening for non-integer n (1738). Lplce proposed the ide of non-integer order differentition by mens of n integrl (1812). Fourier suggested some integrl representtion of frctionl differentition (1822). Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 13 / 25 o

14 Frctionl clculus relly begn with Abel nd Liouville Abel solved the integrl eqution (1823) φ(t)(x t) µ dt = f(x), x >, 0 < µ < 1 Liouville mde the mjor contribution to the theory ( ) D α f(x) = c k α k e kx, for f(x) = c k e kx. k=0 He proposed D α ( f(x) := lim α h f)(x) h 0 h but didn t pursue it. α Riemnn cme up with tody s frctionl integrtion formul in 1847 (when still student, but published in 1876 ten yers fter his deth). Grünwld (1867) nd Letnikov (1868) introduced the definition D α f(x) := lim h 0 ( α hf)(x) h α. k=0 Letnikov proved tht this definition coincides with Riemnn s. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 14 / 25 o

15 Frctionl integrl s n extension of iterted integrls For ny n N, the iterted integrls cn be expressed by I 1 xf(x) := = I n x f(x) := = x z f(y)dy, I 2 xf(x) := (I n 1 y f(y)dydz = z f ) (z)dz = y z ( Iz 1 f ) (z)dz f(y)dzdy = (z y) n 2 f(y)dzdy = 1 (n 2)! Γ(n) (z y) n 2 f(y)dydz (n 2)! (x y)f(y)dy, (x y) n 1 f(y)dy. Here the Gmm function Γ(β) := 0 e t t β 1 dt nd Γ(n) = (n 1)!. For ny β R +, define the left nd right frctionl integrls s Ix β f(x) := 1 Γ(β) xi β 1 b f(x) := Γ(β) b x (x y) β 1 f(y)dy, (y x) β 1 f(y)dy. (20) { Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 15 / 25 o

16 Riemnn-Liouville nd Cputo frctionl derivtives The Riemnn-Liouville frctionl derivtives of order α = n β, 0 < β < 1 RL Dx α f(x) := D n Ix β f(x) = 1 (x y) β 1 f(y)dy, Γ(β) dx n RL x Db α f(x) := ( 1) n D n xi β ( 1)n d n b b f(x) = (y x) β 1 f(y)dy. Γ(β) dx n d n x (21) { The Cputo frctionl derivtives of order α = n β, 0 < β < 1 C Dx α f(x) := Ix β D n f(x) = 1 Γ(β) C x Db α f(x) := ( 1) n xi β b Dn f(x) = ( 1)n Γ(β) (x y) β 1 f (n) (y)dy, b x (y x) β 1 f (n) (y)dy. (22) { Frctionl derivtives defined vi Fourier trnsform D α x f(x) := F 1[ (ik) α ˆf(k) ], xd α f(x) := F 1[ ( ik) α ˆf(k) ]. (23) { Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 16 / 25 o

17 Grünwld-Letnikov frctionl derivtives Integer-order derivtives cn be expressed s limit of difference quotients f 1 1 [ ] (x) = lim (I Bε)f(x) = lim f(x) f(x ε) ε 0 ε ε 0 ε f (n) 1 (x) = lim ε 0 ε (I 1 n (24) { n Bε)n f(x) = lim g (n) ε 0 ε n k f(x kε). with g (n) k k=0 := ( 1) k( n k) being the binormil coefficients. The n in ε n nd ( n k) in (24) counts for the order of the derivtive. The n in n k=0 in (24) counts for the number of summnds. If we replce n in the former by α nd n in the ltter by the number of summnds to the left boundry x =, we obtin the definition of the Grünwld-Letnikov frctionl derivtives of order α (x )/ε GL Dx α 1 f(x) := lim ε 0 + ε α k=0 g (α) k GL x Db α ( 1) α (b x)/ε f(x) := lim ε 0 + ε α k=0 f(x kε), g (α) k f(x + kε). (25) { Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 17 / 25 o

18 Reltionship mong the different definitions of frctionl derivtives Under pproprite smoothness ssumptions, the Riemnn-Liouville frctionl derivtives nd Grünwld-Letnikov frctionl derivtives coincide GL Dx α f(x) = RL Dx α f(x), GL x Db α f(x) = RL x Db α f(x). (26) { The Riemnn-Liouville frctionl derivtives nd the Cputo frctionl derivtives differ by singulr boundry terms. For exmple, for 0 < α < 1, C 0 Dx α f(x) = RL 0 Dx α f(x) f(0)x α Γ(1 α). (27) { All the three frctionl derivtives (with = nd b = ) coincide for rpidly decying f on R nd equl to those defined by Fourier trnsforms (Multidimensionl cses much subtle). Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 18 / 25 o

