LAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS

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1 Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp Open forum LAPLAE TRANSFORM OVEROMING PRINIPLE DRAWBAKS IN APPLIATION OF THE VARIATIONAL ITERATION METHOD TO FRATIONAL HEAT EQUATIONS by Guo-heng WU,b ollege of Mhemics nd Informion Science, Neijing Norml Universiy, Neijing, hin b ollege of Wer Resources nd Hydropower, Sichun Universiy, hengdu, hin Shor pper DOI:.2298/TSI24257W This noe presens Lplce rnsform pproch in he deerminion of he Lgrnge muliplier when he vriionl ierion mehod is pplied o ime frcionl he diffusion equion. The presened pproch is more srighforwrd nd llows some simplificion in pplicion of he vriionl ierion mehod o frcionl differenil equions, hus improving he convergence of he successive ierions. Key words: vriionl ierion mehods, Lplce rnsform, Lgrnge muliplier, frcionl he diffusion equion Inroducion The pplicion of he frcionl clculus is ho opic in he rnsfer llowing solving mny no-liner problems such s Sefn problem [], he herml sub-diffusion model [2], nd he rnsiion flows of complex fluids [3, 4]. Even hough he frcionl models re correcly describing non-liner rel world phenomen, he soluions re quie complex nd he rel cll mong he scieniss o find efficien nlyicl echniques for soluions of such problems in explici forms. The vriionl ierion mehod (VIM) [5, 6] is n nlyicl echnique which hs been widely used in he ps en yer in non-liner problems. The key problem of he VIM is he correc deerminion of he Lgrnge muliplier when he mehod is pplied o frcionl equions describing diffusion of he or mss. This crucil poin of he mehod is solved efficienly in he presen work. Problem formulion The following ineger-order prbolic equion describes rnsien he conducion: u =cu xx, u(, x) =f(x) () Auhor's e-mil: wuguocheng22@yhoo.com.cn

2 Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp The soluion in ccordnce o he VIM rules needs o consruc he correcion funcionl: un un l(, )( un, cun, xx) d, u f ( x) (2) The weighed funcion l(, ) is clled he Lgrnge muliplier which cn be deermined by he vriionl heory looking for sionry condiions of he funcionl (2) [5, 6]. This procedure involves inegrion by prs of he inegrl in (2) h leds o serious problem when he VIM is pplied o differenil equions of frcionl order, nmely: Here D u cu xx, u (, x ) f ( x ), (3) D is he puo derivive [7]. Equion (3) reduces o he clssicl one () for =. onsrucing he correcion funcionl o eq. (3) we ge: u u n n l(, )( D un cun, xx) d, (4) The Lgrnge muliplier in eq. (3) cnno be deermined in srighforwrd wy like h in he ineger-order model (). The principle problem is he exisence of he frcionl derivive under he inegrl sign which does no llow he inegrion by prs o be performed. The secion shows by pplying he Lplce rnsform o eq. (3) he problem cn be voided. Lplce rnsform pproch in he Lngrge muliplier's deerminion Assuming for simpliciy of he explnion c = in eq.(3) nd pplying he Lplce rnsform L o boh sides we ge: where Lplce rnsform of he herm su () s u( ) s Lu [ xx ], (5) m D u holds [7]: L[ D u] s U() s u( k) ( ) s k, m [ ], U() s L[ u( )] (6) k Then, consrucing he correcion funcionl o (5) we hve: U s U s s U s u x s n () n() l( n() (, ) L[ un, xx]) (7) The sionry condiion of he funcionl (7) require he following condiion o be sisfied: du n () s (7b) du n () s This condiion simply defines Lgrnge muliplier s: l (7c) s As resul, he inverse Lplce rnsform of he vriionl ierion formul (7) becomes: un un L s U n s u x s L un xx u L [ l( () (, ) [, ])] Lu s ( [ nxx ]), (8) u u(, x) f ( x)

3 Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp On he oher hnd, we recenly give noher wy o idenify he Lgrnge muliplier in [8-]: u u n n l(, )( Dun un xx, ) d, l(, )= ( ) ( ) (9, b) G( ) Boh (9, 9b) nd (8) cn led o he sme resul. The Lgrnge muliplier (9b) rnsforms he Riemnn inegrl (9) of he ierion funcionl ino he Riemnn-Liouville (R-L) inegrl. This is correc pproch becuse if we pply o he R-L inegrl I boh sides of eq. (3), hn he correc ierion formul should be: u u I n n [ l(, )( D u n un xx, )], () However, lbei he correcness of () he impossibiliy o pply he inegrion by prs in he frcionl inegrl led o simplificion by replcing i by he Riemnn inegrl s i defined by (9). The simplificion ever coninued wih he Lgrnge muliplier s l = [-4]. This chin of simplificions leds o poor convergence of he ierion formul. The Lplce rnsform pproch in he deerminion of l(, ) correcs he second sep of he simplificion chin nd resuls in wh he ierion formul should be. The finl resul is, in fc, he Lgrnge muliplier defined by (9b) is he kernel of he R-L inegrl. This poin ws nlyzed recenly by Hrisov [5] wih wo opions in he inegrion: () ierion formul s i is defined by (9, b) nd (2) ierion formul wih l = nd he R-L inegrl defined by (). Exmple: Frcionl he diffusion equion of he R-L ype In his secion, we pply our mehod o he frcionl he-diffusion equion of he R-L ype, nmely: RL D u u xx, I u ( ) sin( x ), () which cn lso describes rnsien flow in porous medium. Our pproch leds o he following ierion formul: u u L n Lun xx ( [ s, ]) sin( x ) u G( ) The successive ierions re obined s: u u n sin( x ) G( ) sin( x) sin( x ) G( + ) 2 n ( ) k G( +k) k (2) For n, u n rpidly ends o sin(x) E, ( ) which is n exc soluion of (). The diffusion behviors re shown in fig. differen frcionl orders.

