Liao s method for a few space and time fractional reaction-diffusion equations arising in Engineering

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1 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Liao s ehod for a few space and ie fracional reacion-diffsion eaions arising in Engineering R.Rajaraan and G.Hariharan Deparen of aheaics School of Haniies & Sciences SASTRA Universi Thanjavr-63 4 Tailnad India rraja@ahs.sasra.ed hariharan@ahs.sasra.ed Absrac In his paper we have applied an accrae and efficien hooop analsis ehod HAM o find he approiae/analical solions for space and ie fracional reacion- diffsion eaions arising in aheaical cheisr. The ehod provides solions in rapid convergence series wih copable ers. To he bes of or nowledge nil now here is no rigoros HAM solions have been repored for he space and ie fracional reacion-diffsion eaions FRDEs. Soe nerical eaples are presened o deonsrae he validi and applicabili of he ehod. The power of he anageable ehod is confired. Moreover he se of HAM is fond o be accrae efficien siple fleible and less copaion cos. Keword- Hooop analsis ehod fracional derivaives space and ie fracional reacion diffsion eaions I. INTRODUCTION In recen ears noable conribions have been ade o he applicaions of fracional differenial eaions FDEs. These eaions are increasingl applied o efficien odel probles in research areas as diverse as achanical sses dnaical sses conrol chaosconinos ie rando wals anoalos diffsive and sbdiffsive sses wave propagaion and so on. The fracional calcls and is applicaions ha is he heor of inegrals and derivaives of an arbirar real or cople order has gained considerable poplari and iporance dring he pas hree decades or so ainl de o is applicaions in diverse fields of science and engineering [56]. Maheaical odelling of cople processes is a ajor challenge for coneporar scienis. In conras o siple classical sses where he heor of ineger order differenial eaions is sfficien o describe heir dnaics fracional derivaives provide an ecellen and an efficien insren for he descripion of eor and herediar properies of varios cople aerials and sses. The diffsion of wo or ore cheicals a neal raes over a srface reac wih one anoher in order o for sable paerns is represened b reacion diffsion eaion. The nare of he diffsion is characerized b eporal scaling of he ean sare displaceen r.for sandard diffsion whereas in anoalos sb diffsion < and in anoalos sper diffsion >. Sandard diffsion is represened b classical diffsion eaions and sb diffsion and sper diffsion are represened b fracional diffsion eaions. In las few decades fracional differenial eaions are increasingl sing in he odelling of varios phsical and dnaical sses. The os iporan advanage of sing fracional differenial eaions is heir non-local proper. I is well nown ha ineger- order fracional operaor b he fracional order differenial operaor is a nonlocal operaor. This indicaes ha he ne sae of a sse depends no onl pon is crren sae b also pon all of is previos saes. Recenl Das [] had applied he variaional ieraion ehod for fracional order diffsion eaions. Hariharan e al. [45] inrodced he Haar wavele ehod for soe nonlinear parabolic eaions. Recenl varios ieraive ehods are applied for geing nerical and analical solions of linear and nonlinear fracional reacion-diffsion eaions [ ]. The hooop analsis ehod HAM was inrodced b Liao [68-]. The proposed ehod has been sed b an aheaicians and engineers o solve varios eaions based on hooop which is a basic concep in opolog. In recen ears HAM has been sccessfll eploed o solve an pes of nonlinear hoogeneos or nonhoogeneos eaions and sses of eaion as well as probles in science and engineering [47897]. More recenl Hariharan [7] applied he HAM for Kologorov-Perovsii-Pisnov KPP and fracional KPP eaions. The validi of he HAM is independen of wheher or no here eis sall paraeers in he considered eaion. HAM provides s wih a siple wa o adjs and conrol he convergence of solion series. Three pes of fracional diffsion eaions can be prodced when considering non-gassisan disribions. The firs is he space fracional diffsion ISSN : Vol 5 No 3 Jn-Jl 3 377

2 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET U U c < Here he diffsion is Marovian. The second pe is he ie fracional diffsion eaion U U c Here he diffsion is non-marovian and can be frher be divided ino < < which has sb diffsive behavior and < < which has a sper diffsive behavior. The hird pe is ied case wih boh space and ie fracional derivaives U U c 3 The oline of his paper is as follows. In secion we review he basic idea of Liao s ehod. In secion 3 we presen he applicaion of he HAM o space and ie fracional reacion-diffsion eaions FRDEs and also soe nerical eaples are provided for deonsraing he applicabili and validi of he ehod. Also a conclsion given in secion 4. Definiions of fracional derivaives and inegrals In his secion we have given soe noaions definiions and preliinar facs ha will be sed frher in his wor. The Capo fracional derivaive allows he ilizaion of iniial and bondar condiions involving ineger order derivaives which have clear phsicall inerpreaions. Therefore in his paper we shall se he Capo derivaive D proposed b Capo in his wor on he heor of viscoelasici. In he developen of heories of fracional derivaives and inegrals i appears an definiions sch as Rieann-Lioville and Capo fracional differenial-inegral definiion as follows. Rieann-Lioville definiion: d f N R d a D f d f T dt < <. d a T Fracional inegral of order is as follows: R a I f T f T dt >. Capo definiion: D f d f N d f T dt < <. a T c a II. BASIC IDEA OF HAM In his secion he basic ideas of he hooop analsis ehod are inrodced. Here a descripion of he ehod is given o handle he general nonlinear proble. > 4 Where is a nonlinear operaor and is nnown fncion of he independen variable. A. Zero- order deforaion eaion ISSN : Vol 5 No 3 Jn-Jl 3 378