19 Exmple: Let f(x) = 1 for x > 0 Then for 0 < α < 1, C 0 D α x f(x) = 0 but 1 d Γ(1 α) dx = d x 1 α dx Γ(2 α) = x α RL 0 Dx α f(x) := D 0Ix 1 α f(x) = This is consistent with (27). (x y) α dy 0 (28) { Γ(1 α) 0. Let f(s) L[f](s) := 0 e st f(t)dt be the LT of f. It is shown tht L [ C 0 D α t f(t)] = s α f(s) s α 1 f(0), L [ RL 0 D α t f(t)] = s α f(s), 0 < α < 1. (29) { L [ C 0 D α t f(t)] resembles tht of f, nd hs been used in time FPDE. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 19 / 25 o

20 Anomlous (super- or sub-) diffusion It ws found tht the dispersive trnsport of electrons in opertion of photocopiers nd lser printers could not be modeled properly by the clssicl Fickin diffusion PDE (Scher & Montroll 1975). Chrges moving in medi get trpped by locl imperfections nd then get relesed due to therml fluctutions. In groundwter contminnt trnsport, remedition is not so effective s predicted by the integer-order dvection-diffusion PDEs The contminnt in groundwter gets trpped to low pemebility zone nd gets relesed when the contminnt is clened. Einstein nd Person s ssumptions re violted in these processes These ssumptions hold for homogeneous medium, but fil for heterogeneous medium. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 20 / 25 o

21 The current modeling of trnsport process in heterogeneous medi is to use integer-order PDEs (vlid for homogeneous medium), to twek free prmeters tht multiply pre-set integer-order PDEs. Field tests show tht contminnt plumes often exhibit power-lw decying til in heterogeneous medi, integer-order PDE model, chrcterized by n exponentilly decying til, struggles vrible coefficient fit (of the dt t ech loction), FPDE model, chrcterized by power-lw decying til, cn fit ll the dt with constnt coefficient. Mny nomlous diffusion processes were found in vrious disciplines signling of biologicl cells, nomlous electrodiffusion in nerve cells forging behvior of nimls, electrochemistry, physics, finnce fluid nd continuum mechnics, viscoelstic nd viscoplstic flow Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 21 / 25 o

22 Derivtion of frctionl PDE A frctionl Fick s lw ssumes tht the underlying prticles hve globl jumps, i.e., δc(x, t) is incresed by n mount of g (α 1) k c(x kδx, t), i.e., q(x, t) = K α 1 c(x, t) x α 1, 1 < α < 2. (30) { Here the frctionl derivtives re Grünwld-Letnikov type. Since g (α 1) k decy like O(k α ), the prticle jumps hve hevy til. Inserting (30) into (11) yields spce FPDE c t = q x = ( K α 1 c(x, t) ) = K α c(x, t). (31) { x x α 1 x α The second equl sign holds for constnt K. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 22 / 25 o

23 Derivtion of frctionl PDEs In nomlous diffusion prticle s motion my hve very different witing times/jump sizes. Clssicl rndom wlk does not pply. Let X 1, X 2,... be sequence of iid rndom jumps of prticle with E[X i ] = 0 nd P[X > x] = Cx α where C > 0 nd 1 < α < 2. E[X p ] = α/(α p) for 0 < p < α or for p α. Centrl limit theorem (Fickin diffusion or SDE by Brownin motion) fils to pply. The FT of different scling of S ct yields E [ e ikc 1/α S ] [ ct = 1 + (ik)α + O ( c 2/α)] ct e t(ik) α. (32) { c Lévy s continuity theorem concludes tht properly scled S ct converges to n α stble Lévy process Z t e t(ik)α = E [ e ikz ] t = ˆp(k, t) = e ikx p(x, t)dx, i.e., c 1/α S ct Z t. R (33) { Unlike Gussin cse, there is no nlyticl expression for p(x, t) now. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 23 / 25 o

24 The Person s viewpoint gives rise to n SDE driven by n Lévy process dx t = µdt + σdl t. (34) { Einstein s pproch: Note tht ˆp(k, t) = e t(ik)α solves dˆp dt = (ik)α ˆp = α p x α (35) { which inverts to p t = α p x. (36) { α The PDF of finding prticle somewhere in spce stisfies (spce-frctionl) PDE, which decys lgebriclly O(x (α+1) ). This justifies why FPDEs model trnsport processes exhibiting nomlous diffusion, long-rnge time memory or spce interctions more ccurtely thn clssicl integer-order PDEs. Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 24 / 25 o

25 Thnk You for Your Attention! Hong Wng, University of South Crolin (Deprtment FPDE: of introduction Mthemtics, nduniversity model of South Crolin IBM April Visiting 12, 2018 Fellow, Division 25 / 25 o