SOLUTION OF DIFFUSION EQUATION WITH LOCAL DERIVATIVE WITH NEW PARAMETER

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1 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 S31 SOLUTION OF DIFFUSION EQUTION WITH LOCL DERIVTIVE WITH NEW PRMETER by bdon TNGN a and Emile Franc DOUNGMO GOUFO b a Insiue of Groundwaer Sudies, Universiy of he Free Sae, Bloemfonein, Souh frica b Deparmen of Mahemaical Sciences, Universiy of Souh frica, Florida, Souh frica Original scienific paper DOI: 1.98/TSCI15S1S31 local derivaive wih new parameer was used o model diffusion. The modified equaion was solved ieraively. Sabiliy of he used mehod ogeher wih he uniqueness of he special soluion was sudied. n algorihm was proposed o derive he special soluion. Key words: diffusion equaion, local derivaive wih new parameer, sabiliy, convergence Inroducion In he mos recen cenury, mahemaics apparaus were employed o replicae real world problems, which ae place in all branches of sciences. The diffusion equaion is a parial differenial equaion ha porrays densiy dynamics in a maerial underaes diffusion. I is also used o describe progression demonsraing diffusive-lie performance, for example he ransmission of alleles in a populaion geneics [1-4]. The convecion diffusion equaion explains he flow of hea, paricles, or oher physical quaniies in condiions where here is boh diffusion and convecion or advecion. For informaion concerning he equaion, is derivaion, and is heoreical significance and consequences [1-4]. The following convecion diffusion equaion is considered here []: Tx (, ) Tx (, ) Tx (, ) + u = λ + Qx (, ) x (1) This equaion can be wrien in he form: Tx (, ) Tx ( ) Tx ( ) Qx = u, + a, + (, ) where a = λ/ is he diffusion coefficien. One of he mos used mahemaical conceps in modeling is perhaps he concep of derivaive. The conemporary improvemen of calculus is frequenly aribued o Isaac Newon and Leibniz, who provided self-deermining and unified approaches o differeniaion and derivaives [5-7]. Due o he complexiy of he physical problems encounered in our daily basic, he concep of derivaive has been modified. The concep of fracional derivaive was formulaed by Riemann-Liouville, and laer modified by Corresponding auhor; abdonaangana@yahoo.fr ()

2 S3 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 Capuo [8-1]. This concep has been used for modeling real world problems [11-13]. However, hese derivaives do no saisfy basic properies of he Newonian concep, for insance he produc, quoien and he Chain rules ha are being augh o undergraduae sudens. This issue has been a worry for researchers in he field of fracional calculus [14, 15]. To solve his problem, we have inroduced a fracional derivaive called bea-derivaive. The derivaive will be used in his paper o modify he diffusion equaion. We shall presen in he following secion, he definiion and some properies of he local derivaive wih parameer. Definiion and properies of bea-derivaive Le a R and g be a funcion, such ha g: [ a,) R. Then, he β-derivaive of g is defined as: 1 β 1 g + + g () ( β ) β Γ D [ g ( )] = lim (3) for all a, β (, 1]. The funcion g is said o be β-differeniable if he above limi exiss. ssume ha, he given funcion g: [ a,) R is β-differeniable a a given poin a, β (, 1], hen, g is also coninuous a, [16]. ssuming ha f is β-differeniable on an open inerval (a, b) hen, [16]: if D β f() < for all (a, b) hen f is decreasing here, if D β f() > for all (a, b) hen f is increasing here, and if D β f() = for all (a, b) hen f is consan here. ssuming ha, g and f are wo funcions β-differeniable wih β (, 1] hen, he following relaions can be saisfied [16]: β β β Dx D D [ af ( x) + bg( )] = a [ f ( )] + b [ f ( )] for all a and b real number; () c D β = for c any given consan; β β β D D D [ f() g ()] = g () [ f()] + f() [ g ()]; β β f() D = g () [ f()] f() D [ g ()] D g () g () ssuming ha f :[ a,) R, be a funcion such ha f is differeniable and also α is differeniable. Le g be a funcion defined in he range of f and also differeniable, hen we have he rule [16]: as: β 1 [ gof ( x)] f ( ) g [ f ( )] D = + (4) Γ( β ) Le f be a funcion, defined in an open inerval (a, b), hen he β-inegral of f is given β I x x 1 β β 1 β 1 [ f( x)] = + f( )d (5) Γ( β )

