Coupled Method for Solving Time-Fractional Navier-Stokes Equation

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1 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 Coupled Mehod for Solng Tme-Fraconal Naer-Sokes Equaon S. O. Edek, and G. O. Aknlab Absrac Ths paper wnesses he couplng of wo basc ransforms: he He-Laplace ransform (HLT) whch s a blend of Laplace ransformaon and Homoop perurbaon mehods and he fraconal complex ransform (FCT). Ths couplng echnque s appled for he soluons of he me-fraconal Naer-Sokes model equaon. Two examples are consdered n demonsrang he effeceness of he coupled echnque. The exac soluons of he soled problems are obaned wh less compuaonal work, whle sll mananng hgh leel of accurac wh lle knowledge of fraconal calculus beng requred. Thus, he proposed mehod s recommended for handlng lnear and nonlnear fraconal models arsng n pure and appled scences. Kewords Fraconal complex ransform; Analcal soluons; Laplace ransform; HPM; Naer-Sokes model. I I. INTRODUCTION N appled scences, Naer-Sokes equaons (NSEs) ac as al models used n descrbng he phscs of man phenomena of scenfc and engneerng neres. The hae wder applcaons n modellng of weaher, ocean currens, waer flow n a ppe and ar flow around a wng. These NSEs esablsh he connecon beween pressure and exernal forces acng on flud o he response of he flud flow []. In general, we consder he me-fraconal NSE of he form: w + ( w ) w= ρ P+ w (.) w = where w s he flow eloc, w s he eloc, s he knemacs scos, P s he pressure, s he me, ρ s he dens, and s a del operaor. For a one dmensonal moon of a scous flud n a ube; he equaons of moon goernng he flow feld n he ube are Naer-Sokes equaons n clndrcal coordnaes [, ]. These are denoed b: w w w P= + η η η, w( η,) g( η) = (.) Ths work was suppored b Coenan Uners, Oa, Ngera. S. O. Edek s wh he Deparmen of Mahemacs, Coenan Uners, Oa, Ngera (correspondng auhor s emal: soedek@ahoo.com). G. O. Aknlab s wh he Deparmen of Mahemacs, Coenan Uners, Oa, Ngera (grace.aknlab@coenanuners.edu.ng). where P P =. ρ In relaon o sochasc dnamcs, appears a radon o represen he soluons of paral dfferenal equaons assocaed wh he Naer-Sokes models as he expeced funconals of sochasc processes [3, 4]. In ha regard, a coupled forward-backward sochasc dfferenal ssem (FBSDS) s formulaed n spaces of felds for he ncompressble Naer-Sokes equaon n he whole space [3]. Prodng soluons (numercal or exac) o lnear and nonlnear dfferenal equaons has led o he deelopmen and adopon of drec and sem-analcal mehods [5-7]. A lo of sem-analcal, analcal, and approxmae mehods hae been proposed n leraure [-3]. Fraconal Complex Transform (FCT) ransforms fraconal order dfferenal equaons o neger dfferenal equaons wh he help of Remann-Loulle deraes [3-33]. FCT as a soluon mehod for fraconal dfferenal equaons (FDEs) was frs proposed b [34]. The noon of Jumare s fraconal derae s nroduced as follows before he oerew of FCT. In hs work, our am s o prode analcal soluons o he NSEs usng he He-Laplace mehod whch combnes he basc feaures of he Laplace ransform and hose of He s polnomals mehod. II. THE OVERVIEW OF THE HE-LAPLACE METHOD [35, 36] Le Ξ be an negral or a dfferenal operaor on he equaon of he form: Ξ( I ) =. (.) Le H( I, p) be a conex homoop defned b: H( I, p) = pξ( I ) + ( p) G( I ) (.) where G ( I) s a funconal operaor wh I s a known soluon. Thus, we hae: H( I,) = G( I ) and (,) wheneer H( I, p) = s sasfed, and (,] H I =Ξ I (.3) p s an embedded parameer. In HPM, p s used as an expandng parameer o oban: ISSN:

