Stability and convergence of the Crank-Nicolson scheme for a class of variable-coefficient tempered fractional diffusion equations

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1 Qu and Lang Advancs n Dffrnc Equatons 07 07:08 DOI 0.86/s RESEARCH Opn Accss Stalty and convrgnc of th Cran-Ncolson sch for a class of varal-cofficnt tprd fractonal dffuson quatons W Qu, and Yong Lang* * Corrspondnc: ylang@ust.du.o Stat Ky Laoratory of Qualty Rsarch n Chns Mdcns & Faculty of Inforaton Tchnology, Macau Unvrsty of Scnc and Tchnology, Avnda Wa Long, Tapa, Macau, , Chna Full lst of author nforaton s avalal at th nd of th artcl Astract A Cran-Ncolson sch catrng to solvng ntal-oundary valu prols of a class of varal-cofficnt tprd fractonal dffuson quatons s proposd. It s shown through thortcal analyss that th sch s uncondtonally stal and th convrgnc rat wth rspct to th spac and t stp s Oh + τ undr a crtan condton. Nurcal xprnts ar provdd to vrfy th ffctvnss and accuracy of th sch. MSC: 6A33; 65L; 65L0; 65N5 Kywords: Cran-Ncolson sch; tprd fractonal dffuson quatons; uncondtonally stal; convrgnc; varal cofficnts Introducton Ths papr s concrnd wth nurcal thods for solvng th followng ntaloundary valu prol of tprd fractonal dffuson quatons tprd-fdes: ux, t = dx κ a Dxα,λ + κ x Dα,λ ux, t + f x, t, t ua, t =, ux, = u x, u, t =, t [, T], x [a, ], whr f x, t s th sourc tr, dx s th dffuson cofficnt functons, th paratrs κ, κ ar swd paratrs that control th as of th dsprson, s Bnson t al. [ ], and λ s non-ngatv. Hr, th a Dxα,λ ux, t and x Dα,λ ux, t rprsnt th lft and rght Rann-Louvll tprd fractonal drvatvs of th functon ux, t wth ordr α < α <, rspctvly, dfind y s Baura and Mrschart [ ] α,λ α,λ α x ux λα ux a Dx ux = a Dx ux αλ Th Authors 07. Ths artcl s dstrutd undr th trs of th Cratv Coons Attruton 4.0 Intrnatonal Lcns whch prts unrstrctd us, dstruton, and rproducton n any du, provdd you gv approprat crdt to th orgnal authors and th sourc, provd a ln to th Cratv Coons lcns, and ndcat f changs wr ad.

