Weighted Average Finite Difference Methods for Fractional Reaction-Subdiffusion Equation

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1 hp://s.u.ac.h Mahacs Wghd Avag Fn Dffnc Mhods fo Faconal Racon-ubdffuson Euaon Nass Hassn WEILAM * Mohad Mabd KHADER and Mohad ADEL Dpan of Mahacs Faculy of cnc Cao nvsy Gza Egyp Dpan of Mahacs Faculy of cnc Bnha nvsy Bnha Egyp * Cospondng auho s -als: n_sla@yahoo.co ohadbd@yahoo.co Rcvd: 7 July 0 Rvsd: 8 Januay 03 Accpd: Dcb 03 Absac In hs acl a nucal sudy fo faconal acon-subdffuson uaons s noducd usng a class of fn dffnc hods. Ths hods a nsons of h ghd avag hods fo odnay non-faconal acon-subdffuson uaons. A sably analyss of h poposd hods s gvn by a cnly poposd pocdu sla o h sandad John von Nuann sably analyss. pl and accua sably con vald fo dffn dsczaon schs of h faconal dvav abay gh faco and abay od of h faconal dvav a gvn and chcd nucally. Nucal s apls fgus and copasons hav bn psnd fo clay. Kyods: Wghd avag fn dffnc appoaons faconal acon-subdffuson uaon sably analyss Inoducon In h las f yas h hav bn nsv suds of faconal od dffnal uaons FDEs du o h poan applcaons n any val aas of sach such as physcs dcn and ngnng. Moov faconal calculus suds can allo undsandng of any facal phnona hch canno b sudd by odnay ans. Th a any applcaons fo FDEs; s [-7]. Th sudd odls hav cvd a ga dal of anon such as n h flds of vscolasc aals [8] conol hoy [9] advcon and dspson of solus n naual poous o facud da [] and anoalous dffuson. Du o h dffculs n clang h ac soluons fo FDEs appoa and nucal chnus a nsvly usd s fo apl [0] and h fncs cd hn. Rcnly sval nucal hods hav bn adapd o solv faconal dffnal uaons s [-7] and h fncs cd hn. In hs scon h dfnons of Rann-Louvll and h Günald-Lnov faconal dvavs hch ll b usd la a gvn [89]. Dfnon : Th Rann-Louvll dvav of od of h funcon y s dfnd by; n d y D y = d > 0 n 0 n n d h n s h salls ng cdng and. s h Gaa funcon. If = N hn h h dvav y. abov dfnon concds h h classcal Dfnon : Th Günald-Lnov dfnon fo h faconal dvavs of od > 0 of h funcon y s dfnd by; Walala J c & Tch 04; 4:

2 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. [ ] h = l 0 h0 h D y y h h [ ] h dfnd by ans h ng pa of and a h noalzd Günald ghs hch a h =. Th Günald-Lnov dfnon s sply a gnalzaon of h odnay dsczaon foula fo ng od dvavs. Th Rann-Louvll and h Günald-Lnov appoachs concd und lavly a condons; f y s connuous and y s ngabl n h nval [0 ] hn fo vy od 0 < < boh h Rann-Louvll and h Günald-Lnov dvavs s and concd fo any valu nsd h nval [0 ]. Ths fac of faconal calculus nsus h conssncy of boh dfnons fo os physcal applcaons h h funcons a pcd o b suffcnly sooh [49]. Faconal acon-subdffuson uaon Th sandad anfld odl fo h voluon of h concnaons a and b of A and B pacls s gvn by h acon-dffuson uaons; a = D a a b 3 b = D b a b 4 h D s h dffuson coffcn assud n hs pap o ual fo spcs and s h a consan fo h bolcula acon. In od o gnalz h acon-dffuson pobl o a acon-subdffuson pobl h subdffusv oon of h pacls us b dal h. al. [0] and Yus al. [] placd Es. 3 and 4 h a s of acon-subdffuson uaons n hch boh h oon and h acon s a affcd by h subdffusv chaac of h pocss; a = D b = D a a b b a b 5 6 h s h gnalzd dffuson coffcn and D s h Rann-Louvll faconal paal dvav of od. Th faconal acon-subdffuson Es. 5 and 6 a dcoupld hch s uvaln o solv h follong faconal acon-subdffuson uaon; u = D [ u u ] f 0 < T 0 < < L 7 36 Walala J c & Tch 04; 4

