ON THE NUMERICAL SOLUTION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS. Solat Karimi Vanani and Azim Aminataei

Transcription

1 Matheatcal ad Coptatoal Applcatos, Vol. 7, No., pp. 40-5, 0 ON THE NUMERICAL SOLUTION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS Solat Kar Vaa ad Az Aatae Departet of Matheatcs, K. N. Toos Uversty of Techology, P.O. Bo: , Tehra, Ira solatkar@yahoo.co, atae@kt.ac.r Abstract- I ths paper, a techqe geerally kow as eshless ethod s preseted for solvg fractoal partal dfferetal eqatos FPDEs). Soe physcal lear ad olear eperets sch as te-fractoal covectve-dffso eqato, tefractoal wave eqato ad olear space-fractoal Fsher's eqato are cosdered. We preset the advatages of sg the radal bass fctos RBFs) especally where the data pots are scattered. Coparg betwee the ercal reslts obtaed fro or ethod ad the other ethods cofrs the good accracy of the preseted schee. Key Words- Radal bass fctos, Fractoal partal dfferetal eqatos. INTRODUCTION The theory of fractoal calcls was frst rased the year 695 by Marqs de L'Hoptal ad fro ow o ay stdes were doe ad ay portat books were pblshed ths feld where we ca pot ot to the books of Oldha ad Spaer [], Mller ad Ross [], Sako et al. [3] ad Podlby [4]. Most of the scetfc probles ad pheoea are odeled by FPDEs. For stace, atheatcal physcs [4] fld ad cot echacs [5], colored oses [6], bology, chestry, acostcs ad psychology [7]. Soe of FPDEs cldg the fractoal Fokker-Plack eqato [8], the fractoal KdV eqato [9] ad lear ad olear space- ad te-fractoal dffso-wave eqato [0, ] have bee stded ad solved. I ost cases, these probles do ot adt aalytcal solto, so these eqatos shold be solved sg specal techqes. I the last decade, several coptatoal ethods have bee appled to solve FDEs, proet aog whch are the hootopy pertrbato ethod HPM) [, 3], the Adoa decoposto ethod ADM) [4, 5], the varatoal terato ethod VIM) [], the geeralzed dfferetal trasforato ethod GDTM) [6] ad the fractoal dfferece ethod FDM) [7, 8]. I the preset research work, we are desred to preset a trly eshless approato strategy for solvg FPDEs based o RBFs. I the last two decades, the se of RBFs for both terpolato ad for solvg PDEs have receved cosderable atteto varos felds of research ad attracted ay researchers to solve the probles hgher-desoal spaces. Becase the RBFs as a class of esh-free schees avod grd geerato ad the doa of terest ca be cosdered by a set of scattered data pots aog whch there s o pre-defed coectvty. Ths ethod of solto s effectve o scattered data pots ad rreglar geoetres, s easy to pleet ay fte deso, ad s spectrally

2 S. K. Vaa ad A. Aatae 4 accrate. The applcato of RBFs to solve PDEs was frst sed [9, 0] ad the was appled to solve varos probles arsg ay sceces especally physcs ad echacs. For stace; PDEs sg vscoelastc flows [], PDEs sg atral freqeces of coposte plates [], Kle-Gordo eqato sg sold state physcs, plasa physcs, fld dyacs [3], KdV eqato sg pressre, waves, lqd-gas bbble tre, ad wave pheoea haroc crystals [4], delay dfferetal systes arsg varos areas [5] ad etc. The preset paper has bee orgazed as follows: Secto gves otatos ad basc deftos of the fractoal calcls. I Secto 3, we trodce soe basc aspects of RBF terpolato ad ts developet to solve FPDEs. Soe eperets ad ther reslts are preseted ad copared wth several other ethods Secto 4. Fally Secto 5, the stdy s coclded ad the fdgs are sarzed.. BASIC DEFINITIONS OF THE FRACTIONAL CALCULUS I ths Secto, we state soe prelares ad deftos of fractoal calcls [4]. Defto. A real fcto ), 0 s sad to be the space C, R, f there p ests a real ber p sch that ) v), where v) C[0, ) ad t s sad to be the space C ff ) ) C, N. Defto. The Rea-Lovlle fractoal tegral operator of order 0, of a fcto ) C,, s defed as: a a- J ) = - t) t)dt, a > 0, > 0, G a) ò 0 0 J) = ), where s the Gaa fcto. Defto 3. The fractoal dervatve of ) the Capto's sese s defed as: a -a -a- ) D ) = J D ) = - t) t)dt, G -a) ò ) 0 where <, N, > 0, ) C. Defto 4. For to be the sallest teger that eceeds, the Capto's spacefractoal dervatve operator of order 0 s defed as:, t) ) d,, ) 0 D, t) ), t),, ad the te-fractoal dervatve operator of order 0 s defed as: t t ) ) 0 D t, t), t), t, ) d,,. 3)