2 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 p p p. (.4) = I= I =I + I + I + From (.4) he soluon s obaned as p. The conergence of (.4) as p has been consdered n [5]. The mehod consders N ( I ) (he nonlnear erm) as: N where ph = I = (.5) H be compued usng: k ' s are he so-called He s polnomals, whch can H( I, I, I, I3,, I ) = N p, n I! p. = p= (.6) A. The He-Laplace Mehod Le h(, x, ) = f( x) expressed as: p pg ( ) g( x), ( ) be a frs order nal alue problem (IVP), where p p ( x) are coeffcen of and g g( ) a nonlnear funcon and + + = = (.7) x and respecel, g x a source erm. Suppose we defne he Laplace ransform (resp. nerse Laplace. So he Laplace { } ransform) as L ( ) resp. L {( ) } ransform of (.7) s as follows: { } { } { } { } L + L p + L pg = L g x. (.8) Applng lnear proper of Laplace ransform on () elds: { } { } { } { } L + p L + p L g = L g x. (.9) Therefore, b dfferenal proper of Laplace ransform, (.9) s expressed as follows: sl = L g x p L + p L g { } { } { } { } { } ( ) ( + ) ( + ) (.) { } { } L = L g x p L g s p + s p. Thus, b nerse Laplace ransform, (.) becomes: { } L g x ( x) = H( x) + L ( s+ p ) pl { g( ) } (.) (.) L = H( x). (.3) ( s+ p ) Suppose we he soluon ( x ) assumes an nfne seres, hen b conex homoop, (.) can be expressed as: { } L g x p x L = + ( s p = + ) p pl p H ( ) = where g( ) ph ( ) = (.4) = for some He s polnomals H, and p an expandng parameer as defned earler. B. Jumare s Fraconal Derae (JFD) I s noed here ha JFD s a modfed form of he Remann- Loulle deraes []. Hence, he defnon of JFD and s basc properes as follows: Le h( ) be a connuous real funcon of (no necessarl h dfferenable), and Dh= denong JFD of h, of order w.r... Then, d Γ ( ) d d D h = Γ h h d (.5) where Γ denoes a gamma funcon. As summared n [], he basc properes of JFD are saed as P-P5: P: Dk=, >, ( ) ( h( ) h( )) d, (,) ( ) ( ( ) ( )), (,) ( ) d ( ( h ) ( ) ) ( η η ), [ ηη, + ), η, D kh = kd h >, P: Γ ( + ) P3: D Γ ( + ) = P4: =, >, ( ) ( ) D h h D h h P5: ( ) ( ) + h Dh g D h g = Dh D, where k s a consan. Noe: P, P, P3, P4, and P5 are referred o as fraconal derae of: consan funcon, consan mulple funcon, power funcon, produc funcon, and funcon of funcon respecel. P5 can be lnked o Jumare s chan rule of fraconal derae. III. THE FRACTIONAL COMPLEX TRANSFORM AND DTM Here, we brefl nroduce he concep of he FCT and he RDTM., ISSN:

3 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 A. The Fraconal Complex Transform Le us consder a general fraconal dfferenal equaon of he form: f ϖ, Dϖ, Dϖ, Dϖ, Dϖ =, ϖ = ϖ x,,,. (3.) ( x ) Then, he Fraconal Complex Transform [4] s defned as follows: a T =, (, ], Γ ( + ) bx X =, (, ], Γ ( + ) (3.) c Y =, (, ], Γ ( + ) d Z =, (, ], Γ ( + ) where a, b, c, and d are unknown consans. From P3, Γ ( + ) D =, >, Γ + a a DT D D Γ ( + ) Γ ( + ) = = Γ ( + ) a = = a. (3.3) Γ ( + ) Γ + Obousl n a smlar manner, usng properes P-P5, and he FCT n (.3), he followng are easl obaned: Hence, T DT = = a, X Dx X = = b, x Y DY = = c, Z DZ = = d. (3.4) ϖ D ϖ ( x,,, ) = D ϖ ( T( ) ) = DTϖ DT = a, T ϖ Dxϖ ( x,,, ) = Dxϖ ( X( x) ) = DXϖ Dx X= b, X ϖ Dϖ ( x,,, ) = Dϖ ( Y( ) ) = DYϖ DY = c, Y ϖ Dϖ ( x,,, ) = Dϖ ( Z( ) ) = DZϖ DZ = d. Z (3.5) D ϖ x,,, = Dϖ T x ϖ = ϖ =, (3.6) DT DT a T (,,, ) = ϖ Dϖ x D X x x ϖ = ϖ =, (3.7) DX Dx X b X (,,, ) = ϖ Dϖ x D Y ϖ = ϖ =, (3.8) DY DY c Y (,,, ) = ϖ Dϖ x D Z = ϖ = ϖ. DZ DZ d Z IV. APPLICATIONS: (3.9) In hs secon, he proposed mehod s appled o mefraconal Naer-Sokes models as follows: A. Problem : Consder he followng me-fraconal Naer-Sokes model: ψ ( ξ, ) = ξ. Procedure w.r. Problem : Soluon procedure: B FCT, ψ ψ ψ = +, ξ ξ ξ a T = Γ + ( ), whch accordng o secon 3 ges Hence, (4.) becomes: (4.) u Du = for a =. T ISSN:

