Application of Homotopy Analysis Method for Water Transport in Unsaturated Porous Media

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1 Studies i Noliear Scieces 1 (1): 8-1, 1 ISSN 1-91 IDOSI Publicatios, 1 Applicatio of Homotopy Aalysis Method for Water Trasport i Usaturated Porous Media Hossei Jafari ad M.A. Firoozjaee Departmet of Mathematics, Uiversity of Mazadara, Babolsar, Ira Abstract: I this paper, a aalytic techique, amely the Homotopy Aalysis Method (HAM) has bee applied for solvig Richards equatio which is oe of the most well-kow equatios to describe the behavior of the ifiltratio of usaturated zoes i soil as a porous medium. The results show that this method is very efficiet ad coveiet ad ca be applied to a large class of problems. Comparisos of the results obtaied by the HAM with that obtaied by the ADM ad MADM suggest that both the ADM ad MADM are special case of the HAM. Key words : Homotopy aalysis method richards equatio burger equatio adomia decompositio method partial differetial equatio INTRODUCTION May researchers have bee ivolved i modelig water movemet i a usaturated porous material [4, 7, 8]. Richards [] derived a goverig equatio for water flow i soil based o cotiuum mechaics. I this model, the cotiuity equatio was coupled with Darcy s law as a mometum equatio. The followig equatio is kow as the oe-dimesioal form of Richards equatio: = (D K) z z where u is usaturated soil moisture cotet, K is coductivity ad D is soil water diffusivity. The sophisticatio of the aalytical ad umerical methods that are available to solve the goverig equatio of usaturated flow i soils (Richards equatio) makes it ecessary to have suitable models for parameters i the equatio. Several methods are available for the estimatio of such parameters, such as coductivity ad water diffusivity [1]. Basically, there are three commoly used models: (i) Brook-Coreys model [, 6], (ii) the va Geuchte model ad (iii) the epoetial model. The Brooks-Corey model itroduces a welldefied air-etry value that is associated with the largest pore size, assumig complete wet ability Brooks-Corey model soils ca be simplified to the followig equatios by some further cosideratios [6, 6]: k (1) K(u) = K u form 1 () D(u) = D(+ 1)u form 1 () where K, D, ad k are costats represetig soil properties such as pore-size dis tributio, particle shape, etc. I these relatios, u is scaled betwee ad 1 ad the form of diffusivity is ormalized so that 1 D(u)d(u) = 1 Most of the results i this paper ca be applied to the geeral values of k ad. It is believed that the Brook- Corey model is widely used because of its well-defied cofiguratio. There are several aalytical ad umerical solutios to Richards equatio cosiderig the Brook-Corey model. The choice = ad k = i Eqs. () ad () yields the classic Burgers equatios. For geeral values of k ad, the geeralized Burgers equatio, which was the focus of several researchers icludig [11], is obtaied. I the special case =, Richards equatio reduces to a liear equatio. Richards equatio with other models was also solved umerically ad by various iovative ad commo aalytical methods. Some of these solutios are limited to very simple geometrical ad iitial coditios. I the last years may fiite differece ad fiite elemet umerical solutios were developed, eve i D ad D. Aother umerical method, the fiite volume method, looks quite promisig for solvig Richards equatio, especially whe sharp ifiltratio frots develop ad must be approimated o ustructured Correspodig Author: Hossei Jafari, Departmet of Mathematics, Uiversity of Mazadara, Babolsar, Ira 8