3 4 O The Nercal Solto of Fractoal Partal Dfferetal Eqatos 3. RADIAL BASIS FUNCTIONS AND THEIR APPLICATIONS ON FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS To epla the cocepts of RBFs, we eed assert soe prelares ad soe otatos fro [6]. Let the fcto ) s defed as ) : R d R ad sppose that the approato û ) at a arbtrary pot ca be wrtte as a lear cobato of bass fctos the followg for: d ) û) = ), R, 4) = where the set of RBFs { )} = = {, c )} = s chose advace ad the set of { } = s to be fd. Also, s the ber of odal pots falle wth the flece doa of ad c deotes the shape paraeter. The ost portat ad sefl RBFs are as follows:. Ecldea: ) =,. Mltqadrc: ) = c,. Iverse ltqadrc: ) =, c v. Th-plate sple: ) = log, /c v. Gass kerel: ) = e. The deotes the dstace betwee ad the -th odal pot ad the shape paraeter ca be defed as c = d, where > 0 s a factor ad d s the dstace fro the -th collocato pot to the earest eghborg collocato pot [7, 8]. I ths paper, we se two portat RBFs, the ltqadrc RBF MQ-RBF) ad verse ltqadrc RBF IMQ-RBF) as the base fctos to approate the desred probles. The MQ-RBF ad IMQ-RBF as ftely sooth fctos, are aog the ost poplar RBFs. The MQ-RBF was frst trodced by Hardy [9] who sccessflly appled t for approatg srface ad bodes fro feld data. Hardy [30] has wrtte a detaled revew artcle sarzg ts eplosve growth se sce t was frst trodced. I 97, Frake [3] pblshed a detaled coparso of 9 dfferet scattered data schees agast aalytc probles. Of all the techqes tested, he coclded that MQ-RBF perfored the best accracy, vsal appeal, ad ease of pleetato, eve agast varos fte eleet schees. The MQ-RBF ad IMQ- RBF approato schees are grd-free spatal approato schees whch coverge epoetally for the spatal ters of ODEs ad PDEs ad offer ay coptatoal advatages over tradtoal ethods. Soe of the ost advatages of

4 S. K. Vaa ad A. Aatae 43 MQ-RBF ad IMQ-RBF schees are that they are trly esh-free schees whch possess very hgh order rates of covergece [3]. Now, the a goal s to evalate the coeffcets { } = eqato 4). Therefore, we sbsttte the collocato pots { } j j= wth the doa eqato 4). So, for j =,,, ; we obta: û j) = j) = j) j) j). 5) = The operatoal atr for of eqato 5) ca be wrtte the followg aer: û =, 6) where T T û = [û ), û ),, û )], = [,,, ], ad s a coeffcet atr as: ) ) ) ) ) ) ) ) = ) ) ) ). ) ) ) ) s a real syetrc atr. Theoretcal reslts show that the RBF terpolato atr s osglar, whatever the ber of the data stes ad therefore ests [33]. Hece, we ca obta as: = û. Let s cosder as: = [ ), ),, )], we ca rewrte eqato 4) as: û ) = = û = û, where = = [,,, ]. The fctos, =,,, ; are the shape fctos whch satsfy the Kroecker delta codto:, = j, j ) =, j =,,,. 0, j, We ca pleet the RBF approato schee two aers, drect RBF DRBF) ad drect RBF IRBF) [8]. I the DRBF, we frst asse that the approate solto s as eqato 4) ad the the dervatve fctos are calclated drectly by dfferetato of eqato 4). I the IRBF for approatg fcto ad ts

