Adomian Decomposition Method for Solving the Equation Governing the Unsteady Flow of a Polytropic Gas

Transcription

1 Available at Appl. Appl. Math. ISSN: Vol., Issue (Jue 9) pp. 5 6 (Previousl, Vol., No. ) Applicatios ad Applied Mathematics: A Iteratioal Joural (AAM) Adomia Decompositio Method for Solvig the Equatio Goverig the Ustead Flow of a Poltropic Gas M.A. Mohamed Departmet of Mathematics Facult of Sciece, Suez Caal Uiversit Ismailia, Egpt dr_magd_ahmed5@ahoo.com Received: Ma, 8; Accepted: Jauar, 9 Abstract I this article, we have discussed a ew applicatio of Adomia decompositio method o oliear phsical equatios. The models of iterest i phsics are cosidered ad solved b meas of Adomia decompositio method. The behavior of Adomia solutios ad the effects of differet values of time are ivestigated. Numerical illustratios that iclude oliear phsical models are ivestigated to show the pertiet features of the techique. Kewords: Partial Differetial Equatios; Differetial Trasform Method; Approimatio Method MSC () No.: K8; 5G5; K7. Itroductio Noliear pheomea that appear i ma areas of scietific fields such as solid state phsics, plasma phsics, fluid damics, mathematical biolog ad chemical kietics ca be modeled b partial differetial equatio. A broad class of aaltical solutios methods ad umerical solutios methods were used i hadle these problems. The Adomia decompositio method has bee proved to be effective ad reliable for hadlig differetial equatios, liear or oliear. Various methods for seekig eplicit travellig solutios to oliear partial differetial equatios are proposed such as Wadati et al. (99), Wadati et al. (975), Wadati (), Wadati (97), Drazi et al. (997). I the begiig of the 98, a so-called Adomia decompositio method (ADM), which appeared i Adomia (99), Adomia ad Serrao (998), Adomia et al. (995), Deeba ad Khuri (996), Oldham (97), Podlub (999), Wazwaz (), Wazwaz (), ElWakil et al. (i press), Abdou (5), Kaa ad El-Saed (), Seg ad Abbaoui (996), ad Lesic (6) has bee used to solve effectivel, easil, ad accuratel a large class of liear ad oliear equatios, solutios partial, 5

2 AAM: Iter. J., Vol., Issue (Jue 9) [Previousl, Vol., No. ] 5 determiistic or stochastic differetial equatios with approimates which coverge (see Figure ). Ulike classical techiques, the oliear equatios are solved easil ad elegatl without trasformig the equatio b usig the ADM. The techique has ma advatages over the classical techiques, mail, it avoids liearizatio ad perturbatio i order to fid eplicit solutios of a give oliear equatios. To give a clear view to our stud, we have chose the equatio goverig the ustead flow of a poltropic gas to illustrate the aalsis of the Adomia.. The Adomia decompositio For the purpose of illustratio of the methodolog to the proposed method, usig ADM, we begi b cosiderig the differetial equatio LuRu Nu g, () with prescribed coditios, where u is the ukow fuctio, L is the highest order derivative which is assumed to be easil ivertible, R is a liear differetial operator of less order tha L (operator L is liear also), Nu represets the oliear term ad g is the source term. Assumig the iverse operator L eists ad it ca be take as the defiite itegral with respect to t from t to t, i.e. L t () t () dt. Applig the iverse operator L - coditios we fid to both sides of equatio () ad usig the iitial u f L [ Ru Nu], () where the fuctio f (, ) represets the term arisig from itegratig the source term g ad from usig the give iitial or boudar coditios, all are assumed to be prescribed. The oliear operator [ Nu ] ca be decomposed b a ifiite series of polomials give b Nu A ( u, u,, u ), () where A( u, u,, u) the appropriate Adomia s polomials are defied b Adomia G.(99), Adomia G, Serrao SE.(998).

3 5 Mohamed d A u,.! d k k k (5) This formula is eas to compute b usig Mathematica software or b settig a computer code to get as ma polomials as we eed i the calculatio of the umerical as well as eplicit solutios. The Adomia decompositio method assumes a series that the ukow fuctio ut (,, ) ca be epressed b a ifiite series of the form ut (,,) u(, t,). (6) Idetifig the zeros compoet u (,,) the remaiig compoets where ca be determied b usig the recurrece relatio u (, ) f(, ), (7) u t L R u A (,, ) [ ( ) ],. (8) The other polomials ca be geerated i a similar wa. The scheme (8) ca easil determie the compoets u(,, t ) It is i priciple, possible to calculate more compoets i the decompositio series to ehace the approimatio. Oe caot compute a ifiite umber of terms; ol a quite limited umber of terms are determied of the series u (,, ) t ad hece the solutio ut (,, ) is readil obtaied. It is iterestig to ote that we obtaied the solutio b usig the iitial coditio ol.. Applicatio For simplicit, we are iterested to deal with Adomia decompositio solutio associated with the operator L - rather tha the other operators i our eample. The equatio goverig the ustead flow of a poltropic gas i two dimesios is give b Feg X (996), Billigham (), Rogers ad Ames (988). p ut uu vu, (9) p vt uv vv, () u v ( u v ), () t p up vp p( u v ), () t