2 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag of and xd α,λ ux= xd α,λ ux+αλα x ux λ α ux, whr and ad α,λ x ux= λx ad α x λx ux = λx n Ɣn α x n xd α,λ ux=λx xd α λx ux = n λx n Ɣn α x n x a x λξ uξ dξ x ξ α n+ λξ uξ dξ, ξ x α n+ whr Ɣ dnots th gaa functon, and x s th frst ordr partal dffrntal oprator wth rspct to x.truly,whnλ = 0, t rducs to th αth ordr lft and rght Rann- Louvll fractonal drvatvs of ux, rspctvly, thn th aov quaton rducsto fractonal dffuson quatons FDEs. Fro th xstng ltratur, tprd-fdes s an xponntally-tprd xtnson of FDEs, whch has provn to an xcllnt tool n capturng so rar or xtr vnts n gophyscs [ 4] andfnanc[5, 6]. For th FDEs prols, thr ar a varty of nurcal schs, and thr fast algorths dvlopd xtnsvly n th past dcads. W rfr radrs to [7 6] and th rfrncs thrn for th rcnt progrss n ths prols. In rcnt yars, Dng s group hav furthr drvd so hgh ordr dffrnc approxatons for th lft and rght Rann-Louvll tprd fractonal drvatvs n [7], and thr rsults hav n an ntrst n th nurcal sulaton of th tprd fractonal Blac-Schols quaton for Europan doul arrr opton [8]. Morovr, th tprd fractonal dffuson odls ar also usd to sulat xponntal tprng of th powr-law jup lngth of th contnuous t rando wal and th For-Planc quatonofthnw stochastc procss,s [9, 0]. Howvr, n ths cass, th dffuson coffcnts ar usually constants, fw paprs hav focusd on th tprd fractonal dffuson odl wth varal coffcnts. Thrfor, t s th a of ths papr to drv a class of varal-coffcnt tprd fractonal dffuson odls. Th papr s organzd as follows. In Scton, w apply th Cran-Ncolson dscrtzaton for th tprd-fdes, and th dsrd ordr n spac and t s otand. In Scton 3, t s shown that th thod s uncondtonally stal and convrgnt. Nurcal xapls ar prsntd n Scton 4 to vrfy our thortcal analyss. Fnally, Scton 5 prsnts th concluson. Th Cran-Ncolson dscrtzaton for th tprd fractonal dffuson quatons To dvlop th Cran-Ncolson sch for prol, w lt h = a N+ and τ = T th M spac stp and t stp rspctvly, whr N, M ar so gvn postv ntgrs. Thn th spatal and tporal parttons can dfnd y x = a + h, = 0,,...,N +and t = τ, =0,,...,M. Nxt, for convnnc, w ntroduc th followng notaton: t + = t + t +, f + = f x, t +, d = dx.

3 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 3 of Not that L and Dng [7] hav stalshd a scond ordr tprd-wsgd oprators adx α,λ and x D α,λ to approxat th lft and rght Rann-Louvll tprd fractonal drvatvs of uxatpontx, rspctvly, whch gvs and adx α,λ ux αλ α ux = a Dx α,λ ux +O h = h α + xd α,λ ux αλ α ux = x D α,λ ux +O h g α ux + h α φλux +O h = N + g α h α ux + h φλux +O h, α whr φλ=γ hλ + γ + γ 3 hλ hλ α,andthwghtsg α ar gvn as g α = γ w α 0 hλ, =0, γ w α + γ w α 0, =, γ w α + γ w α + γ 3w α hλ,. Hr th wghts w α 0 =andw α = α can valuatd rcursvly,.., w α = +α wα for all, s[7]. It s worth ntonng that th paratrs γ =,,3 satsfy th lnar syst γ + γ + γ 3 =, 3 γ γ 3 = α/. Not that thr ar nfnt any solutons of lnar syst 3. Howvr, th ran th coffcnt atrx s so that a soluton can unquly dtrnd whn on of γ s provdd. Thy can collctd y th followng thr sts: { S α γ, γ, γ 3 = γ s gvn, γ = +α γ, γ 3 = γ α } ; or or { S α γ, γ, γ 3 = γ = +α 4 γ, γ s gvn, γ 3 = α γ } ; 4 { S3 α γ, γ, γ 3 = γ = α + γ 3, γ = α } γ 3, γ 3 s gvn. W rar that th aov approach usd to approxat th lft and rght Rann- Louvll tprd fractonal drvatvs s sply a follow-up usd y L and Dng [7].