3 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 S33 Soluion of he modified diffusion equaion One can find in he lieraure nowadays several mehods o dealing wih linear and non-linear equaions. However, we shall menion he recen and efficien one, ha have been inensively used, hey are homoopy perurbaion mehod [17, 18], domian decomposiion mehod [19, ], homoopy Laplace perurbaion mehod [1], Sumudu homoopy perurbaion mehod [], and homoopy decomposiion mehod [3, 4]. However, in his paper we shall use only wo of hese menioned echniques namely: Laplace homoopy perurbaion mehod, and homoopy decomposiion mehod. Noe ha, he homoopy decomposiion mehod will be used o solve he sysem wih he β-derivaive. We modify eq. () by replacing he local derivaive wih he β-derivaive o obain: β [ ( )] D Tx u a Tx (, ) Tx (, ) Qx (, ), = + + To solve eq. (6), we apply on boh sides he inverse operaor of β-derivaive eq. (5) o obain: ( ) ( ) ( ) ( ) ( ) β Tx, Tx, Qx, T x, T x, = u a I + + (7) We assume ha, he soluion of our equaion is in he following form, wih p an imbedding parameer: Tx (,, p) = pt ( x, ) (8) = However, replacing eq. (8) in eq. (7), we obain: = pt( x, ) Tx (, ) = pt( x, ) pt( x, ) β = = Qx ( ) p u a I, = + + Now, re-arranging and puing ogeher all erms according o heir power of he embedding parameer p, we obain he equaions: ( x, ) ( ) ( ) ( ) ( ), 1( ) β T x, T x, Q T x, = Tx, T x, = u a I + + For any > 1, we have he recursive formula: 1( ) 1( ) ( ) β T x, T x, T x, = u a I + I is herefore imporan o noe ha, if he firs componen is provided, he res are obained jus by inegraion of each above equaion. (6) (9) (1)

4 S34 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 Convergence analysis One of he imporan pars of any ieraion mehod is o prove he uniqueness and he convergence of he mehod; we are going o show he analysis underpinning he convergence and he uniqueness of he proposed mehod for he general soluion for p = 1. ssuming ha X and Y are Banach spaces and V : X Y is conracion non-linear mapping. If he progression engender by he hree dimensional homoopy decomposiion mehods is regarded as: n 1 T( x, ) = VT [ ( x, )] = T( x, ), n= 1,, 3 (11) n n 1 = Then, he following saemens hold: a) T(, ) (, ) n n x Tx ρ Ix (, ) Tx (, ),(< ρ < 1) b) For any n greaer han, T(x, ) is always in he neighborhood of he exac soluion T(x, ) and c) lim T( x, ) = T( x, ) n Proof. The proof of a) shall be achieved via inducion on he naural number n. However, when n = 1, we obain: T( x, ) Tx (, ) = VT ( ( x, )) Tx (, ) 1 However, by hypohesis, we have ha V has a fixed poin, which is he exac soluion. Because if Tx (, ) is he exac soluion, hen: 1 Tx (, ) = T ( x, ) = V T( x, ) = V T( x, ) = T( x, ) = = = since 1 is he same as, herefore we have ha: Then Tx (, ) = VTx [ (, )] T( x, ) Tx (, ) = VT [ ( x, )] VMx [ (, )] 1 VT [ ( x, )] VTx [ (, )] = VT [ ( x, )] Tx (, ) = VT [ ( x, )] VTx [ (, )] n 1 n 1 Using he fac ha V is a non-linear conracive mapping we have he inequaliy: VT [ ( x, )] VTx [ (, )] < ρ T ( x, ) VTx [ (, )] n 1 Furhermore using he inducion hypohesis, we arrive a: n 1 n 1 ρρ ρ T ( x, ) VTx [ (, )] < T( x, ) Tx (, ) and he proof is compleed. gain we shall proof his by employing inducion echnique on m. For m =, we have: n 1 m 1 fwh, T ( x, ) = Ix (, ) = wh!! w= h=