5 44 O The Nercal Solto of Fractoal Partal Dfferetal Eqatos dervatves, the hghest order dervatve fcto f s frst approated ters of RBFs. I ths paper, we apply the DRBF. Now, order to apply the RBF approato schee for solvg FPDEs, let s cosder a FPDE the for: d D L = f, R, 7) B = g, o, 8) where d s the deso, deotes the bodary of the doa, L s the dfferetal operator, D s the fractoal dfferetal operator of order that operates o the teror, ad B s a operator that specfes the bodary codtos. Both the f ad g : R d R are kow fctos. Let { } j j= be the collocato pots. We asse the collocato pots are arraged sch a way that the frst pots ad the last B pots are ad o, respectvely. Sbstttg the collocato pots to eqatos 7) ad 8), we obta: D L) j) = f j), j =,,,, 9) = B j) = g j), j =, =,,. 0) Therefore, we have the followg syste:,b = f,g), ) where D L) ) D L) ) D L) ) D L) ) D L) ) D L) ) = D L) =, D L) ) D L) ) D L) ) B ) B ) B ) B ) B ) B ) B = B =, B ) B ) B ) T T f = [f ),f ),,f )] ad g = [g ),g ),,g )]. Therefore, the syste of eqatos wth kows s avalable. The, we st solve ths syste to ake dstct the kow coeffcets. Hece, we have sed the Gass elato ethod wth total pvotg to solve sch a syste. Reark. It s otceable that collocatg pots ca be scattered. Ths s oe of the ost portat advatages of the RBF collocato ethod [0, 33]. I Secto 4, the ercal reslts show ths sse easly, ad the applcablty of the MQ-RBF ad IMQ- RBF approato schees ths sese, s observable.

6 S. K. Vaa ad A. Aatae ILLUSTRATIVE EXAMPLES I ths Secto, three eaples cldg lear ad olear physcal probles o reglar ad rreglar doas are solved sg MQ-RBF ad IMQ-RBF. If the eact solto of the proble ests, the accracy of a approate solto s easred by eas of the dscrete relatve L or defed as: [û ) )] = N e =, ) ) = where ad û are the eact ad copted soltos, respectvely; ad s the ber of kow odal vales of. Sce, ost of FDEs have ot eact solto, therefore we st copare the wth kow ercal ethods sch as HPM, VIM ad GDTM. I all eperets, we asse that = 0. The reslts ad ther coparso wth several powerfl ethods llstrate the valdty ad capablty of these RBFs. The coptatos assocated wth the eperets were perfored Maple 3 o a PC, CPU.8 GHz. Eaple 4. Cosder the followg lear te-fractoal covectve-dffso eqato []: = t ), t > 0, R, 0 <, t wth the tal codto:,0) =. The eact solto s ot kow. We have solved ths proble by MQ-RBF wth 5 pots, = 0.75, 0.85, 0.95 ad have copared t wth HPM to show the effcecy of the MQ-RBF. The reslts are gve Table. Table : Coparso of the HPM ad MQ-RBF approato schee for dfferet t ad of the eaple 4. t / t = t = HPM MQ RBF = 0.75 = 0.85 = 0.95 = 0.75 = 0.85 =

7 46 O The Nercal Solto of Fractoal Partal Dfferetal Eqatos t = t = t = The copted reslts Table, show a good agreeet betwee the reslts obtaed by MQ-RBF approato schee ad HPM, aga. Ths deostrate the valdty of MQ-RBF approato schee for FPDEs. Fgre. The graph of approate solto of eaple 4. wth = ad 5.

8 S. K. Vaa ad A. Aatae 47 Fgre. The graph of approate solto of eaple 4. wth = ad 5. Fgre 3. The graph of approate solto of eaple 4. wth = ad 5. Eaple 4. Cosder the followg lear te-fractoal wave eqato [, 34]: =, t > 0, R, <, t sbject to tal codtos:,0),0) =, ad =. t The eact solto s ot kow. We have solved ths proble by IMQ-RBF approato schee wth 5 scattered data pots, =.5 ad have copared t wth VIM.