4 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 w( η, ) = η. w w w = +, T η η η We ake he Laplace ransform (LT) of (3.) as follows: L ψ ψ ψ = + T ξ ξ ξ (4.) (4.3) L{ ψ } ψ ψ ψ = + L +. (4.4) s s ξ ξ ξ of B applng he nerse Laplace ransform, L ( ) L {( ) } on boh sdes of (4.4), we hae: ψ ψ L L + s ξ ξ ξ w = ξ + L, ψ = ψ ( ξ, ) s ψ ψ L { } = ξ + L +. (4.5) s ξ ξ ξ B Conex Homoop approach (4.5) becomes: = + ψ ψ pψ = ξ + L L p +. s ξ ξ ξ (4.6) Thus, comparng he coeffcens of he p powers n (4.6) ges: ( ) p : w = η ( ) ψ ψ = L L +, s ξ ξ ξ ψ ψ = L L +, s ξ ξ ξ ( 3) ψ ψ 3 = L L +, s ξ ξ ξ ( 4) ψ3 ψ3 4 = L L +, s ξ ξ ξ ( 5) ψ4 ψ4 5 = L L +, s ξ ξ ξ ( ) ψ ψ L L = +,. s ξ ξ ξ So, for ψ = ξ, we hae he followng: T ψ = ξ, ψ =, 3 ξ T 9845 T ψ4 =, ψ 7 6 =,, 8 ξ 6 ξ 3 T 3 T ψ =, ψ3 =, 5 ξ ξ T T ψ5 =, ψ 9 7 =, 3 8 ξ 6 ξ, (4.7). (4.8) (3.8) Thus, he soluon of (4.) s as follows: 3 4 T T 3 T 75 T ξ ξ ξ ξ 8 ξ w( ξ, T) = T 9845 T T ξ 6 ξ 6 ξ 3 5 ( ι 3) = ξ +! T ι.(4.9) ι ι= ξ ι Hence, he exac soluon of (4.) s: ι 3 5 ( ι 3) u( ξ, ) = ξ + ι ι = ξ ι! Γ ( + ) (4.) Our soluon (4.) s er much n lne wh hose obaned n [, 3]. B. Problem : Consder he followng me-fraconal Naer-Sokes model: w w w = p + + (4.) subec o: wr, = r. (4.) Soluon procedure: B FCT, a T = Γ +, whch accordng o secon 3 ges Hence, (4.5) and (4.6) become: w w w = p + +, T r r r wr (,) = r. u Du = for a =. T We ake he Laplace ransform (LT) of (4.3) as follows: (4.3) ISSN:

5 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 L w w w = ρ + + T r r r (4.4) L{ w} w w w = + L ρ + + (4.5) s s of B applng he nerse Laplace ransform, L ( ) L {( ) } on boh sdes of (4.5), we hae: w w L L ρ + + s w = r + L, ψ = ψ ( r, ) s ( ) w w ρ { } = r + L L + + (4.6) s B Conex Homoop Approach (4.6) becomes: ( r ) pw = + w w. = + L L p ρ + + s (4.7) Thus, comparng he coeffcens of he p powers n (4.7) ges: ( ) : ( = ) p w r ( ) w w p : w = L L ρ, s w w p : w = L L ρ, s ( 3) w w p : w3 = L L ρ, s ( 4) w3 w3 p : w4 = L L ρ, s ( 5) w4 w4 p : w5 = L L ρ, s ( ) w w p : w L L ρ = + +,. s Thus, applng he He-Laplace Mehod n secons. and. o (4.7) ges he soluon of (4.7) as: So, smplfng he process for w ( r ) followng: (, ) ( ) ( 4) =, we hae he w rt = r + p T. (4.8) Hence, he exac soluon of (3.) s: (, ) ( r ) wr ( p ) ( ) 4 = + Γ +. (4.) Our soluon (4.) s er much n lne wh hose obaned n [, 3]. Remark: for p =, he soluon s: 3 wr (, ) = ( r ). (4.) Γ + For =, and wr, = r 3 as p = we hae he correspondng exac soluon. Here, we presen n Fg. hrough Fg. 6, he relaonshp beween he exac soluons of he neger cases =, and he fraconal cases for as regards example II. Fg. : The soluon graph for =.5 & r [,] ISSN:

6 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 Fg. 4: The soluon graph for =.5 & r [,] Fg. : The soluon graph for =.5 & r [,] Fg. 5: The soluon graph for =.5 & r [,] Fg. 3: The soluon graph for =.5 & r [,] Fg. 6: The soluon graph for =.5 & r [,] Remark: In Fg. 7 and Fg. 8, he exac and approxmae soluons of problem are dsplaed respecel. These are consdered a =. ISSN:

