A DECOMPOSITION METHOD FOR SOLVING DIFFUSION EQUATIONS VIA LOCAL FRACTIONAL TIME DERIVATIVE

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1 S13 A DECOMPOSITION METHOD FOR SOLVING DIFFUSION EQUATIONS VIA LOCAL FRACTIONAL TIME DERIVATIVE by Hossen JAFARI a,b, Haleh TAJADODI c, and Sarah Jane JOHNSTON a a Deparen of Maheacal Scences, Unversy of Souh Afrca, Florda, Souh Afrca b Deparen of Maheacs, Unversy of Mazandaran, Babolsar, Iran c Deparen of Maheacs, Unversy of Ssan and Baluchesan, Zahedan, Iran Orgnal scenfc paper DOI: 10.98/TSCI15S1S3J In hs paper a decoposon ehod based on Dafardar-Jafar ehod s appled for solvng dffuson equaons nvolvng local fraconal e dervaves. The convergence of hs ehod for solvng hese ype of equaons s proved. Key words: erave ehod, local fraconal dervave, dffuson equaon, Dafardar-Jafar ehod Inroducon Fraconal calculus and fraconal dynacs have becoe ho opcs of research n whch here s rapd developen and pleenaon n varous felds of engneerng and scence [1-5]. Analyss of he dffuson equaon n aheacal physcs have been of consderable neres n he leraure. Fraconal dffuson equaon has poran applcaons o aheacal physcs. Ngaulln [6] has eployed he fraconal dffuson equaon o descrbe dffuson n eda wh fracal geoery. Also hs equaon s used o odel anoalous dffuson n plasa ranspor. In he recen years several auhors, Jafar e al. [7], Chuna [8], and Saha Ray [9, 10], have solved he fraconal dffuson-wave equaons usng dfferen ehods. Recenly, local fraconal calculus heory has been used o odel soe nondfferenable probles for aheaca physcs [11-17] and he references heren. Our an a n hs paper s o apply a new erave ehod o solve dffuson equaons nvolvng local fraconal e dervaves. Ths ehod s nroduced by Dafardar and Jafar [18], laer referred o as he Dafardar-Jafar ehod (DJM) []. The core of hs approach s o solve non-lnear equaons whou usng Adoan polynoals. In [, 19] he convergence of hs ehod s dscussed. Basc defnons Defnon 1. A real funcon f( x ) s sad local fraconal connuous on he nerval ( ab, ) f x ( a, b) l x x f( x) f( x 0 0) whch s denoed by f( x) Ca ( ab, ). Defnon. Le f( x) Ca ( ab, ), he local fraconal dervave of f( x) of order a x x 0 s defned as: Correspondng auhor; e-al: jafarh@unsa.ac.za; jafar@uz.ac.r

2 S14 x d ( ) D f x f x0 0 x x d x 0 x x 0 ( x x0 ) D f( x ) f( x) f ( x) l [ ( ) ( )] (1) where [ f( x) f( x0)] Γ ( [ f( x) f( x0)]. Noe ha he local fraconal dervave of hgh order and he local fraconal paral dervave of hgh order are wren n he for: respecvely. (( k es ( ( k ( k) x x x D f( x) f ( x) D D f( x) () (( ((, k es k f( xy, ) f( xy, ) (3) k Defnon 3. The local fraconal negral operaor s defned as: b N 1 a 1 a 1 a aib f( x) f( )(d ) l f( j)( j) Γ ( a ( 1) 0 a Γ a + j 0 (4) where j j+ 1 j, ax{ 0, 1,, j, }, [ j, j+ 1], j 0,, N 1, 0 a, and N b, s a paron of he nerval [ ab, ]. The followng properes hold: x x x D[ f( xgx ) ( )][ D f( x)] gx ( ) + f( x)[ D gx ( )] a ( a) x a ( a) a x a a x I f( xg ) ( x)[ f( xgx ) ( )] I f ( xgx ) ( ) D x k ( k 1) x x Γ (1 + k) Γ [1 + ( k 1) ] k x 0 Ix ( k+ 1) x Γ (1 + k) Γ [1 + ( k ] (5) Defnon 4. In fracal space, he Mage-Leffler funcon, sne and cosne funcons are defned on Canor ses are defned as: n0 n x E ( x ), 0 < 1 (6) Γ (1 + n) n0 (n+ 1) n x sn ( x ) ( 1), 0 < 1 (7) Γ [1 + (n ] n0 n n x cos ( x ) ( 1), 0 < 1 (8) Γ [1+ n ]

