On Tempered and Substantial Fractional Calculus

Transcription

1 On Tempered nd Subsnil Frcionl Clculus Jiniong Co,2, Chngpin Li nd YngQun Chen 2, Absrc In his pper, we discuss he differences beween he empered frcionl clculus nd subsnil frcionl operors in nomlous diffusion modelling, so h people cn beer undersnd he wo frcionl operors. We rs inroduce he de niions of empered nd subsnil frcionl operors, nd hen nlyze he properies of wo de niions. A ls, we prove h he empered frcionl derivive nd subsnil derivive re equivlen under some condiions. A diffusion problem de ned by using empered derivive is lso given o illusre he slow convergence of n nomlous diffusion process. Keywords Frcionl clculus, Tempered frcionl clculus, Subsnil frcionl clculus, Anomlous diffusion I. INTRODUCTION The process of nomlous diffusion is one of common phenomen in nure, nd he coninuous ime rndom wlks (CTRWs frmework [] is useful ool o describe his phenomenon. The CTRWs re ofen governed by he wiing ime probbiliy densiy funcion (PDF nd jump lengh PDF. When he PDFs re power lw, he nomlous rnspor process cn be depiced by frcionl diffusion equions. While he PDFs re eponenilly empered power lw, hen empered nomlous diffusion models re derived in [2 8]. As Meerscher [9] poined ou, empered sble processes re he limis of rndom wlk models where he power lw probbiliy of long jumps is empered by n eponenil fcor. These rndom wlks converge o empered sble sochsic process limis, whose probbiliy densiies solve empered frcionl diffusion equions. Tempered power lw wiing imes led o empered frcionl ime derivives, which hve proven useful in geophysics. Meerscher e l. proposed empered diffusion model o cpure he slow convergence of subdiffusion [6]. Beumer nd Meerscher sudied empered sble Lévy moion in [2], hey proposed nie difference nd pricle rcking mehods o solve he empered frcionl diffusion equion wih drif. In view of he ef ciency of empered frcionl clculus in describing eponenilly empered power lw behvior nd is vrins, i hs rced mny reserchers o sudy numericl mehods o solve hese problems. Beumer nd Meerscher [2] derived nie difference nd pricle rcking mehods. Cre e l. [] presened generl nie difference scheme o numericlly solve Blck-Meron-Scholes model wih empered frcionl derivives. Momoni nd Deprmen of Mhemics, Shnghi Universiy, Shnghi 2444, Chin (lcp@shu.edu.cn 2 School of Engineering, Universiy of Cliforni, Merced, CA 95343, USA (yqchen@ieee.org, yngqun.chen@ucmerced.edu Corresponding uhor. Tel. ( ; F: ( Momoni [] compre he numericl soluions of hree kinds of frcionl Blck-Meron-Scholes equions wih empered frcionl derivives. Recenly, high order numericl scheme for empered diffusion equion is presened in [2]. However, numericl lgorihms for solving hese problems re limied. As n eension of he concep of CTRWs o phse spce, Friedrich e l. derived new frcionl Krmers Fokker Plnck equion [3], which involved frcionl subsnil derivive, i hs imporn nonlocl couplings in boh ime nd spce. In 2, bsed on he CTRW models wih coupling PDFs, Crmi nd Brki obined deerminisic equion by using frcionl subsnil derivive [4]. The properies nd numericl discreizions of he frcionl subsnil operors re recenly discussed in [5]. To our bes knowledge, wheher he empered frcionl operors [3] or he frcionl subsnil operors [3] is origined from he empered funcion spce. Moived by his, we ry o le people know he relionship of hese wo frcionl operors. The work is orgnized s follows. In Secion II, we inroduce hree common used de niions of frcionl inegrls nd derivives. Two clsses of frcionl operors clled empered nd subsnil operors re inroduced in Secion III. Numericl eperimen is crried ou o show he effeciveness of empered model in describing eponenilly empered powerlw behvior. Finlly, we conclude he pper in he ls secion. II. PRELIMINARIES In his secion, we give some preliminries bou frcionl clculus. There re sverl differen de niions of frcionl derivives, bu he mos frequenly used re he following hree de niions, i.e. Grünwld Lenikov derivive, he Riemnn Liouville derivive nd he Cpuo derivive [6 2]. We inroduce he de niions in he following wy. De niion II.. The frcionl inegrl of order α> for funcion f( is de ned by D α f( Γ(α ( s α f(sds, ( where Γ( is he Euler s funcion. De niion II.2. The lef nd righ Grünwld Lenikov derivives of order α> of f( re de ned s GLD α, f( lim h Nh h α N j ( j ( α j f( jh, ( /4/$3. 24 IEEE