9 48 O The Nercal Solto of Fractoal Partal Dfferetal Eqatos Table : Coparso of the soltos of VIM ad IMQ-RBF approato schee for scattered ad t of the eaple 4. t=0 t=0.06 t=0.3 t=0.9 t= VIM IMQ RBF t=0 t=0.06 t=0.3 t=0.9 t= Here, the agreeet of all data of ad t s observable for scattered data pots. Ths s the bggest advatage of RBFs. Fgre 4. The graph of approate solto of eaple 4. wth =.5 ad = 5. Eaple 4.3 Cosder the followg olear space-fractoal Fsher's eqato [6, 35 ]:.5, t), t)) =, > 0,.5 t wth the tal codto:,0) =. The eact solto s ot kow. We have solved ths eperet by IMQ-RBF approato schee ad have obtaed the approate solto wth =49. Coparso of the IMQ-RBF approato schee wth GDTM [6] ad VIM [35] for soe pots are gve Table 3.

10 S. K. Vaa ad A. Aatae 49 Table 3: Coparso of the soltos of GDTM, VIM ad IMQ-RBF approato schee for dfferet t ad of the eaple 4.3 t =0. t =0. GDTM VIM IMQ RBF GDTM VIM IMQ RBF t =0.3 t =0.4 GDTM VIM IMQ RBF GDTM VIM IMQ RBF Fro the above coparso, a good agreeet betwee IMQ-RBF approato schee wth the other ethods s observable. Fgre 5. The graph of approate solto of eaple 4.3 wth = CONCLUSION I ths paper, the RBFs are appled for solvg FPDEs sg egeerg ad physcs. As the ercal eaples show, the RBFs ca deal wth lear ad olear PDE probles wth ease. For ths, oel lear ad olear eaples are ercally solved. Coparg betwee or reslts ad other sefl ethods sch as HPM, VIM ad GDTM show the versatlty, the capablty ad the effcecy of the RBFs. As we

11 50 O The Nercal Solto of Fractoal Partal Dfferetal Eqatos have observed, the ethod works ecelletly for scattered data pots ad rreglar doas. Hece, these schees are ot deped o the selecto of pots. Also, ths stdy has show that the RBFs schees ca copete wth the tradtoal ethods as far as accracy s cocered. It s hoped that the sghts preseted ths stdy wll attract the atteto of researchers to vestgate the RBFs for the ercal solto of FPDEs. 6. REFRENCES. K.B. Oldha ad J. Spaer, The fractoal calcls, Acadec Press, New York ad Lodo, K.S. Mller ad B. Ross, A trodcto to the fractoal calcls ad fractoal dfferetal eqatos, Joh Wley, New York, S.G. Sako, A.A. Klbas ad O.I. Marchev, Fractoal tegrals ad dervatves: theory ad applcatos, Gordo ad Breach Scece Pblshers, USA, I. Podlby, Fractoal dfferetal eqatos, Acadec Press, New York, F. Maard, Fractals ad fractoal calcls cot echacs, Sprger Verlag, 9-348, H.H. S, A.A. Abdelvahab ad B. Oaral, Lear approato of trasfer fcto wth a pole of fractoal order, IEEE Tras. Atoat. Cotrol 9, , W.M. Ahad ad R. El-Khazal, Fractoal-order dyacal odels of love. Chaos, Soltos & Fractals 33, , S. Che, F. L, P. Zhag ad V. Ah, Fte dfferece approatos for the fractoal Fokker-Plack eqato, Appled Matheatcal Modellg 33, 56-73, S. Moa, A eplct ad ercal soltos of the fractoal KdV eqato, Math. Copt. Sl. 70, 0-8, S. Moa, Z. Odbat ad V.S. Ertrk, Geeralzed dfferetal trasfor ethod for solvg a space ad te-fractoal dffso-wave eqato, Physcs Letters A 370, , H. Jafar ad S. Sef, Hootopy aalyss ethod for solvg lear ad olear fractoal dffso-wave eqato, Co. Nolear Sc. Ner. Slat. 4, 006-0, S. Moa ad Z. Odbat, Coparso betwee the hootopy pertrbato ethod ad the varatoal terato ethod for lear fractoal partal dfferetal eqatos, Copters ad Matheatcs wth Applcatos 54, 90-99, H. Jafar ad S. Sef, Hootopy aalyss ethod for solvg lear ad olear fractoal dffso-wave eqato, Co. Nolear Sc. Ner. Slat. 4, 006-0, A.M.A. El-Sayed ad M. Gaber, The Adoa decoposto ethod for solvg partal dfferetal eqatos of fractal order fte doas, Physcs Letters A 359, 75-8, S. Moa ad Z. Odbat, Aalytcal solto of a te-fractoal Naver-Stokes eqato by Adoa decoposto ethod, Appl. Math. Copt. 77, , S. Moa ad Z. Odbat, A ovel ethod for olear fractoal partal dfferetal eqatos: Cobato of DTM ad geeralzed Taylor's forla, Joral of Coptatoal ad Appled Matheatcs 0, 85-95, 008.