4 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 4 of Our an contrutons to ths papr ar to drv a Cran-Ncolson sch for a class of varal-coffcnt tprd-fdes and to llustrat th stalty and convrgnc of ths sch. Lt u ux, t, w can thrfor consdr th Cran-Ncolson tchnqu to th tprd-fdes wth th tprd-wsgd approxatons to th tprd fractonal drvatvs and th scond ordr cntral dffrnc approxaton to u,s[7, ]. Thn, x nglctng th truncaton rrors, w drv th followng scond ordr fnt dffrnc sch: u + u τ = d κ h α + g α u + + κ h α + g α u κ N + h α + κ N + g α h α u + κ + κ u + d φλ + u + h α + d αλ α κ κ g α u + u + + u + + u + u + f +, 4 h h for =,,...,N and =,,...,M.Wdnotε = τ h α 0, η = αλα τ 0. In a atrx 4h for, t s gvn as I Au + =I + Au + τf, 5 n whch A = D εκ G + εκ G T + ηκ κ W, 6 whr D = dagd, d,...,d N, u =[u, u,...,u N ]T, f + =[f +, f +,...,f + N ] T,th trdagonal atrx W = trdag, 0,, I s th dntty atrx, and G [g,j ]sann N Topltz atrx dfnd as 0, j +, g α 0, j = +, g,j = g α φλ, j =, g α, j =, g α j+, j. 3 Stalty and convrgnc of th Cran-Ncolson fnt dffrnc sch To show th uncondtonal stalty and convrgnc of th Cran-Ncolson fnt dffrnc sch 4, th followng rsults gvn n [7, 4] ar rqurd. La L and Dng [7] Lt S th soluton st of lnar syst 3. If <α <, λ 0, γ, γ, γ 3 S, and. ax{ α +3α 4 α +3α+, α +3α α +3α+4 } < γ < 3α +3α α +3α+ ; or 7

5 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 5 of. α 4α +3α++4 α +3α+ 3. ax{ αα +α 8 α +3α+ w hav and < γ < n{ α α +3α+4+6 α +3α+4, αα +α } < γ α +3α+4 3 < αα +α 3 α +3α+ g α <0, g α + g α 0 >0, g α >0, 3, g α = γ hλ + γ + γ 3 hλ hλ α., α 6α +3α++48 }; or α +3α+, La Quartron t al. [] If A C n n, lt H = A+A th hrtan part of A, thn for any gnvalu λ of A, th ralpart RλA<0 satsfs λ n H R λa λ ax H, whr λ n H and λ ax H ar th nu and axu of th gnvalus of H, rspctvly. La 3 Quartron t al. [] A ral atrx A of ordr N s ngatv dfnt f and only f ts sytrc part H = A+A s ngatv dfnt; H s ngatv dfnt f and only f th gnvalus of H ar ngatv. Dfnton ThnurcalrangofatrxA s dfnd as WA { v Av : v C n, v v = }. La 4 HornandJohnson[3] Lt A, B C n n, f B s postv dfnt, thn σ AB WAWB, whr σ AB s th spctru of AB. La 5 Quartron and Vall [4]; Dscrt Gronwall s nqualty Assu that { n } and {p n } ar non-ngatv squncs, and th squnc {φ n } satsfs n n φ 0 c 0, φ n c 0 + p l + l φ l, n, l=0 l=0 whr c 0 0. Thn th squnc {φ n } satsfs n φ n c 0 + p l xp n l=0 l=0 l, n. Nxt,wanalyzthstaltyoffntdffrncsch5, w hav th followng thor. Thor For all α,, f th paratrs γ, γ, and γ 3 satsfy th hypothss gvn n La, thn th dffrnc sch 5 s uncondtonally stal.

6 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 6 of Proof Lt M := εκ G + εκ G T + ηκ κ W,thnA = DM,notthat M + M T = ε κ + κ G + G T + η κ κ W + W T = ε κ + κ G + G T. Fro [7], w not that th atrx G s ngatv dfnt. Snc κ 0, κ 0andε 0, thn w hav M+MT s also ngatv dfnt. Thans to La 3,thatrxM s ngatv dfnt. Dnot y μa an gnvalu of A = DM.SncD s non-ngatv, con- ng th ngatv dfnt proprts of M and La 4,wotanthat σ A=σ DM WDWM { δ C Rδ<0 }. Hnc RμA < 0, whch pls that th nqualty +μa μa holds for any α,. Thrfor, th nurcal sch 5 s uncondtonally stal. In th squl, w consdr th convrgnc of th nurcal sch 4. Lt V h = { v v =v 0, v, v,...,v N, v N+, v 0 = v N+ =0 } spac grd functons dfnd on {x = h} N+ =0. For any u, v V h,wdfn u, v=h N u v = andthcorrspondngdscrtl nor v = v, v= N h v. = Th followng thor dscrs th convrgnc of th Cran-Ncolson thod whn A = DM s ngatv dfnt. Thor Lt U th xact soluton of tprd-fdes and u th soluton of dscrt Eq.4 at sh pont x, t, rspctvly, whr =,,...,Nand=,,...,M. For all α,, f A = DM s ngatv dfnt, and th paratrs γ, γ, and γ 3 satsfy th hypothss gvn n La, w hav U u C h + τ, =,,...,M, whr C s a postv constant, and U =[U, U,...,U N ]T.