3 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. h < < assud as follos; 0 and s a posv consan. Th Dchl bounday condons fo hs pobl a u0 = u L = 0 < T 8 h an nal condon; u 0 = 0 L. 9 In h las f yas any paps hav sudd h poposd odl 7-9 [50-7]. Th an a of hs pap s o adap h faconal ghd avag fn dffnc hod FDM o sudy hs odl. Th plan of h pap s as follos; n scon 3 so appoa foula of h faconal dvavs and nucal fn dffnc sch a gvn. In scon 4 a sably analyss and an accuacy analyss of h psnd hod a gvn. In scon 5 nucal suds fo faconal acon-subdffuson odl pobl a psnd. Th pap nds h so conclusons n scon 6. Wghd avag sch fo h faconal acon-subdffuson uaon In hs scon h ghd avag fn dffnc hod s usd o oban h dsczaon fn dffnc foula of h acon-subdffuson E. 7. Fo so posv consan nubs M and N follong noaons and a usd a -sp lngh and spac-sp lngh spcvly. Th coodnas of h sh pons a = = 0... N and = = 0... M L and h valus of h soluon u on hs gd pons a u u h = N T and =. Fo o dals on h dsczaon n faconal calculus s [89]. M In h fs sp h odnay dffnal opaos a dsczd as follos [7]; u and u u u = u O O 0 u u u = u O O. In h scond sp h Rann-Louvll opao s dsczd as follos; D u = u O p h p s h od of h appoaon hch dpnds on h choc of and; Walala J c & Tch 04; 4 363

4 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. [ ] u = u h u 3 h [ ] ans h ng pa of. Th a any chocs of h ghs [98] so h abov foula s no unu. Dnong h gnang funcon of h ghs by z..; z = If z. z = z 4 hn gvs h bacad dffnc foula of h fs od hch s calld h Günald-Lnov foula. Th coffcns can b valuad by h follong foula; = 0 =. 5 Fo = h opao 0 us = and 0 0 D bcos h dny opao so ha h conssncy of Es. and 3 0 = fo 5 fo hch n un ans ha z0 =. No h fn dffnc sch of h faconal acon-subdffuson E. 7 s oband. In hs sudy a = =. To achv hs a E. 7 s valuad a h pons of h gd ; [ u D u D u ] = f. 6 Thn n h abov E.6 h fs od -dvav s placd by h foad dffnc foula 0 and h scond od spac-dvav by h h-pon cnd foula h spc o h ghd avag foula a h s and ; 7 u [ u u ] u f = T h [0] bng h gh faco. foula s gvn by; T s h sulng uncaon o. Th sandad dffnc 8 u [ u u ] u f = 0. No by subsung fo h dffnc opaos gvn by 0 and 3 n E.8 h follong sch can b oband. = 9 R 364 Walala J c & Tch 04; 4

5 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. h and = = 0 = R = [ f ][ =... N. E. 9 s h faconal ghd avag fn dffnc sch consdd n hs pap. Founaly E. 9 s a dagonal sys ha can b solvd usng h Thoas algoh [9]. In h cas of = and = h bacad Eul faconal uadau hod and h Can-Ncholson faconal uadau hods a avalabl spcvly hch hav bn sudd.g. n [30] [3] bu a = 0 h sch s calld fully plc. No o sudy h solvably of h poposd fn dffnc hod l; ] 0 T T = [... N and = [... N ] = 0... M spcvly. Thfo h plc dffnc appoaon sch 9- can b n n a fo: ] A h = b A = 0 0 and b = R. Tho Th dffnc E. s unuly solvabl. Poof Bcaus > 0 hn h coffcn a of h dffnc uaon s a scly dagonally donan a. Thfo A s a nonsngula a; hs povs Tho. La Th coffcns = 0... sasfy; = ; = 0 ; < 0 = 3... ; Walala J c & Tch 04; 4 365