5 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 S35 ccording o he idea of he homoopy decomposiion mehod, previous equaion is he conribuion of he iniial condiions. More imporanly, he above is nohing more han Taylor series of he exac soluion of order nml, hus his leads us o he siuaion ha, we can find a posiive real number r such ha: T ( x, ) Tx (, ) < r This is rue, because he conribuion of he iniial condiions is in he same neighborhood of he exac soluion. Then he propery is verified for m =, le us assume ha, he propery is also rue for m 1, ha is we assume ha, we can find a posiive real number r such ha: Tn 1 ( x, ) Tx (, ) < r We now wan o show ha he propery is also rue for m. In fac: T ( x, ) Tx (, ) = VT [ ( x, )] VTx [ (, )] n 1 n 1 using he fac ha V is a non-linear conracive mapping leads us o obain: VT [ ( x, )] VT [ ( x, )] = VT [ ( x, )] VT [ ( x, )] < r n 1 n 1 Since ρ < 1, we finally have: Tm ( x, ) Tx (, ) < r and his complees he proof. The proof of c) is direcly achieved using he a) according o: Then lim T( x, ) Tx (, ) lim ρ Ix (, ) Tx (, ) = n n n lim T( x, ) = Tx (, ) n n n Uniqueness analysis To parially show he efficiency of he used mehod, we presen in his secion he uniqueness of he special soluion for using he above echnique. To achieve his, we assume ha he exac soluion of eq. (6) exiss and ha, he special soluion converge o he exac soluion for larger naural number N. Le assume by conradicion ha, here exis wo differen special soluions T 1 ( x, ) and T ( x, ) hen consider he Hilber space: T1, T H = (1) T, T < 1 Proof. lso consider he following operaor: β D Tx (, ) Tx (, ) Qx (, ) PTx [ (, )] = [ Tx (, )] = u + a + The aim of our proof is o show, using he inner produc ha: (13) T T1 < (14)

6 S36 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 To achieve his, we evaluae: However ( PT ( ) PT ( ), D) for D H= 1 u, v uv < = + [ T( x, ) T( x, )] [ T( x, ) T( x, )] PT ( ) PT ( ) u a Thus ( T1 T) ( T T1) ( PT ( ) PT ( 1), D) = u, D + a, D x (15) (16) (17) We shall evaluae case by case, hus we sar wih he firs componen: ( T1 T) u, D Bu using he Schwarz inequaliy, we obain: ( T1 T) u, D < D ( T1 T) x u Bu using he properies of he inner produc and using he coninuiy of he parial derivaive, we can find a posiive consan w such ha: ( T1 T) u, D < w D ( T1 T) u Using he same rouine, we obain he inequaliy for he second componen: ( T T1) a, D < ao 1O D ( T T1) where O 1 and O are posiive consans. Now puing eqs. (19) and () ino eq. (17) o obain: (18) (19) () ( PT ( ) PT ( ), D) < ( w + aoo) D ( T T) (1) Now since T is he exac soluion of our modified equaion, hen T 1 and T converge o T, hus, we can find wo large naural number N and M such ha: Now ae m = max(n, M), hen: T T1 ( w + ao1o ) D for N () T T ( w + ao O ) D for M (3) 1 T T1 for M (5) ( w + ao O ) D 1