7 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 7 of Proof For =,,...,N and =0,,,...,M, th rror satsfs th followng quaton: + τd [ κa D α,λ x + + κ x D α,λ + + αλ α κ κ x + ] = + τd [ κa Dx α,λ + κ x D α,λ + αλ α κ κ x + τo h + τ, 8 ] whr = U u,and 0 =0, 0 = N+ =0for =,,...,N and =,...,M. Dnot E =[,,..., N ]T, thn th atrx for of Eq. 8canwrttnas I AE + =I + AE + τρ, E 0 =0, =0,,...,M. Through arrangnt, w otan E + E A E + + E = τρ, E 0 =0, =0,,...,M, 9 whr ρ =[ρ, ρ,...,ρ N ]T wth ρ = Oh + τ. Multplyng 9yh and tang E + + E T on oth sds, w hav h E + + E T E + E h E + + E T A E + + E = h E + + E T τρ. 0 Rcall that th atrx A s ngatv dfnt, thn E + + E T A E + + E <0, and fro Eq. 0whav h N = + τh N = + + ρ. Sung up for all 0 =,,...,Mladsto h N N τh + + ρ = = =0 N = τh ρ + ρ N + τh τh + τh = = N = = N = + τh + τh = = = ρ N ρ + ρ N ρ. =

8 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 8 of Snc ρ = Oh + τ, t ans that ρ Ch + τ forc > 0. It follows fro La 5 that E C h + τ + τ E = C h + τ xp τ = C h + τ T. Thrfor, E = E = U u Ch + τ =,,...,M, whch s th dsrd rsult. 4 Soxapls In ths scton, two nurcal xapls ar gvn to show th ffctvnss and convrgnc ordrs of th proposd schs. In th tst, w coput th axu nor rrors twn th xact and th nurcal solutons at th last t stp y h, τ= ax ux, t M u M N, whr ux, t M s th xact soluton and u M s th nurcal soluton wth th sh stp szs h and τ at th grd pont x, t M. Th ordr n th followng tals s calculatd y Ordr = log h,τ/h, τ. Exapl Consdr th followng two-sdd tprd fractonal dffuson prol: ux, t t = x 0D α,λ x + x D α,λ ux, t+f x, t, u0, t=0, u, t=0, t [0, ], ux,0= λx x 4 x 4, x [0, ]. Th xact soluton s gvn y ux, t= t λx x 4 x 4. Thn th sourc tr s gvn as [ 4 4 Ɣ5 + f x, t t x λx 4 x 4 + Ɣ5 + α x5+ α =0 ] λ α x 5 x 4 50 x λx t λ j j=0 j! 4 4 Ɣ5 + + j =0 xj+4+ α Ɣ5 + + j α Exapl Ths xapl s a odfcaton of Exapl. W rplac th dffuson coffcnt functon dxyx, thn th sourc tr s gvn as [ 4 4 Ɣ5 + f x, t t x λx 4 x 4 + Ɣ5 + α x6+ α =0 ] λ α x 6 x 4 50 x λx t λ j j=0 j! 4 4 Ɣ5 + + j =0. xj+4+ α Ɣ5 + + j α.