6 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon Nass Hassn WEILAM al. hp://s.u.ac.h Walala J c & Tch 04; = ; N n < = n. Poof []. La. = O Poof []. ably analyss In hs scon h John von Nuann hod s usd o sudy h sably analyss of h ghd avag sch 9. In hs sudy h souc.. 0 = f s nglcd. Poposon Assung ha = hn = 0. ] [ sn 4 ] sn 4 [ 3 Poof By usng E. 9 can b n n h follong fo ]. ][ [ = 4 In h faconal John von Nuann sably pocdu h sably of h faconal WAM s dcdd by pung =. Insng hs psson no h ghd avag dffnc sch 4 h follong s oband. ] ][ [ = 5 subsung by = and dvdng 5 by suls n; = 0. ] ][ [ 6

7 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. sng h non Eul s foula [ cos ] ][ cos ] = cos sn n E.6 oban; [ = 0. 7 nd so splfcaons h abov uaon 7 can b n n h ud fo 3. Ths copls h poof of h poposon. Poposon Assung n poposon ha = hn h sch ll b sabl as long as: 4 sn [ 4 sn ]. 8 Poof Th sably of h sch 3 s dnd by h bhavo of. In h John von Nuann hod h sably analyss s cad ou usng h aplfcaon faco dfnd by =. 9 Of cous dpnds on. Bu fo h on assu ha as n [30] s ndpndn of. Thn nsng hs psson 9 no E. 3 on gs; sn [ 4 ] 4 sn [ ] = 0 30 dvdng E. 30 by o oban h follong foula of ; 4 sn = [ 4 sn ]. 3 Th sch ll b sabl as long as usng E. 3 hch copls h poof of poposon and 8. Walala J c & Tch 04; 4 367

8 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Poposon 3 Assung n poposon ha = sn and ha; L = 3 4{ [ ] } = hn h sch ll b sabl hn L. 33 Poof. By consdng h -ndpndn l valu = and snc; 4 sn > 0 hn fo E. 8 hav; 4 sn 4 sn [ ]. 34 Fo h abov nualy 34 4 sn 4 [ sn ] sng h assupon ha = sn fo 35 s found ha; [ ] 0. W can E. 36 n h follong fo; 4 4 [ ] = sng h assupon fo Es n 37 on fnds ha h od s sabl hn. Ths L nds h poof of h poposon. Tho Th faconal ghd avag fn dffnc sch dvd n 9 s sabl a 0 und h follong sably con; 368 Walala J c & Tch 04; 4

9 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al Poof nc L dpnds on L nds oads s l valu; L = l L. 39 In hs l 39 h sably condon s; 4{ [ = ] l } 40 bu fo E. 4 h z = on ss ha ] l = 4{ [ = } so ha; L 4 and by placng sn by s hghs valu so as sn and l = 0 hn fo 40-4 s found ha h suffcn condon fo h psnd hod s sabl and hs copls h poof of E.38 and ho. Ra Fo < h sably condon 9 can b sasfd und spcfc valus of. = Nucal suls In hs scon h poposd hod s sd by consdng h follong nucal apls. Eapl Consd h nal-bounday valu pobl of faconal acon-subdffuson uaon yp h a non-hoognous ; u = D [ u u ] 0 < < 0 T 4 Walala J c & Tch 04; 4 369

10 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. h h follong bounday condons condon u 0 = 0 0. u0 = u = 0 T and h nal Th ac soluon of hs pobl s u =. Th bhavo of h analycal soluon and h nucal soluon of h poposd faconal acon-subdffuson E. 4 by ans of h ghd avag FDM h dffn valus of and fnal T a psnd n Fgus - 5. Fgu Th bhavo of h ac soluon and h nucal soluon of 4 by ans of h poposd hod a = 0 fo = 0.8 = = and T = Walala J c & Tch 04; 4

11 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Fgu Th bhavo of h ac soluon and h nucal soluon of 4 by ans of h poposd fo = 0.3 = = and T = hod a = 0.5 Fgu 3 Th bhavo of h ac soluon and h nucal soluon of 4 by ans of h poposd hod a = fo = 0.5 = = and T = Walala J c & Tch 04; 4 37