3 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Le denoe he iniial gess of he eac solion of E. 4 h an ailiar paraeer an ailiar fncion and L an ailiar linear operaor wih he proper. when. 5 The ailiar paraeer h he ailiar fncion and he ailiar linear operaor pla iporan roles wihin he HAM o adjs and conrol he convergence region of solion series. Liao [68] consrcs sing as an ebedding paraeer he so-called zero-order deforaion eaion. 6 Where is he solion which depends on h and. when he zero-order deforaion E.4 becoes 7 and when since h and he zero-order deforaion E.6 redces o 8 h So is eacl he solion of he nonlinear E.6. Define he so-called order deforaion derivaives. 9! If he power series E.6 of converges a hen we ges he following series solion:. Where he ers can be deerined b he so-called high order deforaion described below. B. High- order deforaion eaion Define he vecor n {... n } Differeniaing E.6 ies wih respec o ebedding paraeer he seing and dividing h he b! we have he so-called order deforaion eaion. Wih iniial condiion Where and! For an given nonlinear operaor he er h Now he solion of he order deforaion E. for 3 4 can be easil epressed b E.4. becoes [ R ] c hh r J χ 5 where c is he inegraion consan deerined b he given iniial condiion and n J { f } ξ f ξ dξ. 6 ISSN : Vol 5 No 3 Jn-Jl 3 379

4 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Ths we can gain. b eans of solving he linear high-order deforaion Eaion one h afer he oher order in order. The order approiaion of is given b. 7 III. APPLICATIONS OF LIAO S METHOD Eaple: We consider he ie fracional Gas dnaic eaion Wih iniial condiion a a consan 9 8 We will appl he Liao s ehod o solve E.8 sbjec o he iniial condiion E.9 We define he nonlinear operaor as N[ ] and linear operaor [ ] L Wih he proper Lc Where c is he inegraion consan. Now b sing E. we have R [ U ] and he solion of he h order deforaion E. for becoes [ hh r R U ] χu L 4 Since χ and nder he rle of solion epression sggesed b Liao [6] we se he ailiar fncion Hr and also his eaion has sb diffsive behavior we obain he following sccessive approiaions as a a a a a a The final solion in a closed for is 4 a 4 a a a a a a a a a 6a 6 a 3 a a a a a... When we ge he closed for solion as 5 3 ISSN : Vol 5 No 3 Jn-Jl 3 38

5 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET ae a ae 6 Eaple: We consider anoher nonlinear hoogeneos gas dnaic eaion 7 Wih iniial condiion e 8 We will appl he Hooop analsis o solve E.7 sbjec o he iniial condiion E.8 We define he nonlinear operaor as N[ ] 9 and linear operaor [ ] L 3 Wih he proper Lc 3 Where c is he inegraion consan. Now b sing E. we have R [ U ] 3 and he solion of he h order deforaion E. for becoes χ U L [ hh r R U ] 33 Since χ and nder he rle of solion epression sggesed b Liao [6] we se he ailiar fncion Hr and also his eaion has sb diffsive behavior we obain he solion as follows n e 34 When we ge he solion in closed fors e e Fig Coparison of solions of E.7 for soe vales of and.5 sing 4 h er HAM approiaion wih.5ble crve.5green crve.75red crve ISSN : Vol 5 No 3 Jn-Jl 3 38

6 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Fig. The srface area shows for he E.7 sing forh er approiaion of HAM wih.5 for soe vales of and Fig.3 The srface area shows for he E.7 sing forh er approiaion of HAM wih.75 for soe vales of and Eaple 3: Consider he biological poplaion eaion as follows: > < wih iniial condiion ISSN : Vol 5 No 3 Jn-Jl 3 38