9 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 9 of Tal Th axu rrors at t t = and convrgnc ordrs n spatal and tporal drctons wth h = τ = N+ for Exapl wth dffrnt α,andλ =.0,thparatrs γ = 0.7, 0.75, 0.80, rspctvly, and γ and γ 3 ar slctd n th st S α γ, γ, γ 3 α N + γ =0.70 γ =0.75 γ =0.80 h, τ Ordr h, τ Ordr h, τ Ordr Tal Th axu rrors at t t = and convrgnc ordrs n spatal and tporal drctons wth h = τ = N+ for Exapl wth dffrnt α,andλ =.0,thparatrs γ = 0.7, 0.75, 0.80, rspctvly, and γ and γ 3 ar slctd n th st S α γ, γ, γ 3 α N + γ =0.70 γ =0.75 γ =0.80 h, τ Ordr h, τ Ordr h, τ Ordr Othr data ar th sa as thos n Exapl. Fro Tals and, w can osrv th scond ordr convrgnc rat n oth spatal and tporal drctons for dffrnt α n L nor, whch s consstnt wth our thortcal analyss. It s rard that w nurcally tst th gnvalus of atrx DM + M T D n Exapls and, rspctvly. W fnd that all gnvalus of th atrx DM + M T D n Exapl ar ngatv, whch pls that th DM s ngatv dfnt, hnc our assupton n th convrgnc analyss s vald, and th nurcal rsults ar consstnt wth Thor. Howvr, whn th assupton n Thor s not satsfd, s Exapl, w also gt th dsrd scond ordr convrgnc rat n oth spatal and tporal drctons.

10 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag 0 of Fgur Th rror curv fgurs wth h = τ = γ =0.8andλ =.0 for Exapl. 56 lft and h = τ = 5 rght at t M =whnα =.5, Fgur Th rror curv fgurs wth h = τ = γ =0.8andλ =.0 for Exapl. 56 lft and h = τ = 5 rght at t M =whnα =.8, In Fgurs and, w plot th curv fgurs of th approxatng rrors ux, t M u M wth dffrnt sh szs at th fnal t stp t M = va a t-archng procdur, whr γ =0.8andλ =.0whnα =.5 for Exapl and α =.8 for Exapl, rspctvly. Ths fgurs show that th axunor rror, dfnd n Eq., cos rlatvly sallr as th sh sz cos sallr, whch provds th valdaton of our rsults onc agan. 5 Concluson In ths papr, th Cran-Ncolson thod s proposd for solvng a class of varalcoffcnt tprd-fdes. Th thod s provn to uncondtonally stal and convrgnt undr a crtan condton wth rat Oh +τ. Nurcal xapls show good agrnt wth th thortcal analyss.