2 Studies i Noliear Sci., 1 (1): 8-1, 1 multidimesioal grids. For istace, a few researchers have tried to solve the equatio by the Adomia decompositio method (ADM) as a aalytical series solutio [-4]. I this study, a simplified Brooks- Corey model (Eqs. () ad ()) was applied o Richards equatio [Eq. (1)]. However, two cases for coductivity epoets (with equal soil water diffusivity) ad the solutios were tried. The hyperbolic taget fuctio is commoly applied to solve these trasform equatios [5]. The geeral form of Burgers equatio i the order of (,1) is: t u + a(u ) + bu =, 1,a,b 1 (4) ad its eact solutio is 1 c c( 1) u(,t) = ( (1+ tah[ ( ct)]) 1 (5) a b I these cases, Richard equatio coicides with Burger equatio (,1), where u is water cotet, is depth (cm) ad c is a arbitrary coefficiet.i 199, Liao [16] employed the basic ideas of the homotopy i topology to propose a geeral aalytic method for oliear problems, amely Homotopy Aalysis Method (HAM), [16-18]. This method is successfully applied to solve may types of oliear problems [, 9, 1, 14, 15, 19]. BASIC IDEA OF HAM We cosider the followig differetial equatio N[u( τ )] = (6) where N is a oliear operator, τ deotes idepedet variable, u (τ) is a ukow fuctio, respectively. For simplicity, we igore all boudary or iitial coditios, which ca be treated i the similar way. By meas of geeralizig the traditioal homotopy method, Liao [17] costructs the so-called zero-order deformatio equatio (1 p)l[ φτ (;p) u()] τ = phh()n[(;p)] τ φτ (7) where p [,1] is the embeddig parameter, h is a o-zero auiliary parameter, H(τ) is a auiliary fuctio, L is a auiliary liear operator, u (τ) is a iitial guess of is a ukow fuctio, respectively. It is importat, that oe has great freedom to choose auiliary thigs i HAM. Obviously, whe p = ad p = 1, it holds 9 φτ (;) = u(), τ φτ (;1) = u( τ ) (8) respectively. Thus, as p icreases from to 1, the solutio φ(τ;p) varies from the iitial guess u (τ) to the solutio u(τ). Epadig φ(τ;p) i Taylor series with respect to p, we have Where m φτ (;p) = u ( τ+ ) u m( τ)p m1 = m 1 φτ (;p) u m( τ ) = m! m p p= (9) (1) If the auiliary liear operator, the iitial guess, the auiliary parameter h ad the auiliary fuctio are properly chose, the series (9) coverges at p = 1, the we have u( τ ) = u ( τ ) + u m() τ m= 1 (11) which must be oe of solutios of origial oliear equatio, as proved by liao [17]. As h =-1 ad H(τ) = 1, Eq. (7) become (1 p)l[ φτ (;p) u()] τ + pn[ φτ (;p)] = (1) Which is used mostly i the homotopy perturbatio method, where as the solutio obtaied directly, without usig Taylor series [1, 1]. Accordig to the defiitio (1), the goverig equatio ca be deduced from the zero-order deformatio equatio (7). Defie the vector u = {u ( τ ),u 1(), τ,u ( τ )} Differetiatig equatio (7) m times with respect to the embeddig parameter p ad the settig p = ad fially dividig them by m!, we have the so-called mthorder deformatio equatio L[u m( τ) χmu m 1( τ )] = hh( τ )R m(u m 1) (1) subject to iitial coditio Where m(,) u m (,) =, = (14)

3 Studies i Noliear Sci., 1 (1): 8-1, 1 Ad m 1 1 N[ φτ (;p)] R m(u m1 ) = (m 1)! m1 p p= m 1 χ m = 1m > 1 (15) (16) It should be emphasized that u m (τ) for m 1 is govered by the liear equatio (1) with the liear boudary coditios that come from origial problem, which ca be easily solved by symbolic computatio software such as Mathematica. If Eq. (6) admits uique solutio, the this method will produce the uique solutio. If equatio (6) does ot possess uique solutio, the HAM will give a solutio amog may other (possible) solutios. For the covergece of the above method we refer the reader to Liao s work [17]. APPLICATIONS I this sectio we apply HAM for solvig Richards equatio. Further we compare the gaied result with both eact ad MADM. Case 1: I this case a coductivity term is selected as a fuctio of cubic water cotet ad costat u K = cm/h ad D = 1cm /h. Therefore, Richard equatio becomes: with iitial coditio: t u + uu u = (17) 1 u(,) = (1+ tah[ ]) Therefore i Eq.(4) a = 1/, =,b=-1 ad we choose c = 1/. For applicatio of homotopy aalysis method, we choose the iitial approimatio: 1 u (,t) = u(,) = (1+ tah[ ]) (18) ad the liear operator: L[ φ (,t;p) = which possesses the property (19) 1 L(c 1) = () where c 1 is a itegral costat to be determied by iitial coditio. Usig Eq. (17), we defie oliear operator as N[( φ (,t,;p)] = +φ (,t,;p) I view of Eq. (15), we obtaied m 1(,t) u m1 (,t) R m(u m1 ) = m1 m 1 (,t) + u(,t) j = Now usig Eq.(1), The HAM would lead to: htsech ( )tah( ) htsech ( ) u(,t) 1 = 18 (1 tah( )) 18 (1 tah( )) h t sech ( )tah( ) httah( )sech ( ) u (,t) = + 16 ( tah( )) 18 (1 tah( )) h t sech ( ) h tsech ( ) 648 ( tah( )) 18 ( tah( )) htsech ( ) 18 (1 tah( )) Thus u(,t) = u (,t) = Figure 1 ad show the 5th-order approimate solutio HAM ad eact solutio, respectively. The behavior of the HAM ad the eact solutio ad MADM i two trasitio poits ( =1,5) are plotted versus time i Fig. ad 4. I [7] this case has bee solved by MADM usig HAM for h =-.85 Case : I this case a coductivity term is selected as a fuctio of quadric water cotet ad costat value

