Research Note GREEN AND AMPT INFILTRATION EQUATION: COMPARISON OF TWO ANALYTIC DIRECT METHODS *

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1 Iraa Joural o Scece & Techology, Trasaco B, Egeerg, Vol. 3, No. B5, pp Pred The Islamc Republc o Ira, 8 Shraz Uversy Research Noe GREEN AND AMPT INILTRATION EQUATION: COMPARISON O TWO ANALYTIC DIRECT METHODS * M. NASSERI * *, Y. DANESHBOD AND H. SEYYEDIAN Dep. o Cvl Egeerg, Shraz Uversy, Shraz, I. R. o Ira Dep. o Cvl Egeerg, Islamc Azad Uversy, Arsaja Brach, I. R. o Ira Emal: mm_asser@yahoo.com Absrac The explc soluos o he Gree ad Amp lrao equao, by developg a moded Adoma Decomposo Mehod (MADM ad Varaoal Ierao Mehod (VIM soluo wh a exac soluo, are compared. These soluos were compared wh he exac mplc soluo ad a good red bewee our approaches ad he exac soluo was oud. The bes resul s gaed by oly he rs hree erms he seres soluos o MADM. As he umber o erms creases, he error he lower boud o he compuaoal doma grows ad he upper boud coverges aer he rs hree erms o MADM. O he oher had, varaoal erao coverges aer he rs wo erms. Boh mehods addressg hs pheomeo seem que promsg, ad by cosderg oly a ew erms he resuls are accurae, makg boh mehods as ad ece. Keywords Gree-Amp lrao equao, moded adoma decomposo mehod varaoal erao mehod. INTRODUCTION There are wo schools o hough wh he sol-waer lrao ssue. Oe s kow as he Horoa mechasm ad he oher as he Due Mechasm reeze []. The Gree ad Amp [] equao, Eq. (, whch belogs o he ormer class, s oe o he mos popular models ad praccal relaos o expla lrao rae o me evoluo cumulave lrao deph ad rae. Is mplemeao s relavely easy; s physcally based ad has bee used he ollowg applcaos: rase raall codos [3], me-varyg deph o podg [4], ad sols wh alerg hydraulc coducvy [5-6]. Wh regard o mplc behavor ad oleary o he equao, o use he equao, oe has o employ a erave procedure o calculae he me,, or dere values o he cumulave lrao,, ad he he lrao rae,, ca be derved rom. So, he cumulave lrao versus lrao rae curve may be cosruced wh regard o a specc sol ad raall esy. The Gree-Amp equao explas he lrao mechasm by he ollowg equao: ( θ ψ ( = K( + ( p = b ( ( where ( s lrao rae, ( s cumulave lrao deph ((=d(/, K s sauraed sol coducvy, ψ s pressure head a he weg ro, θ s al sol volumerc waer coe, s he sol porosy, ad b s he cumulave lrao he podg me. Equao ( has a mplc closed orm soluo as ollows: [] Receved by he edors Ocober 3, 6; Acceped Jauary, 8. Correspodg auhor

2 558 M. Nasser e al. ( p + ψ ( θ K ( p = ( ( p + ψ ( θl( ( + ψ ( θ ( where p s me o pod, ad ( p s correspodg cumulave lrao head. The Varaoal Ierao Mehod (VIM ad he Adoma Decomposo Mehod (ADM, as as covergece approaches o acheve he aalyc soluo o olear PDE's or ODE s, have bee appled o solve hs o-lear dereal equao. I hs regard, Serraos' approach [7-8] s based o he smplcao o he mplc o-lear exac soluo o he Gree-Amp equao by usg ADM. I he curre arcle, wo approaches, MADM ad VIM, are compared wh he exac soluo o solve he Gree-Amp equao.. ANALYSIS There are may echques such as perurbao mehods, he Cole-Hope rasormao, he Varaoal Ierao Mehod, ec. avalable o solve he olear paral dereal equaos or make hem easer o solve [8-]. The ADM was rs proposed by G. Adoma he 98's []. He oed ha he ADM ca be employed a drec maer whou ay eed o rasorm ormula or blear orms. By usg he recursve egrao ormula, hese mehods, he approxmao wll be closer o he exac soluo. Cosderg he olear operaor Ou = g (3 where O u represes a geeral orm o he olear ordary operaor cludg pars L lear, N olear, ad R as he remader o he lear ad olear erms, g s a kow uco ad u s he exac soluo o Eq. (3. Respecvely equal o he lear erm, u(β=α s cosdered as a al codo, Lu + R + N = g y y u y y L ( L = L ( R L ( N + L ( g I he ADM, he N y ca be smpled by seres' erms ( φ = u, u = u, where u = u( β + L ( g u L ( R L ( A = u N y = = = = u( β = α (4 = A ( u =, L (5 d A = Ny( u( λ (! dλ λ N y, he olear uco Eq. (4, ca be replaced by A (or more deals o A see he Adoma book [9]. There are some modcaos o mprove hs approxmaed soluo [-6]. The ex used ovel approach, VIM, s proposed by He [7]. Ths mehod, as he prevous mehod, preses a seres soluo o descrbe he equao behavor, boh o whch are hghly depede o he operao uco o he dereal equao. Applyg he varaoal calculus ad recursve egrao order o reduce errors he compuaoal rage, hs mehod mproves s approxmaed soluo. I oher words, hs mehod mproves he soluo by usg he evoluoary egrao algorhm. To show he basc cocep o he VIM, he ollowg paral dereal equao s cosdered: Iraa Joural o Scece & Techology, Volume 3, Number B5 Ocober 8