7 We will appl he HAM o solve E.36 sbjec o he iniial condiion E.37 We define he nonlinear operaor as [ ] N 38 and linear operaor [ ] L 39 Wih he proper Lc 4 Where c is he inegraion consan. Now b sing E. we have [ ] R U 4 and he solion of he h order deforaion eaion E. for becoes [ ] U R r hh L U χ 4 Since χ we se h - Hrand also he eaion has sb diffsive behavior we obain he final solion as 43 Eaple:4 Consider he ie fracional advecion dispersion eaion < > λ b D 44 Where D b and λ Wih iniial condiion e 45 Now we will appl he Liao s ehod o solve E.43 sbjec o he iniial condiion E. 45. We define he nonlinear operaor as [ ] N 46 and linear operaor [ ] L 47 Wih he proper Lc 48 Where C is he inegraion consan. Now b sing E. we have [ ] U R [ ] N 49 and he solion of he h order deforaion E. for becoes [ ] U R r hh L U χ 5 Since χ we se h - Hr we obain he final solion as R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET ISSN : Vol 5 No 3 Jn-Jl 3 383

8 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET e Fig. 4 The srface shows he solion for he above E.44 a and Fig.5 The srface shows he solion for E.44 a and.5 Eaple 5: Consider anoher ie fracional advecion dispersion eaion D b λ > < 5 Where D b and λ wih iniial condiion sin 53 ISSN : Vol 5 No 3 Jn-Jl 3 384

9 We will appl he Liao s ehod o solve E.5 sbjec o he iniial condiion E.53 We define he nonlinear operaor as [ ] N 54 and linear operaor [ ] L 55 Wih he proper Lc 56 Where c is he inegraion consan. Now b sing E. we have [ ] U R [ ] N 57 and he solion of he h order deforaion eaion E. for becoes [ ] U R r hh L U χ 58 Since χ we se h - Hrand also he eaion has sb diffsive behavior we obain he final solion as sin -cos 59 Fig 6. The srface shows he solion for E.5 a and R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET ISSN : Vol 5 No 3 Jn-Jl 3 385

10 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Fig.7 The srface shows he solion for E.5 a and.5 Eaple 6. Consider he space fracional advecion-dispersion eaion.5.5 < < π > 6 Wih iniial condiion cosh 6 We will appl he Liao s ehod o solve E.6 sbjec o he iniial condiion E.6 We define he nonlinear operaor as N.5 [ ].5 and linear operaor L[ ] Wih he proper Lc 64 Where c is he inegraion consan. Now b sing E. we have R[ U ] and he solion of he h order deforaion E. for becoes χu L [ hh r R U ] 66 Since χ and nder he rle of solion epression sggesed b Liao [6] we se he ailiar fncion Hr and we ae h- and we obain he following sccessive approiaions as cosh cosh ISSN : Vol 5 No 3 Jn-Jl 3 386

11 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET cosh The final solion is cosh n.5! When and he eac solion is.5 67 cosh e Fig.8 Coparison of solions of E.6 for soe vales of and.5 sing 4 h er HAM approiaion wih.5ble crve.5green crve.75red crve Fig.9 The srface area shows for he E.6 sing forh er approiaion of HAM wih.5 for soe vales of and Fig. The srface area shows for he E.6 sing forh er approiaion of HAM wih.75 for soe vales of and ISSN : Vol 5 No 3 Jn-Jl 3 387

12 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Fig. The srface area shows for he E.6 wih for soe vales of and Eaple. 7 Consider he space and ie fracional diffsion eaion c > < 69 Here we ae c wih iniial condiion R cos π 7 We will appl he Liao s ehod o solve E.69 sbjec o he iniial condiion E.7 We define he nonlinear operaor as N[ ] c 7 and linear operaor [ ] L 7 Wih he proper Lc 73 Where c is he inegraion consan. Now b sing E. we have R[ U ] c 74 and he solion of he h order deforaion E. for becoes χu L [ hh r R U ] 75 Since χ and nder he rle of solion epression sggesed b Liao [6] we se he ailiar fncion Hr and we ae h- and also his eaion has sb diffsive behavior we obain he following sccessive approiaions as cosπ ISSN : Vol 5 No 3 Jn-Jl 3 388

13 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET π π cos π π π cosπ π π cos π 3 The final solion is n π π cosπ 3 8π When we ge he closed for as e cosπ Fig. The srface area shows for he E.69 sing forh er approiaion of HAM wih.5 for soe vales of and Fig.3 The srface area shows for he E.69 sing forh er approiaion of HAM wih.75 for soe vales of and ISSN : Vol 5 No 3 Jn-Jl 3 389