2 nd GLD α,b f( lim h Nhb h α N j ( j ( α j f( + jh, (3 respecively. De niion II.3. Suppose h f( be (n -imes coninuously differenible on (,, nd is n-imes derivives be inegrble on ny subinervl [,. Then he lef Riemnn- Liouville derivive of order α> of f( is de ned by [ ] RLD, α dn f( d n D (n α f( d n (4 Γ(n α d n ( s n α f(sds, nd he righ Riemnn-Liouville frcionl derivive is de ned s d n RLD,bf( α ( n Γ(n α d n (s n α f(sds, (5 respecively, where n is nonnegive ineger nd n α<n. De niion II.4. Assume h f( be (n -imes coninuously differenible on (,, nd is n-imes derivives be inegrble on ny subinervl [,. Then he lef Cpuo frcionl derivive of order α> for f( is de ned s [ ] CD,f( α D (n α f (n ( (6 ( s n α f (n (sds, Γ(n α b nd he righ Cpuo derivive is de ned by b CD,b α f( ( n (s n α f (n (sds, (7 Γ(n α respecively, where n is nonnegive ineger nd n < α<n. III. DEFINITIONS OF TEMPERED AND SUBSTANTIAL FRACTIONAL CALCULUS In his secion, we will inroduce he de niions nd noions of empered nd subsnil frcionl operors. Then we discuss he relions beween hem. A. De niions of he empered frcionl operors De niion III.. [3, 2] Suppose h f( is piecewise coninuous on [, nd inegrble on ny nie subinervl of [,, α>, λ. Then The lef Riemnn-Liouville empered frcionl inegrl of order α of funcion f( is de ned by RLD α,λ, f( Γ(α e λ( τ ( τ α f(τdτ. 2 The righ Riemnn-Liouville empered frcionl inegrl of order α for f( is de ned s b RLD α,λ b, f( e λ(τ (τ α f(τdτ. Γ(α De niion III.2. [3, 2] Le f( be (n -imes coninuously differenible on (,, nd is n-imes derivives be inegrble on ny subinervl [,. Then he lef empered frcionl derivive of order α> for given funcion f( is de ned s, f( (e λ RLD, α eλ f( e λ d n Γ(n α d n ( τ n α e λτ f(τdτ, (8 nd he righ empered frcionl derivive is de ned s,b f( (eλ RLD,be α λ f( ( n e λ d n b Γ(n α d n (τ n α e λτ f(τdτ, (9 respecively, where n is nonnegive ineger nd n α<n. Remrk III.. If λ, he lef nd righ Riemnn-Liouville empered frcionl derivives reduce o he lef nd righ Riemnn-Liouville frcionl derivives de ned in De niion II.3. Remrk III.2. The vrins of he lef nd righ Riemnn- Liouville empered frcionl derivives re de ned s [2, 2, 22] {, f( λ α f(, <α<,, f(, f( αλα f( λ α f(, <α<2, ( nd,b f( {,b f( λα f(, <α<,,b f( αλα f( λ α f(, <α<2, ( respecively, where represens he clssicl rs derivive operor. B. Subsnil frcionl operors De niion III.3. [3, 5] Le f( be piecewise coninuous on [, nd inegrble on ny nie subinervl of [,. Then he frcionl subsnil inegrl of order α> for f( is de ned by Ds α f( e λ( τ ( τ α f(τdτ, Γ(α where λ cn be consn or funcion no reled o. De niion III.4. [3, 5] Suppose h α>, f( be (n - imes coninuously differenible on (,, nd is n-imes derivives be inegrble on ny subinervl [,. Then he