4 Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp Figure. VIM soluions o he sub-diffusion equion vrious ( < ) onclusions This scienific noe presens he pplicion of Lplce rnsform in correc deerminion of Lgrnge muliplier when he VIM is pplied o frcionl he-diffusion equions. The pproch is exemplified by soluions of frcionl he diffusion equions wih he puo derivive nd he R-L derivive, respecively. The resuls show h he new pproch is more efficien nd srighforwrd o idenify he Lgrnge muliplier here nd ye give pproxime soluions of high ccurcies. The VIM now cn be relible ool o nlyiclly invesige he models wih frcionl derivives. Nomenclure c specific he cpciy, [Jkg ] D u ime-frcionl puo derivive RL D u ime-frcionl Riemnn-Liouville derivive E,b Mig-Leffler funcion wih prmeers nd b I Riemnn-Liouville inegrl of order L Lplce rnsform m ineger beween nd + n order of he pproxime soluions s complex rgumen of Lplce rnsform ime, [s] U s Lplce rnsform of u() u emperure, [K] u n n-h order pproxime soluion x spce co-ordine, [m] Greek symbols frcionl order [ ] d vriion operor G gmm funcion l(, ) Lgrnge muliplier ime, [s] References [] Meilnov, R., Shbnov, M., Akhmedov, E., A Reserch Noe on Soluion of Sefn Problem wih Frcionl Time nd Spce Derivives, In. Rev. hem. Eng., 3 (2), 6, pp [2] Hrisov, J., He Blnce Inegrl o Frcionl (Hlf-Time) He Diffusion Sub-Model, Therml Science, 4 (2), 2, pp [3] Siddique, I., Vieru, D., Sokes Flows of Newonin Fluid wih Frcionl Derivives nd Slip he Wll, In. Rev. hem. Eng., 3 (2), 6, pp

5 Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp [4] Qi, H., Xu, M., Some Unsedy Unidirecionl Flows of Generlized Oldroyd-B Fluid wih Frcionl Derivive, Appl. Mh. Model., 33 (29),, pp [5] He, J.-H., Approxime Anlyicl Soluion for Seepge Flow wih Frcionl Derivives in Porous Medi, ompu. Mehod. Appl. M., 67 (998), -2, pp [6] He, J.-H., Vriionl Ierion Mehod Kind of Non-Liner Anlyicl Technique: Some Exmples, In. J. Nonliner Mech., 34 (999), 4, pp [7] Podlubny, I., Frcionl Differenil Equions, Acdemic Press, New York, USA, 999 [8] Wu, G.-., Vriionl Ierion Mehod for Solving he Time-Frcionl Diffusion Equions in Porous Medium, hin. Phys. B., 2 (22), 2, 254 [9] Wei, M. B., Wu, G.-., Vriionl Ierion Mehod for Sub-Diffusion Equions wih he Riemnn-Liouville Derivives, He. Trns. Res. cceped, 22 [] Wu, G.-., Applicions of he Vriionl Ierion Mehod o Frcionl Diffusion Equions: Locl Versus Nonlocl Ones, In. Rev. hem. Eng., 4 (22), 5, pp [] Momni, S., Odib, Z., Anlyicl Approch o Liner Frcionl Pril Differenil Equions Arising in Fluid Mechnics, Phys. Le., A 355 (26), 4-5, pp [2] Inc, M., The Approxime nd Exc Soluions of he Spce- nd Time-Frcionl Burgers Equions wih Iniil ondiions by Vriionl Ierion Mehod, J. Mh. Anl. Appl., 345 (28),, pp [3] Molliq, R. Y., Noorni, M. S. M., Hshim, I., Vriionl Ierion Mehod for Frcionl He- nd Wve-Like Equions, Nonliner Anlysis: Rel World Applicions, (29), 3, pp [4] Skr, M. G., Erdogn, F., Yildirim, A., Vriionl Ierion Mehod for he Time-Frcionl Fornberg-Whihm Equion, ompu. Mh. Appl., 63 (22), 9, pp [5] Hrisov, J., An Exercise wih he He s Vriion Ierion Mehod o Frcionl Bernoulli Equion Arising in Trnsien onducion wih Non-Liner He Flux he Boundry, In. Rev. hem. Eng., 4 (22), 5, pp Pper submied:ocober 9, 22 Pper cceped: Ocober 23, 22