7 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 8 Fg. 7: The exac soluon graph for problem. Fg. 8: The approxmae soluon graph for problem V. CONCLUDING REMARKS In hs paper, exac soluons of me-fraconal Naer- Sokes model equaon were proded n seres form wh easl compuable componens. The mehod of soluons noled he couplng of wo basc ransforms: he He-Laplace ransform (HLT) whch s a blend of Laplace ransformaon and Homoop perurbaon mehods and he fraconal complex ransform (FCT) wh he requremen of lle knowledge of fraconal calculus whle sll mananng hgh ACKNOWLEDGMENT The auhors hank Coenan Uners for fnancal suppor and proson of good workng enronmen. Also, sncere hanks o he anonmous referee(s) for consruce suggesons. REFERENCES [] D. Kumar, J. Sngh, S. Kumar, A fraconal model of Naer Sokes equaon arsng n unsead flow of a scous flud, Journal of he Assocaon of Arab Unerses for Basc and Appled Scences 7, (5): 4 9. [] S. Kumar, D. Kumar, S. Abbasband, M.M. Rashd, Analcal soluon of fraconal Naer Sokes equaon b usng modfed Laplace decomposon mehod, An Shams Engneerng Journal 5, (4): [3] F. Delbaen, J. Qu, S. Tang, Forward-Backward Sochasc Dfferenal Ssems Assocaed o Naer-Sokes Equaons n he Whole Space, Mahemacal Phscs, 3, hps://arx.org/abs/ [4] S. O. Edek, E. A. Owoloko, E. A., O. O. Ugbebor. The Modfed Black-Scholes Model a Consan Elasc of Varance for Sock Opons Valuaon, AIP Conference proceedngs, 75, 4 (6); do:.63/ [5] A.M. Wawa, M.S. Mehanna, The combned Laplace Adoman mehod for handlng sngular negral equaon of hea ransfer, In J Nonlnear Sc. (): [6] Y. Khan, N. Fara, Applcaon of modfed Laplace decomposon mehod for solng boundar laer equaon, J Kng Saud Un, 3 ():5 9. [7] N. Fara, Sud of he effecs of he Renolds number on crcular porous slder a araonal eraon algorhm-ii, Compu Mah Appl. 6(8), ():99 4. [8] M.M. Rashd, The modfed dfferenal ransform mehod for solng MHD boundar-laer equaons, Compu Phs Commun, 8, (9): 7. [9] G.O. Aknlab, S.O. Edek, Solng Lnear Schrodnger Equaon hrough Perurbaon Ieraon Transform Mehod, Proceedngs of he World Congress on Engneerng, London, UK, 7. [] J. Sngh, D. Kumar and S. Rahore, Applcaon of Homoop Perurbaon Transform Mehod for Solng Lnear and Nonlnear Klen-Gordon Equaons, Journal of Informaon and Compung Scence, 7 (), (): [] S.O. Edek, I. Adna, O.O. Ugbebor, The Effec of Sochasc Capal Resere on Acuaral Rsk Analss a an Inegrodfferenal equaon, IAENG Inernaonal Journal of Appled Mahemacs, 44 (), (4): [] J.H. He, A couplng mehod of homoop echnques and perurbaon echnque for nonlnear problems, Inernaonal Journal of Non-Lnear Mechancs, 35() (): [3] N.H. Swelam, M.M. Khader, Exac soluons of some coupled nonlnear paral dfferenal equaons usng he homoop perurbaon mehod, Compuers & Mahemacs wh Applcaons, 58: (9): [4] M. Shakl, T. Khan, H.A. Wahab, S. Bha, A Comparson of Adoman Decomposon Mehod (ADM) and Homoop Perurbaon Mehod (HPM) for Nonlnear Problems, Inernaonal Journal of Research n Appled, Naural and Socal Scences, (3), (3): [5] G.O. Aknlab and S.O. Edek Perurbaon Ieraon Transform Mehod for he Soluon of Newell-Whehead-Segel Model Equaons, Journal of Mahemacs and Sascs, 7, hps://arx.org/abs/ [6] A. Ghorban and J. S. Nadf, He s homoop perurbaon mehod for calculang Adoman s polnomals, In. J. Nonln. Sc. Num. Smul. 8 () (7): [7] A. Ghorban, Beond Adoman s polnomals: He polnomals, Chaos, Solons & Fracals, (7), n press. [8] H.A. Wahab, A. Jamal, S. Bha, M. Naeem, M. Shahad, S. Hussan, Applcaon of homoop perurbaon mehod o he ISSN:

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