3 subsnil frcionl derivive of order α>for f( is de ned by ( Ds α f( Ds n Ds (n α f(, where Ds n ( d d + λ n. Remrk III.3. If λ, i is cler h De niion III. is equivlen o De niion III.3. Theorem III.. In De niion III.4, if λ is posiive consn, hen he empered nd subsnil derivives re equivlen. Proof: Wihou loss of generliy, we ke <α< in De niion (8 nd De niion III.4, hen he lef Riemnn- Liouville empered derivive is, f( e λ d Γ( α d e λτ ( τ α f(τdτ, (2 nd he subsnil derivive becomes ( Ds α d f( Γ( α d + λ e λ( τ ( τ α f(τdτ d (e λ e λτ ( τ α f(τdτ Γ( α d λ + e λ( τ ( τ α f(τdτ Γ( α λ e λ( τ ( τ α f(τdτ Γ( α + e λ d e λτ ( τ α f(τdτ Γ( α d λ + e λ( τ ( τ α f(τdτ Γ( α e λ d e λτ ( τ α f(τdτ. Γ( α d (3 The proof ends. De niion III.5. [23, 24] The lef nd righ generlized frcionl inegrl of order α > of funcion f( wih respec o noher funcion z( nd weigh w( re de ned in he following wy (I α,+;[z;w] f ( [w(] Γ(α w(τz (τf(τ dτ, (4 [z( z(τ] α nd (I b, ;[z;w] α f ( [w(] b w(τz (τf(τ dτ, (5 Γ(α [z(τ z(] α respecively. Remrk III.4. If we ke z(, w( e λ, hen he lef nd righ generlized inegrls reduce o he lef nd righ empered frcionl inegrls. IV. NUMERICAL SIMULATION In his secion, bsed on he discussion of empered nd subsnil derivives, we use nie difference mehod o solve empered diffusion problem. Emple IV.. Solve he following empered frcionl diffusion equion u(, RL D.5,λ, u(, +f(,, <<, <<, (6 wih he iniil nd boundry condiions: u(,, u(,,u(,e λ,wheref(, 2e λ 2.5 Γ( e λ 2. The nlyicl soluion of Eq. (6 is u(, e λ For he numericl soluion of Eq. (6, le k kδ, k,, 2,..., N, i ih, i,, 2,..., M, whereδ T N nd h L M re he ime, spce seps, respecively. The rs-order derivive u(, mesh poin ( i, k cn be pproimed by he following bckwrd difference mehod u( i, k u( i, k u( i, k + O(Δ, (7 Δ hen, we use shifed Grünwld Lenikov formule o pproime frcionl subsnil derivive erm in Eq. (6 i+ Ds α u( i, k h α m g,α m u( i m+, k +O(h. (8 Le u k i be he pproime soluion of u( i, k,ndfi k f( i, k, subsiuing he Eqs. (7 nd (8 ino Eq. (6, nd denoing η K Δ h, we obin he following implici nie α difference scheme for Eq. (6 u k i u k i + η i+ m gm,α u k i m+ +Δf k i, i M, k N u i, i M, u k φ ( k, u k M φ 2 ( k, k N. (9 Theorem IV.. The locl runcion error of difference scheme (9 is O(τ + h. Proof: According o (6, (7 nd (8, we de ne he locl runcion error Ri k of difference scheme (9 s below: R k i u( i, k u( i, k Δ f( i, k [ u(i, k i+ Kh α m u( ] i, k u( i, k Δ i+ + K(Ds α u( i, k h α m O(τ+KO(h O(τ + h. g,α m u( i m+, k g,α m u( i m+, k