4 Studies i Noliear Sci., 1 (1): 8-1, 1 Fig. 1: The 5th-order approimatio Fig. : Compariso of eact solutio Fig. : Eact solutio of burger (/1) 4 u K = cm/h 4 ad D = 1cm /h Eq.(1). Cosequetly, Richards equatio is coverted to: with iitial coditio: t u + uu u = (1) u(,) = 1 (1+ tah[ ] 8 Therefore, i Eq.(4) a = ¼, = 4, b =-1 ad we choose c = 1/4. We choose the liear operator: L[ φ (,t;p) = which possesses the property: () L(c 1) = () 11 Fig. 4: Compariso of eact solutio ad 5th-order of HAM ad MADM ad 5th-order of HAM ad MADM We choose 1 u (,t) = u(,) = (1+ tah[ ] 8 From Eq. (1) the oliear operator is defied as N[( φ (,t,;p)] = +φ (,t,;p) The by applicatio Eq. (15), we have: m 1(,t) u m1 (,t) R m(u m1 ) = m 1 j m 1 (,t) + u j k(,t) u k(,t) = j= k= Therefore by usig Eq. (1), we obtai terms of HAM:

5 Studies i Noliear Sci., 1 (1): 8-1, 1 Fig. 5: Compariso of eact solutio Remark: I Fig. -6, we show the comparisos betwee the 4-term HAM solutios ad the eact solutios. We observe that the results of the 4-term HAM are very close to the eact solutios which cofirm the validity of the HAM. All the umerical results obtaied by the 4-term HAM are eactly same as the ADM solutios ad HPM solutios for special case h =-1, H(t) = 1. So its meas that the ADM is a special case of HAM. But HAM is more geeral ad cotais the auiliary parameter h, which provides us with a simple way to adjust ad cotrol the covergece regio of solutio series. As poited out by Abbasbady i [1] oe had to choose a proper value of h to esure the covergece of series solutio for strogly oliear problems. So it is more importat to show that the HAM gives coverget series solutio for ay larger values of t by choosig proper values of h. Fig. 6: Compariso of eact solutio ad 4th-order of HAM ad MADM ad 4th-order of HAM ad MADM htsech ( )tah( ) htsech ( ) u(,t) 1 = (1 tah( )) (1 tah( )) 6 6 5h t sech ( )tah( ) 15h t sech ( ) u (,t) = (1 tah( )) 48 (1 tah( )) 6 4 5h t sech ( ) 5h t tah ( )sech ( ) (1 tah( )) 48 (1 tah( )) Thus u(,t) = u (,t) = Now, Richards equatio is approimated by repeatig the HAM. The behavior the 4thorder of HAM ad the eact solutio MADM i two trasitio poits ( = 1;7) are plotted versus time i Fig. 5 ad 6. By choosig h =-1, we will reach ADM solutio give by [17]. 1 CONCLUSION I this paper, we have successfully developed HAM for solvig model Broke-Corey i Richards equatio for all values of ad various itervals of t. It is apparetly see that HAM is very powerful ad efficiet techique i fidig aalytical solutios for wide classes of oliear problems. It is worth poitig out that this method presets a rapid covergece for the solutios. HAM provides accurate umerical solutio for oliear problems i compariso with other methods such as MADM. They also do ot require large computer memory ad discrimiatio of the variables t ad. The results show the validity ad great potetial of the homotopy aalysis method for oliear problems i sciece ad egieerig. Mathematica has bee used for computatios i this paper. REFERENCES 1. Abbasbady, S., 6. The applicatio of homotopy aalysis method to oliear equatios arisig i heat trasfer. Phys. Lett. A, 6: Ayub, M., A. Rasheed ad T. Hayat,. Eact ow of a third grade uid past a porous plate usig homotopy aalysis method. It. J. Eg. Sci., 41: Brooks, R.H. ad A.T. Corey, Hydraulic Properties of Porous Media, Hydrol Paper, Colorado State Uiversity, Fort Collis. 4. Biazar Jafar, Ayati Zaiab ad Ebrahimi Hamideh, 9. Homotopy Perturbatio Method for Geeral Form of Porous Medium Equatio. Joural of Porous Media.

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