12 S. K. Vaa ad A. Aatae 5 7. S. Moa ad Z. Odbat, Nercal coparso of ethods for solvg lear dfferetal eqatos of fractoal order, Chaos, Soltos & Fractals 3, 48-55, A. Ghorba, Toward a ew aalytcal ethod for solvg olear fractoal dfferetal eqatos,copt. Methods Appl. Mech. Eg. 97, , C. Frake ad R. Schaback, Solvg partal dfferetal eqatos by collocato sg radal bass fctos, Appl. Math. Copt. 93, 73-8, E.J. Kasa, Mltqadrc a scattered data approato schee wth applcatos to coptatoal fld dyacs II. Copt. Math. Appl. 9, 47-6, T. Tra-Cog, N. Ma-Dy ad N. Pha-The, BEM-RBF approach for vscoelastc flow aalyss, Egeerg Aalyss wth Bodary Eleets 6, , 00.. A.J.M. Ferrera ad G.E. Fasshaer, Aalyss of atral freqeces of coposte plates by a RBF-psedospectral ethod, Coposte Strctres 79, 0-0, M. Dehgha ad A. Shokr, Nercal solto of the olear Kle-Gordo eqato sg radal bass fctos, J. of Coptatoal ad Appled Matheatcs 30, , Q. She, A eshless ethod of les for the ercal solto of KdV eqato sg radal bass fctos, Egeerg Aalyss wth Bodary Eleets 33, 7-80, S. Kar Vaa ad A. Aatae, Mltqadrc approato schee o the ercal solto of delay dfferetal systes of etral type, Matheatcal ad Copter Modellg 49, 34-4, X. L, G. R. L, K. Ta ad K.Y. La, Radal pot terpolato collocato ethod for partal dfferetal eqatos, Copters ad Matheatcs wth Applcatos 50, 45-44, N. Ma-Dy ad T. Tra-Cog, Approato of fcto ad ts dervatves sg radal bass fcto etworks, Appled Matheatcal Modellg 7, 97-0, A. Aatae ad M.M. Mazare, Nercal solto of Posso's eqato sg radal bass fcto etworks o the polar coordate, Copters ad Matheatcs wth Applcatos 56, , R.L. Hardy, Mltqadrc eqatos of topography ad other rreglar srfaces, J. Geophys. Res. 76, , R.L. Hardy, Theory ad applcatos of the ltqadrc b-haroc ethod: 0 years of dscovery, Copt. Math. Appl. 9, 63-67, 990). 3. R. Frake, Scattered data terpolato: Tests of soe ethods, Math. Copt , A.H.D. Cheg, M.A. Golberg, E.J. Kasa ad G. Zato, Epoetal covergece ad h-c ltqadrc collocato ethod for partal dfferetal eqatos, Ner. Methods Partal Dfferetal Eqatos 9, , C.A. Mcchell, Iterpolato of scattered data: dstace atrces ad codtoally postve defte fctos, Costr. Appro., -, S. Moa ad Z. Odbat, Aalytcal approach to lear fractoal partal dfferetal eqatos arsg fld echacs, Physcs Letters A 355, 7-79, Z. Odbat ad S. Moa, A relable treatet of hootopy pertrbato ethod for Kle Gordo eqatos, Phys. Lett. A 365, , 007.