3 Gree ad Amp lrao equao: comparso o 559 L + N = g (6 y where L y s a lear erm, N y s equal o he olear erm, ad g s a homogeeous erm. So, he correcg ucoal s cosruced as show below [7-8], u + x ( y = u + λ Lu ( ζ + Nu~ ( ζ g( ζ dζ (7 where u, λ are respecvely a al codo ad lagraga mulpler, addresses he h approxmao erm, whch s assumed as a resrco, ad x s a depede varable, whch Eq. ( s deed as me [7]. Here s a example o show he ably ad accuracy o hese mehods. I hs example, he raall esy s. cm/hr ad he sol speccaos are ced rom Chow [9]. I order o use ADM ad VIM, some smplcaos are appled o Eq. ( ( a ( = +K ( a K ψ ( θ = (8 I Table, hree paramerc erms o ADM ad VIM are preseed. gures -3 represe he rs hree ADM approxmaos o he equao ( φ, φ, φ 3. I may be cosdered ha by creasg he order o ADM approxmao, he absolue error ear he al value creases, ad he me o lrao approaches large values, excelle covergece appears. To acheve a beer resul, oe should reduce he oscllao ad deergecy by applyg he [/] order o Pade approxmao oφ (Moded ADM. The resul s show gs. 4 ad 5; errors reduce a maer whch he Pade approxmao s compleely ed oφ. Aer applyg VIM o he smpled Eq. (, s covered o seres soluo wh dere covergece as show (Table. I order o sasy he saoary codo o Eq. (8, λ wll produce a complex expoeal uco erms o,, ad p, so a hgh order egral ca o be smply compued eravely, hereore we assume λ as - [7,-], bu he soluo by hs assumpo coverges less rapdly ha he exac orm. The behavor o soluos or he rs wo ad hree erms o VIM s show gs. 6 ad 7. Ths mehod coverges well o he exac soluo (excep he eghborhood o al values. Table. rs hree paramerc erms o ADM ad VIM soluos ADM VIM ( = c+ K ( = b al ( c+ K a ( = ( = b+ ( + K K b a (+ l( c+ K l( b + ( a+ bk ( = ( = b+ K+ ab kc ( + K ( a+ bk M M ( p s he al codo( ( p = b ad ( c = b K p 3. SUMMARY AND CONCLUSIONS A explc soluo o he Gree ad Amp lrao equao was derved by developg ADM ad s moded verso versus VIM. VIM ad ADM boh are oher ypes o seres' soluos used o solve lear or olear PDEs accuraely. A ew rs erms o seres expressos o he meoed Horoa equao or lrao rae were proposed. The soluos o boh mehods were compared wh he exac soluo. I was oud ha he ADM has he leas error whe here are oly wo erms. The cluso o addoal Ocober 8 Iraa Joural o Scece & Techology, Volume 3, Number B5