3 S15 Noe ha: d E ( x ) ( E x ) dx d E ( kx ) ( ke kx ) dx (9) (10) Analyss of he ehod In hs secon, a decoposon ehod based on Dafardar-Jafar ehod s appled for solvng dffuson and hea equaon nvolvng local fraconal dervave [18]. Ths decoposon of he non-lnear funcon s que dfferen fro ha of Adoan decoposon. we nvesgae he general for of non-lnear local fraconal paral equaon: n 1 ux (, ) D u( x, ) ψ( x, ) + ϕ( x, ) u ( x, ) (11) ux (,0) gx ( ) (1) where 0, 1,,, x ( x1, xn), and ψ ( x, ) C a. For 0or 1, and 0 < < 1, eq. (11) represens lnear fraconal dffuson equaon (hoogeneous f ϕ( x, ) 0 and nonhoogeneous oherwse). I represens non-lnear dffuson equaon for, 3, When ( L ) D ( ) and by applyng L o eq. (11), we ge: n ( ) ( ) ux (, ) ux (, ) ux (,0) L [ ϕ( xu, ) ( x, )] L + + ψ ( x, ) (13) 1 Followng Dafardar-Jafar ehod, he unknown funcon can be shown n ers of an nfne seres as: u u (14) 0 For 0 we se: ( ) f( x) ux (,0) + L [ ϕ( x, )] n ( ) 1 Nu ( ) L ux (, ) ψ ( x, ) (15) and for oher values of 1,, we se: f( x) ux (,0) ( n ) ( ) ϕ + ψ 1 ux (, ) N( u) L [ ( x, ) u ( x, )] L ( x, ) (16) The non-lnear funcon N can be decoposed as: 1 N u N( u0) + N uj N uj 0 0 j0 j0 (17)

4 S16 Subsung eqs. (14) and (17) n eq. (13), we have: hen we have he recurrence relaons: and are calculaed. u f + N( u ) + N u N u 1 0 j j 0 0 j0 j0 u un+ 1 Nu0 + + un Nu0 + + un 1 n f u Nu ( ) ( ) ( ), 1,,. u f u 1 (18) + (19) Theore 1. If he seres soluon defned n eq. (19) s convergen, hen converges o an exac soluon of eq. (11). Proof. The runcaed seres eq. (11), usng he above we have: u s used as an approxaon o he soluon u () of 0 Takng ls of eq. (0), gans: u f + N u (0) 0 0 u l u l f + N u l f + l N u f + N l u f + N( u) Hence u s he soluons of eq. (11) and he proof s coplee. Illusrave exaples To deonsrae he effecveness of he ehod we consder here soe fraconal dffuson and hea equaons wh local fraconal e dervave. Exaple 1. Consder he dffuson equaon wh local fraconal e dervave: ux (, ) Dux (, ), < x<, >0 (1) By applyng ( ) I L x ux (,0) e, (0,1) on boh sde of eq. (1) and usng nal condon, we ge: ux (, )e x I ux (, ) Followng hs ehod, ux (, ) s represened as: ()

5 S17 and we ge: u Nu ( ) I u 0 (3) ux (, ) where he non-lnear funcon N can be decoposed as: 1 N u N( u0) + N uj N uj 0 0 j0 j0 (4) (5) Accordng o he Dafardar-Jafar ehod and applyng, we oban: x u0( x, )e u1( x, ) Nu ( 0) n n 1 un+ 1( x, ) N uj N uj n 1,,... j0 j0 In he frs eraon we have: u 0 1 x x u1 ( x, ) I e (d ) e x Γ ( (6) 0 Γ ( (, )e x u x Γ ( (7) In slar anner, we can derve he oher approxaon. Fnally, he copac soluon becoes: where ses. Exaple. Consder he fraconal dffuson equaon wh local fraconal e dervave: a a a n x x ( ) x ux (, )e 1 + e 1 + e E ( ) (8) Γ ( a Γ (a Γ ( na n1 n n E ( kz ) ( k z )/ Γ ( n denoes Mag-Leffler funcon defned on Canor n0 ux (, ) Dux (, ) + ux (, ) ux (, 0) cos(π x), (0,1) (9) Accordng slar procedure as n he prevous exaple, we can srucure he sae local fraconal eraon procedure of eq. (18):