14 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET Fig.4 The srface area shows for he E.69 wih in E.5 for soe vales of and Eaple: 8 Consider he space and ie fracional diffsion eaion c > < 78 Here we ae c Wih iniial condiion 3 R We will appl he Liao s ehod o solve E.78 sbjec o he iniial condiion E.79 We define he nonlinear operaor as N c [ ] and linear operaor L [ ] Wih he proper Lc 8 Where c is he inegraion consan. Now b sing E. we have R[ U ] c 83 and he solion of he h order deforaion E. for becoes χu L [ hh r R U ] 84 Since χ and nder he rle of solion epression sggesed b Liao [6] we se he ailiar fncion Hr and we ae h- and also his eaion has sb diffsive behavior we obain he following sccessive approiaions as ISSN : Vol 5 No 3 Jn-Jl 3 39

15 R.Rajaraan e.al / Inernaional Jornal of Engineering and Technolog IJET 3 6 The final solion is n All he nerical eperiens presened in his secion were coped in doble precision wih soe MATLAB codes on a personal coper Sse Vosro 4 Processor 86 Fail 6 Model 5 Sepping 3 Genine Inel ~596 Mhz IV. CONCLUSION In his paper he hooop analsis ehod HAM has been sccessfll applied o obain he approiae/analical solions of he space and ie fracional reacion-diffsion eaions. This wor shows ha HAM has significan advanages over he eising echnies. I avoids he need for calclaing he Adoian polnoials which can be difficl in soe cases. The reliabili of he ehod and redcion in he size of copaional doain give his ehod wider applicabili. The resls show ha HAM is a powerfl aheaical ool for finding he eac and approiae solions of he nonlinear fracional RDEs. ACKNOWLEDGMENT I a ver graefl o he reviewers for heir sefl coens ha led o iproveen of anscrip. REFERENCES [] R.Hilfer Applicaions of Fracional Calcls in Phsics Acadeic Press Orlando 999. [] S. Das Analical solion of a fracional diffsion eaion b variaional ieraion ehod Copers Mah. Appl [3] Z.Odiba S.Moani Applicaion of variaional ieraion ehod o nonlinear differenial eaions of fracional order In. J. Nonlinear Sci. Ner. Sil [4] H.X S.Liao X.Yo Analsis of nonlinear fracional parial differenial eaions wih he hooop analsis ehod Con. Nonlinear Sci. Ner. Sil [5] B.J.Wes M. Bolognab P. Grigolini Phsics of Fracal Operaors Springer New Yor 3. [6] K.S. Miller B. Ross An Inrodcion o he Fracional Calcls and Fracional Differenial Eaions Wile New Yor 993. [7] S. Kar A. YildiriYasir Khan L. Wei A fracional odel of he diffsion eaion and is analical solion sing Laplace ransfor Scienia Iranica [8] H. Jafari S. Seifi Hooop analsis ehod for solving linear and nonlinear fracional diffsion-wave eaion Con Nonlinear Sci Ner Sila [9] Najeeb Ala Khan Nasir-Uddin Khan Asa Ara Mhaad Jail Approiae analical solions of fracional reaciondiffsion eaions Jornal of King Sad Universi Science4 8. [] Vipl K. Baranwal Ra K. Pande Manoj P. Tripahi O P. Singh An analic algorih for ie fracional nonlinear reacion diffsion eaion based on a new ieraive ehod Con Nonlinear Sci Ner Sila [] S. Saha Ra K.S. Chadhri R.K. Bera Applicaion of odified decoposiion ehod for he analical solion of space fracional diffsion eaionapplied Maheaics and Copaion [] Veis Tr Nran Gzel Coparing Nerical Mehods for Solving Tie-Fracional Reacion-Diffsion Eaions ISRN Maheaical Analsis Vol. Aricle ID 7376 doi:.54//7376. [3] G.Hariharan K.Kannan Haar wavele ehod for solving Fisher s eaion Appl.Mah.Cop [4] G.Hariharan K.Kannan Haar wavele ehod for solving nonlinear parabolic eaions J.Mah.Che [5] G.Hariharan K.Kannan A Coparaive Sd of a Haar Wavele Mehod and a Resricive Talor's Series Mehod for Solving Convecion-diffsion Eaions In. J. Cop. Mehods in Engineering Science and Mechanics : [6] S.J.Liao Beond Perrbaion: Inrodcion o Hooop Analsis Mehod. CRC Press/Chapan and Hall Boca Raon. 4 [7] G.Hariharan The hooop analsis ehod applied o he Kologorov Perovsii Pisnov KPP and fracional KPP eaions J Mah Che 3 5:99. DOI.7/s [8] S.J.Liao Coparision beween he Hooop analsis ehod and Hooop perrbaion ehod Appl.ah.cop [9] S.J.Liao Noes on Hooop analsis ehod:soe definiions and heores Con Nonlinear sci.ner.sil [] S.J.Liao On he Hooop analsis ehod for non linear probles Appl. Mah. Cop [] S.Abbasband The applicaion of Hooop analsis ehod o nonlinear eaions arising in hea ransfer Ph.Leer.A ISSN : Vol 5 No 3 Jn-Jl 3 39