4 AAM: Iter. J., Vol., Issue (Jue 9) [Previousl, Vol., No. ] 55 where is the desit, p the pressure, u ad v the velocit compoets i the ad directios, respectivel, ad the adiabatic ide is the ratio of the specific heats. With the iitial data: u (,,) e, () v (,,) e, () (,,) e, (5) p(,,) c. (6) Note that the selectio of equatios (9-) that are obtaied from Billigham () the fluid is icompressible ad ivisid (o viscose), because we assumed the seteces u u u u, that appear i equatio (9), which is hard to solve. Equatios (9-) ca be writte i a operator form as Lu [ N ( u, u ) K ( v, u ) H (, p )], (7) Lv[ N ( u, v ) K ( v, v ) H (, p )], (8) L [ N ( u, ) K ( v, ) H (, u ) G (, v )], (9) Lu [ N ( u, p ) K ( v, p ) H ( p, u ) G ( p, v )], () where L. t The Adomia Decompositio Method (ADM) assumes a series solutio of the ukow fuctios ut (,, ) u( t,, ), () vt (,, ) v( t,, ), () (, t, ) ( t,, ), ()

5 56 Mohamed pt (,,) p(, t,). () Substitutig Equatios (-) with iitial coditios ito Equatios (7-) ields ut (,, ) u (,,) L N( uu, ) K( vu, ) H(, p), (5) vt (,, ) v (,,) L N( uv, ) K( vv, ) H(, p), (6) (, t, )= (,,) L N( u, ) K( v, ) H(, u) G(, v), (7) p t p L N up K vp H pu G pv (,, ) (,,) (, ) (, ) (, ) (, ), (8) where the fuctios N ( u, u ), K ( v, u ), H (, p ), N ( u, v ), K ( v, v ), H (, p ), N ( u, ), K ( v, ), H (, u ), G (, v ), N ( u, p ), K ( v, p ), H ( p, u ) ad G ( p, v ) are: m ( m) m N( uu, ) uu A ( uu, ) uu uu uu u u (9) K(, vu ) vu B (, vu ) vu vu vu v u () m ( m) m p p p H p p C p () (, ) (, ) m ( m) m N( uv, ) uv A ( uv, ) uv uv uv u v () K (, vv ) vv B (, vv ) vv vv vv v v () m ( m) m p p p H p p C p () (, ) (, )

6 AAM: Iter. J., Vol., Issue (Jue 9) [Previousl, Vol., No. ] 57 m ( m) m N ( u, ) u A ( u, ) u u u u (5) K (, v v ) v B (, v ) v v v v (6) m ( m) m m ( m) m H (, u ) u C (, u ) u u u u (7) G (, v ) v D (, v ) v v v v (8) m ( m) m m ( m) m N( up, ) up A ( up, ) up up up u p (9) K(, vp) vp B (, vp) vp vp vp v p () m ( m) m H ( pu, ) pu C ( pu, ) pu pu pu p u. m ( m) m Idetifig the zeros compoets of u, v, ad p the remaiig compoets u( t,, ), v( t,, ), ( t,, ) ad p( t,, ), ca be determied b usig recursive relatios give b u (,,) e, v (,,) e, (,,) e, p(,,) c, () u (,, t) L N( u, u) K( v, u) H(, p), () vt (,, ) L N( u, v) K( v, v) H(, p), () ( t,, ) L N( u, ) K( v, ) H(, u) G(, v ), (5) pt (,, ) L N( u, p) K( v, p) H( p, u) G( p, v). (56)

7 58 Mohamed The remaiig compoets u, v, ad pca be completel determied such that each term that determied b usig the previous terms, ad the series solutios thus etirel evaluated. t t t t u( t,, ) e, v( t,, ) e, ( t,, ) e (7)!!! t t t t u(,, t) e, v(,, t) e, (,, t) e (8)!!! t t t t u(,, t) e, v(,, t) e, (,, t) e (9)!!! t t t t u(,, t) e, v(,, t) e, (,, t) e (5)!!! p (,, ),,,,, t (5) etc. I geeral we have t u (,, t) e (5)! t v (,, t) e (5)! t (, t, ) e (5)! p (,, ),,,, t (55) The solutio of ut (,, ), vt (,, ), ( t,, ) ad p( tare,, ) u (,,) e t, (56) v (,,) e t, (57) t (,,) e, (58) p(,,) c. (59) Adomia solutios coicides with the eact solutio