4 56 M. Nasser e al. erms he ADM seres creases he error, especally he eghborhood o al values. However, usg Pade approxmao [/] reduces he error a maer whch modedφ machesφ. The secod mehod, VIM, also produces sasacory resuls wh dere approxmaed seres' erms. Addg more erms hs mehod, coras wh ADM, decreases error ad coverges rapdly. Boh mehods address hs pheomeo very well ad, cosderg oly a ew erms, resul a accurae soluo, makg boh mehods as ad ece. The accuracy o he rs hree erms o he MADM ad VIM seres soluo seem adequae or mos praccal calculaos, versus s mplc exac soluo... ADM [φ ] Exac ADM [φ ] Exac g. Comparso o ADM [φ ] ad exac soluo. ADM [φ 3 ] Exac g. 3. Comparso o ADM [φ 3 ] ad exac soluo g.. Comparso o ADM [φ ] ad exac soluo. Pade[/](φ Exac g. 4. Comparso o Pade [/] (φ ad exac soluo.. ADM [φ ] Pade[/]( φ VIM- Exac g. 5. Comparso o Pade [/] (φ ad ADM [φ ] g. 6. Comparso o VIM- ad exac soluo.. VIM-3 Exac VIM-3 ADM [φ ] Pade[/](φ g. 7. Comparso o VIM-3 ad exac soluo g. 8. Comparso o VIM-3, Pade [/] (φ ad ADM [φ ] Iraa Joural o Scece & Techology, Volume 3, Number B5 Ocober 8

5 Gree ad Amp lrao equao: comparso o 56 REERENCES. reeze, R. A. (98. A sochasc-cocepual aalyss o raall-ruo processes. Waer Resour. Res., Vol. 6, pp Gree, W. H. & Amp, G. A. (9. Sudes o sol physcs : The low o ar ad waer hrough sols. J. Agrc. Sc., Vol. 4, No., pp Chu, S. T. (978. Ilrao durg a useady ra. Waer Resour. Res., Vol. 4, pp reyberg, D. L., Reeder, J. W., az, J. B. & Remso, I. (98. Applcao o Gree ad Amp model o lrao uder me-depede surace waer deph. Waer Resour. Res., Vol. 6, pp Beve, K. (984. Ilrao o a class o vercally o-uorm sols. Hydrologcal Sc. J., Vol. 9, pp Salem, H. R. & Sepaskhah, A. R. (3. Modcao o emprcal equaos or seepage loss esmao small earh caals. Iraa Joural o Scece ad Techology, Trasaco B, Vol. 7, No. B, pp Serrao, S. E. (. Explc soluo o Gree ad Amp lrao equao. J. Hydrologc Eg., vol. 6, No. 4, pp Serrao, S. E. (3. Improved decomposo soluo o Gree ad Amp equao. J. Hydrologc Eg., Vol. 8, No. 3, pp Adoma, G. (994. Solvg roer problems physcs-the decomposo mehod. Kluwer Academc, Boso.. Syam, M. & Hamda, A. (6. A ece mehod or solvg Brau equaos. Appled Mahemacs ad Compuao, Vol. 76, No., pp Adoma, G. (988. A revew o he decomposo mehod appled mahemacs. J. Mah Aal Appl., Vol. 35, p. 5.. Adoma, G. & Rach, R. (996. Moded Adoma polyomals. Mah. Compu. Modellg, Vol. 4, pp Wazwaz, A. M. (997. Necessary codos or he appearace o ose erms decomposo soluo seres. Appl. Mah. Compu. Vol. 8, pp Wazwaz, A. M. (999. The moded decomposo mehod ad Pade approxmas or solvg he Thomas- erm equao. Appl. Mah. Compu. Vol. 5, pp Wazwaz, A. M. (. A ew algorhm or calculag Adoma polyomals or olear operaors. Appl. Mah. Compu., Vol., pp Chrysos, M., Sachez, A. & Cherruaul, Y. (. Improveme o covergece o Adoma's mehod usg Pade approxmas. Kyberes, Vol. 3, pp He, J. (997. A ew approach o olear paral dereal equao. Commu. Nolear Sc. Numer. Smul., Vol., pp Abdou, M. A. & Solma, A. A. (5. Varaoal erao mehod or solvg Burger's equaos. J. Compu. Appl. Mah., Vol. 8, pp Chow, V. T., Madme, D. R. & Mays, L. W. (988. Appled Hydrology. McGraw-Hll.. He, J. (999.Varaoal erao mehod- a kd o o-ler aalycal echque: some examples. I. J. No- Lear Mech., Vol. 34, pp He, J. (. Varaoal erao mehod or auoomous ordary dereal sysems. Appl. Mah. Comp., Vol. 4, pp Ocober 8 Iraa Joural o Scece & Techology, Volume 3, Number B5