4 The proof ends. Le λ,.5,., he nlyicl nd numericl soluions re displyed in Fig.. I cn be seen h he numericl soluions he nlyicl soluions very well. When λ, he equion (6 reduces o he Riemnn Liouville diffusion equion, Fig. ( nd (b show h soluion pek is high. For λ.5 nd λ., he soluion re ploed in Fig. (c, (d nd Fig. (e, (f, respecively. From Fig., we cn see h he pek of he soluions of empered diffusion equion becomes more nd more smooh s eponenil fcor λ increses. V. CONCLUSION In his pper, we inroduce wo clsses of frcionl operors for nomlous diffusion, nd furher discuss he properies of empered nd subsnil derivives. We obin heorem on wo de niions under some condiions. I is esy o conclude h empered nd subsnil frcionl clculus re he generlizion of frcionl clculus, nd boh of hem re specil cses of generlized frcionl clculus. Alhough subsnil derivive is equivlen o empered derivive when he prmeer λ, hey re inroduced from differen physicl bckgrounds. Mhemiclly he frcionl subsnil clculus is ime-spce coupled operor bu he empered frcionl clculus is no. However, he empered frcionl operors re he more commonly used in runced eponenil power lw descripion. ACKNOWLEDGMENT The work ws prilly suppored by he Nurl Science Foundion of Chin under Grn No. 3727, he Key Progrm of Shnghi Municipl Educion Commission under Grn No. 2ZZ84 nd Chin Scholrship Council. REFERENCES [] Rlf Mezler nd Joseph Klfer, The rndom wlk s guide o nomlous diffusion: frcionl dynmics pproch, Physics repors, vol. 339, no., pp. 77, 2. [2] Boris Beumer nd Mrk M. Meerscher, Tempered sble Lévy moion nd rnsien super-diffusion, Journl of Compuionl nd Applied Mhemics, vol. 233, no., pp , 2. [3] Álvro Cre nd Diego del Csillo-Negree, Fluid limi of he coninuous-ime rndom wlk wih generl Lévy jump disribuion funcions, Physicl Review E, vol. 76, no. 4, pp. 45, 27. [4] I. M. Sokolov, A. V. Chechkin, nd J. Klfer, Frcionl diffusion equion for power-lw-runced Lévy process, Physic A: Sisicl Mechnics nd is Applicions, vol. 336, no. 3, pp , 24. [5] A Kullberg nd D del Csillo-Negree, Trnspor in he spilly empered, frcionl Fokker Plnck equion, Journl of Physics A: Mhemicl nd Theoreicl, vol. 45, no. 25, pp. 255, 22. [6] Mrk M. Meerscher, Yong Zhng, nd Boris Beumer, Tempered nomlous diffusion in heerogeneous sysems, Geophysicl Reserch Leers, vol. 35, no. 7, 28. [7] Ariji Chkrbry nd Mrk M. Meerscher, Tempered sble lws s rndom wlk limis, Sisics & Probbiliy Leers, vol. 8, no. 8, pp , 2. [8] Dumiru Blenu, Frcionl Clculus: Models nd Numericl Mehods, vol. 3, World Scieni c, 22. [9] Frzd Sbzikr, Mrk M Meerscher, nd Jinghu Chen, Tempered frcionl clculus, Journl of Compuionl Physics, 24. [] Alvro Cre nd Diego del Csillo-Negree, Frcionl diffusion models of opion prices in mrkes wih jumps, Physic A: Sisicl Mechnics nd is Applicions, vol. 374, no. 2, pp , 27. [] O. Mrom nd E. Momoni, A comprison of numericl soluions of frcionl diffusion models in nnce, Nonliner Anlysis: Rel World Applicions, vol., no. 6, pp , 29. [2] Cn Li nd Weihu Deng, High order schemes for he empered frcionl diffusion equions, rxiv preprin rxiv:42.64, 24. [3] R. Friedrich, F. Jenko, A. Bule, nd S. Eule, Anomlous diffusion of ineril, wekly dmped pricles, Physicl review leers, vol. 96, no. 23, pp. 236, 26. [4] Shi Crmi nd Eli Brki, Frcionl Feynmn-Kc equion for wek ergodiciy breking, Physicl Review E, vol. 84, no. 6, pp. 64, 2. [5] Minghu Chen nd Weihu Deng, Discreized frcionl subsnil clculus, rxiv preprin rxiv:3.386, 23. [6] Keih B. Oldhm nd Jerome Spnier, The frcionl clculus: heory nd pplicions of differeniion nd inegrion o rbirry order, vol., Acdemic press New York, 974. [7] Igor Podlubny, Frcionl differenil equions: n inroducion o frcionl derivives, frcionl differenil equions, o mehods of heir soluion nd some of heir pplicions, vol. 98, Acdemic Press, 998. [8] Anolii Aleksndrovich Kilbs, Hri Mohn Srivsv, nd Jun J. Trujillo, Theory nd pplicions of frcionl differenil equions, vol. 24, Elsevier Science Limied, 26. [9] Chngpin Li nd Weihu Deng, Remrks on frcionl derivives, Applied Mhemics nd Compuion,vol. 87, no. 2, pp , 27. [2] Chngpin Li, Deling Qin, nd YngQun Chen, On riemnn-liouville nd cpuo derivives, Discree Dynmics in Nure nd Sociey, vol. 2, 2. [2] Chngpin Li nd Fnhi Zeng, Finie difference mehods for frcionl differenil equions, Inernionl Journl of Bifurcion nd Chos, vol. 22, no. 4, 22. [22] Mrk M. Meerscher nd Frzd Sbzikr, Tempered

5 .8.8 Numericl soluion Ec soluion (c (d.8.8 Numericl soluion Ec soluion (e Fig (b Numericl soluion Ec soluion (.6 The comprision beween ec nd numericl soluions.8.5 (f

6 frcionl Brownin moion, Sisics & Probbiliy Leers, vol. 83, no., pp , 23. [23] Virgini S Kirykov, Generlized frcionl clculus nd pplicions, CRC Press, 993. [24] Om P. Agrwl, Some generlized frcionl clculus operors nd heir pplicions in inegrl equions, Frcionl Clculus nd Applied Anlysis, vol. 5, no. 4, pp. 7 7, 22.