11 Qu and LangAdvancs n Dffrnc Equatons 07 07:08 Pag of Coptng ntrsts Th authors dclar that thy hav no coptng ntrsts. Authors contrutons All authors rad and approvd th fnal anuscrpt. Author dtals Stat Ky Laoratory of Qualty Rsarch n Chns Mdcns & Faculty of Inforaton Tchnology, Macau Unvrsty of Scnc and Tchnology, Avnda Wa Long, Tapa, Macau, , Chna. School of Mathatcs and Statstcs, Shaoguan Unvrsty, Shaoguan, 5005, Chna. Acnowldgnts W would l to than th anonyous rvwrs for provdng us wth constructv conts and suggstons. Ths wor s supportd y th Macau Scnc and Tchnology Dvlopnt Funds Grant No. 099/03/A3 fro th Macau Spcal Adnstratv Rgon of th Popl s Rpulc of Chna, th Natonal Natural Scnc Foundaton of Chna undr Grant No , and th Natural Scnc Foundaton of Shaoguan Unvrsty undr Grant No. SY04KJ07. Pulshr s Not Sprngr Natur rans nutral wth rgard to jursdctonal clas n pulshd aps and nsttutonal afflatons. Rcvd: 4 Dcr 06 Accptd: March 07 Rfrncs. Bnson, DA, Whatcraft, SW, Mrschart, MM: Th fractonal-ordr govrnng quaton of Lévy oton. Watr Rsour. Rs. 36, Baura, B, Mrschart, MM: Tprd stal Lévy oton and transnt supr-dffuson. J. Coput. Appl. Math. 33, Mrschart, MM, Zhang, Y, Baur, B: Tprd anoalous dffusons n htrognous systs. Gophys. Rs. Ltt. 35, L7403-L Mtzlr, R, Klaftr, J: Th rstaurant at th nd of th rando wal: rcnt dvlopnts n th dscrpton of anoalous transport y fractonal dynacs. J. Phys. A 37, R6-R Carr, P, Gan, H, Madan, DB, Yor, M: Stochastc volatlty for Lévy procsss. Math. Fnanc 3, Wang, WF, Chn, X, Dng, D, L, SL: Crculant prcondtonng tchnqu for arrr optons prcng undr fractonal dffuson odls. Int. J. Coput. Math. 9, Mrschart, MM, Tadjran, C: Fnt dffrnc approxatons for fractonal advcton-dsprson flow quatons. J. Coput. Appl. Math. 7, Sousa, E, L, C: A wghtd fnt dffrnc thod for th fractonal dffuson quaton asd on th Rann-Louvll drvatv. Appl. Nur. Math. 90, Pang, HK, Sun, HW: Multgrd thod for fractonal dffuson quatons. J. Coput. Phys. 3, Pan, JY, K, RH, Ng, MK, Sun, HW: Prcondtonng tchnqus for dagonal-ts-topltz atrcs n fractonal dffuson quatons. SIAM J. Sc. Coput. 36, A698-A L, SL, Sun, HW: A crculant prcondtonr for fractonal dffuson quatons. J. Coput. Phys. 4, Qu, W, L, SL, Vong, SW: Crculant and sw-crculant splttng traton for fractonal advcton-dffuson quatons. Int. J. Coput. Math. 9, Wang, H, Basu, TS: A fast fnt dffrnc thod for two-dnsonal spac-fractonal dffuson quatons. SIAM J. Sc. Coput. 34, A444-A Ln, FR, Yang, SW, Jn, XQ: Prcondtond tratv thods for fractonal dffuson quaton. J. Coput. Phys. 56, Chn, MH, Dng, WH: Fourth ordr accurat sch for th spac fractonal dffuson quatons. SIAM J. Nur. Anal. 53, Tan, WY, Zhou, H, Dng, WH: A class of scond ordr dffrnc approxatons for solvng spac fractonal dffuson quatons. Math. Coput. 84, L, C, Dng, WH: Hgh ordr schs for th tprd fractonal dffuson quatons. Adv. Coput. Math. 4, Zhang, H, Lu, FW, Turnr, I, Chn, S: Th nurcal sulaton of th tprd fractonal Blac-Schols quaton for Europan doul arrr opton. Appl. Math. Modl. 40, Zhng, M, Karnadas, GE: Nurcal thods for SPDEs wth tprd stal procsss. SIAM J. Sc. Coput. 37, A97-A Charaarty, Á, Mrschart, MM: Tprd stal laws as rando wal lts. Stat. Proa. Ltt. 8, Sazar, F, Mrschart, MM, Chn, J: Tprd fractonal calculus. J. Coput. Phys. 93, Quartron, A, Sacco, R, Salr, F: Nurcal Mathatcs, nd dn. Sprngr, Brln Horn, RA, Johnson, CR: Topc n Matrx Analyss. Cardg Unvrsty Prss, Cardg Quartron, A, Vall, A: Nurcal Approxaton of Partal Dffrntal Equatons. Sprngr, Brln 997