8 AAM: Iter. J., Vol., Issue (Jue 9) [Previousl, Vol., No. ] 59 t t t ( uv,,, p) ( e, e, e, c). (6) With differet values of time t, it is show from Figures a ad b, the eact solutio (6). Also the behavior of the solutio is show i Figure Figure a. Eact solutio of ut (,, ) ad vt (,, ) for t:5, :5,, c 5 5 Figure b. Eact solutio of (, t, ) ad pt (,, ) for t:5, :5,, c The followig figures show the differece betwee umerical ad eact solutio which show the depedec of the error to the umber of terms M sice from the followig figures, whe we icrease the umber of terms the solutio coverges to the eact solutio

9 6 Mohamed Figure. compariso betwee the eact solutio ad the behavior of the solutio obtaied b ADM method B the same maer the fuctios (,, t) ad p(,, t) ca be obtaied to which gives us the same results that we got it from the last figures.. Coclusio I this article, Adomia decompositio method for approimatig the solutios of the equatio goverig the ustead flow of a poltropic gas is implemeted. B usig this scheme, eplicit eact solutios arisig i oliear phsics are calculated i form of a coverget power series with easil computable compoets. To illustrate the applicatio of this method, umerical results were derived b usig the calculated compoets of the decompositio series. Numerical illustratios are ivestigated to show the pertiet features of the techique. The results reported here provide further evidece of the usefuless of Adomia decompositio method (ADM). The ADM was clearl ver efficiet ad powerful techique i fidig the solutios the equatio goverig the ustead flow of a poltropic gas sice we ca retch to the eact solutio after few iteratios of usig Adomia decompositio method (ADM). It is clear that this method avoids liearizatio ad biologicall urealistic assumptios, ad provides a efficiet umerical solutio. Ackowledgemets The author is thakful to aomous referees for their useful suggestios, which led to the preset form of the paper ad I am highl grateful to Professor Dr. A. M. Haghighi for his costructive commets.

10 AAM: Iter. J., Vol., Issue (Jue 9) [Previousl, Vol., No. ] 6 REFERENCES Abdou MA.(5). Adomia decompositio method for solvig the telegraph equatio i charged particle trasport. JQSRT Vol. 95 pp. 7. Adomia, G., Rach, R., Shawagfeh, N.T. (995). O the aaltic solutio of the Lae-Emde equatio. Phs Lett., Vol. 8(), pp Adomia G., Serrao S.E. (998). Stochastic Cotamiat Trasport Equatio i Porous Media. Appl Math Lett Vol., No., pp Adomia G. (99). The decompositio method. Bosto: Kluwer Acad. Publ. Billigham (). Damics of a strogl olocal reactio-diffusio populatio model. J. Noliearit Vol. 7(), pp. -6. Debtath L. (997). Noliear partial differetial equatios for scietist ad egieers. Bosto: Birkhauser. Deeba, E.Y., Khuri, S.A. (996). A decompositio method for solvig the oliear Klei- Gordo equatio. J Comput Phs Vol., pp. -8. Drazi, P.G., Johso, R.S. (989). Solitos: a itroductio. Cambridge: Cambridge Uiversit Press. ElWakil, S.A., Abdou, M.A., Elhabal, A. (i press) Adomia decompositio method for solvig the diffusio-covectio reactio equatios. Appl Math ad Comput. Feg, X. (996). Eact wave frot solutios to two geeralized coupled oliear phsical equatios. Phs Lett A, Vol., pp. I Kaa, D., El-Saed, M. (). O a geeralized fifth order KdV equatios. Phs Lett A, Vol., pp. 5. Lesic, D. (6). Blow-up solutios obtaied usig the decompositio method. Chaos, Solitos & Fractals Vol. 8, pp Oldham, K.B. (97). Spaier J. The fractioal calculus. New York: Academic Press. Podlub, I. (999). Fractioal differetial equatio. New York: Academic Press. Rogers, C., Ames, W.F. (988). Noliear Boudar value. Problems i sciece ad Egieerig, Academic Press, New York. Seg, V., Abbaoui, K.Y. (996). Adomia s Polomials for Noliear Operators. Math Comput Model Vol., No., pp Wadati, M., Sauki, H., Koo, K. (975). Relatioships amog Iverse Method, Bäcklud Trasformatio ad a Ifiite Number of Coservatio Laws. Prof Theor Phs. Vol. 5 No., pp Wadati, M., Segur, H., Ablowitz, M.J. (99). A New Hamiltoia Amplitude Equatio Goverig Modulated Wave Istabilities. J. Phs. Soc. Jp., Vol. 6, pp Wadati, M. (97). The Eact Solutio of the Modified Korteweg-de Vries Equatio. J Phs Soc Jp Vol., pp. 68. Wadati, M. (). Itroductio to solitos. J Phs. Vol. 57(5 6): 8 7. Wazwaz, A.M. (). New Solitar wave special solutios with compact support for the oliear dispersive K(m,) equatios. Appl Math Comput Vol. pp.. Wazwaz, A.M. (). Eact specific solutios with solitar patters for the oliear dispersive K(m,) equatios. Chaos, Solitos & Fractals Vol., pp..