7 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 S37 Conclusion Replacing eq. (5) in eq. (1), we arrive a: ( PT ( ) PT ( ), D) (6) 1 Wih exremely very small, we have ha T T 1 =, his implies T = T 1. novel local derivaive was used in his wor o model he convecion-diffusion. This derivaive is a local derivaive wih fracional order. Some useful properies of he new derivaive were presened. The modified equaion was solved ieraively using he well nown echnique homoopy decomposiion mehod, which is he modificaion of homoopy perurbaion mehod. To show he efficiency of he used mehod, we presened a deail analysis of he sabiliy and he uniqueness of he special soluion. Nomenclaure c specific hea, [Jg 1 K 1 ] Q(x, ) source erm, [Wm 3 ] T emperaure, [K] ime, [s] u velociy, [ms 1 ] Gree symbols porosiy, [ ] λ hermal conduciviy, [WK 1 m 1 ] ρ mass densiy, [g] References [1] Kilbas,., e al., Theory and pplicaions of Fracional Differenial Equaions, Elsevier, 6 [] Li, B., Q., Disconinuous Finie Elemens in Fluid Dynamics and Hea Transfer, Springer-Verlag, London, 6 [3] Verseeg, H., Malalaseera, W., n Inroducion o Compuaional Fluid Dynamics, Prenice Hall, Upper Saddle River, N. J. US, 7, pp [4] Smoluchowsi, M. v., bou Brownian Moion under he cion of Exernal Forces and he Relaionship wih he Generalized Diffusion Equaion (in German), nn. Phys. 353 (1915), pp [5] Yang, X. J. e al., Canor-Type Cylindrical-Coordinae Mehod for Differenial Equaions wih Local Fracional Derivaives, Physics Leers, 377 (13), 8-3, pp [6] Florian C., The Hisory of Noaions of he Calculus, nnals of Mahemaics, 5 (193), 1, pp [7] Leonid, P., Cloud, M. J., pproximaing Perfecion: a Mahemaician s Journey ino he World of Mechanics, Princeon Universiy Press, Princeon, N. J., US, 4 [8] posol, T. M., One-Variable Calculus wih an Inroducion o Linear lgebra, Wiley, New Yor, US, 1967 [9] angana,., ydin, S., Noe on Fracional Order Derivaives and Table of Fracional Derivaives of Some Special Funcions, bsrac and pplied nalysis, 13 (13), ID [1] Capuo, M., Linear Models of Dissipaion Whose Q is lmos Frequency Independen, Par II, Geophysical Journal Inernaional, 13 (1967), 5, pp [11] angana,., Boha, J. F., Generalized Groundwaer Flow Equaion Using he Concep of Variable Order Derivaive, Boundary Value Problems, 13 (13), pp. 53 [1] Magin, R. L., Fracional Calculus in Bioengineering, Begell House, Redding, Conn., US, 6 [13] Chechin,. V. e al., Fracional Diffusion in Inhomogeneous Media, Journal of Physics, 38 (5), 4, pp. L679-L684 [14] Granville, S., The Numerical Soluion of Ordinary and Parial Differenial Equaions, cademic Press Professional, Inc., San Diego, Cal., US, 1988 [15] Yang, X. J., Local Fracional Inegral Transforms, Progress in Non-linear Science, 4 (11), pp. 1-5 [16] Merdan, M., Mohyud-Din, S. T., New Mehod for Time-Fracional Coupled-KDV Equaions wih Modified Riemann-Liouville Derivaive, Sudies in Non-linear Science, (11),, pp [17] Tan, Y., bbasbandy S., Homoopy nalysis Mehod for Quadraic Riccai Differenial Equaion, Communicaions in Non-linear Science and Numerical Simulaion, 13 (8), 3, pp

8 S38 angana,., e al.: Soluion of Diffusion Equaion wih Local Derivaive wih THERML SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S31-S38 [18] angana,., Drawdown in Prolae Spheroidal-Spherical Coordinaes Obained via Green s Funcion and Perurbaion Mehods, Communicaions in Non-linear Science and Numerical Simulaion, 19 (14), 5, pp [19] angana,., Noe on he Triple Laplace Transform and is pplicaions o some Kind of Third- Order Differenial Equaion, bsrac and pplied nalysis, 13 (13), ID 7691 [] Duffy, D. G., Transform Mehods for Solving Parial Differenial Equaions, CRC Press, New Yor, US, 4 [1] Brychov, Y.., Mulidimensional Inegral Transformaions, Gordon and Breach Science Publishers, Philadelphia, Penn., US, 199 [] Kilicman,., Gadain, H. E., On he pplicaions of Laplace and Sumudu Transforms, Journal of he Franlin Insiue, 347 (1), 5, pp [3] Yang, X. J., Baleanu, D., Fracal Hea Conducion Problem Solved by Local Fracional Variaion Ieraion Mehod, Thermal Science, 17 (13),, pp [4] Chen, Y. Q., Moore, K. L., Discreizaion Schemes for Fracional-Order Differeniaors and Inegraors, IEEE Transacions on Circuis and Sysems I, 49 (), 3, pp Paper submied: November 15, 14 Paper revised: February, 15 Paper acceped: March 4, 15