6 S18 u0( x, ) ux (,0) u1( x, ) Nu ( 0) n n 1 un+ 1( x, ) N uj N uj n 1,,... j0 j0 (30) Makng use of eq. (30), we presen: u0 ( x, ) cos(π x ) (31) 1 (, ) [1 π ]cos(π ) u x x Γ ( (, ) [1 π ] cos( ) u x π x Γ ( (3) (33) Proceedng n hs anner, we can derve he oher approxaon. Thus, he fnal soluon reads: ux (, ) cos(π x) + [1 π ]cos(π x) [1 π ] cos(π x) ( 1) + ( 1) + Γ + Γ + [1 π ] [1 π ] cos(π x) cos(π xe ) ([1 π ] ) Γ ( Γ ( (34) where E ( kz ) denoes he Mag-Leffler funcon defned on Canor ses. Concluson In hs work, we appled he Dafardar-Jafar ehod o solve he dffuson and hea equaons nvolvng local fraconal dervave. The resuls show ha hs ehod s accurae and effecve and can be used for non-lnear local fraconal dfferenal equaons. Ths ehod have an advanage over he Adoan decoposon ehod n ha he Dafardar-Jafar ehod can solve non-lnear probles whou usng Adoan polynoals. Maheaca has been used for copuaons and prograng n hs paper. References [1] Baleanu, D., e al., Fraconal Calculus Models and Nuercal Mehods, (Seres on Coplexy, Nonlneary and Chaos), World Scenfc, Sngapore, 01 [] Jafar, H., An Inroducon o Fraconal Dfferenal Equaons (n Persan), Mazandaran Unversy Press, Babolsar, Iran, 013 [3] Podlubny, I., Fraconal Dfferenal Equaons, Acadec Press, San Dego, Cal., USA, 1999 [4] Aangana, A., Bldk, N., The Use of Fraconal Order Dervave o Predc he Groundwaer Flow, Maheacal Probles n Engneerng, 013 (013), ID [5] Aangana, A., Drawdown n Prolae Spherodal Sphercal Coordnaes Obaned va Green s Funcon and Perurbaon Mehods, Councaons n Non-lnear Scence and Nuercal Sulaon, 19 (014), 5, pp [6] Ngaulln, R. R., The Realzaon of he Generalzed Transfer Equaon n a Medu wh Fracal Geoery, Phys Saus Sold B, 133 (1986), 1, pp

7 S19 [7] Jafar, H., Dafardar-Gejj, V., Solvng Lnear and Non-lnear Fraconal Dffuson and Wave Equaons by Adoan Decoposon, Appled Maheacs and Copuaon, 180 (006),, pp [8] Chuna, C., e al., Nuercal Mehod for he Wave and Non-lnear Dffuson Equaons wh he Hooopy Perurbaon Mehod, Copuers and Maheacs wh Applcaons, 57 (009), 7, pp [9] Saha Ray, S., Bera, R. K., An Approxae Soluon of a Non-lnear Fraconal Dfferenal Equaon by Adoan Decoposon Mehod, Appled Maheacs and Copuaon, 167 (005), 1, pp [10] Saha Ray, S., Bera, R. K., Analycal Soluon of a Fraconal Dffuson Equaon by Adoan Decoposon Mehod, Appled Maheacs and Copuaon, 174 (006), 1, pp [11] Cao, Y., e al., Local Fraconal Funconal Mehod for Solvng Dffuson Equaons on Canor Ses, Absrac and Appled Analyss, 014 (014), do: /014/ [1] Wang, S., e al., Local Fraconal Funcon Decoposon Mehod for Solvng Inhoogeneous Wave Equaons wh Local Fraconal Dervave, Absrac and Appled Analyss, 014 (014), do: /014/ [13] Yang, Y., e al, A Local Fraconal Varaonal Ieraon Mehod for Laplace Equaon whn Local Fraconal Operaors, Absrac and Appled Analyss 013 (013), do: /013/0650 [14] Yang, A., e al., Analycal Soluons of he One-Densonal Hea Equaons Arsng n Fracal Transen Conducon wh Local Fraconal Dervave, Absrac and Appled Analyss, 013 (013), do: /013/46535 [15] Yang, X. J., Local Fraconal Funconal Analyss and Is Applcaons, Asan Acadec Publsher, Hong Kong, 011 [16] Yang, X. J., Advanced Local Fraconal Calculus and Is Applcaons, World Scence Publsher, New York, USA, 01 [17] Yang, X. J., e al., Fracal Boundary Value Probles for Inegral and Dfferenal Equaons wh Local Fraconal Operaors, Theral Scence, (013), DOI: 10.98/TSCI Y [18] Dafardar-Gejj, V., Jafar, H., An Ierave Mehod for Solvng Non-lnear Funconal Equaons, J. Mah. Anal. Appl., 316 (006),, pp [19] Bhalekar, S., Dafardar-Gejj, V., Convergence of he New Ierave Mehod, Inernaonal Journal of Dfferenal Equaons, 011 (011), ID Paper subed: Ocober 10, 014 Paper revsed: January 0, 015 Paper acceped: February 1, 015