12 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Fgu 4 Th bhavo of h nucal soluon of 4 by ans of h poposd hod a = 0.5 = = and T = 0.5 h dffn valus of. 0 0 Fgu 5 Th bhavo of h nucal soluon of 4 by ans of h poposd hod a = 0.5 = = = 0.7 h dffn valus of T Walala J c & Tch 04; 4

13 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Eapl Consd h follong nal-bounday pobl of h faconal acon-subdffuson uaon: u = D [ u u ] f 43 on a fn doan < < 0 h 0 T 0 < < and h follong souc ; f = sn und h bounday condons u 0 = u = 0 and h nal condon u 0 = 0. Th ac soluon of E. 43 n hs cas s u = sn. Th bhavo of h ac soluon and h nucal soluon of h poposd faconal aconsubdffuson E. 43 by ans of h faconal ghd avag FDM h dffn valus of and fnal T a psnd n Fgus 6-9. Fgu 6 Th bhavo of h ac soluon and h nucal soluon of 43 by ans of h poposd hod a = 0 fo = 0.5 = = and T = Walala J c & Tch 04; 4 373

14 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Fgu 7 Th bhavo of h ac soluon and h nucal soluon of 43 by ans of h poposd hod a = 0.5 fo = 0. = = and T = Fgu 8 Th bhavo of h ac soluon and h nucal soluon of 43 by ans of h poposd hod a = fo = 0.9 = = and T = Walala J c & Tch 04; 4

15 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Fgu 9 Th bhavo of h nucal soluon of h poposd pobl 43 by ans of h poposd hod fo = = 0.4 = = and T = Fo h pvous fgu can b sn ha h nucal soluon s unsabl snc h sably condon 38 s no sasfd. Tabls and sho h agnud of h au o bn h nucal soluon and h ac soluon oband by usng h faconal ghd avag FDM dscussd abov a = 0 and = 0.5 spcvly h dffn valus of and h fnal T. Tabl Th au o h dffn valus of a = 0 = 0.5 and = 0. T Mau o Walala J c & Tch 04; 4 375

16 Wghd Avag FDM fo Faconal Racon-ubdffuson Euaon hp://s.u.ac.h Nass Hassn WEILAM al. Tabl Th au o h dffn valus of a = 0. 5 = 0.3 and = 0. T Mau o Conclusons Ths pap psns a class of nucal hods fo solvng faconal acon-subdffuson uaons. Ths class of hod s vy clos o h ghd avag fn dffnc hod. pcal anon s gvn o sudy h sably of h faconal ghd avag FDM. To cu hs a John von Nuann sably analyss s usd. Fo h hocal sudy can b concludd ha hs pocdu s suabl fo h faconal fn ghd avag FDM and lads o vy good pdcons fo h sably bounds. Th sably of h faconal fn ghd avag FDM psnd dpnds songly on h valu of h ghng paa. Nucal soluons and ac soluons of h poposd pobl a copad and h dvd sably condon s chcd nucally. Fo hs copason can b concludd ha h nucal soluons a n clln agn h h ac soluons. All copuaons n hs pap a un usng Malab pogang. Acnoldgns Th auhos a vy gaful o h do and fs fo cafully adng h pap and fo h cons and suggsons hch hav povd. Rfncs [] DA Bnson W Whacaf and MM Mscha. Th faconal-od govnng uaon of Lévy oon. Wa Rsou. Rs. 000; [] M Chang Chn F Lu I Tun and V Anh. A Fou hod fo h faconal dffuson uaon dscbng sub-dffuson. J. Cop. Phys. 007; [3] E Cusa and J Fna. Iag pocssng by ans of a lna ngo-dffnal uaons. In: Pocdng of h Innaonal Assocaon of cnc and Tchnology fo Dvlopn Bnaládna pan 003 p [4] F Lu V Anh and I Tun Nucal soluon of h spac faconal Fo-Planc uaon. J. Cop. Appl. Mah. 004; [5] R Mzl and J Klaf. Th ando al s gud o anoalous dffuson: a faconal dynacs appoach. Phys. Rp. 000; [6] KB Oldha and J pan. Faconal Calculus: Thoy and Applcaons Dffnaon and Ingaon o Abay Od. Acadc Pss N Yo Walala J c & Tch 04; 4

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