New Applications of Adomian Decomposition Method. Emad A. Az-Zo'bi

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1 Middle-Eas Joral of Scieific Research (): ISSN 99-9 IDOSI Pblicaios 5 DOI:.589/idosi.mejsr.5... New Applicaios of Adomia Decomposiio Mehod Emad A. Az-Zo'bi Deparme of Mahemaics ad Saisics Mah Uiversiy Mah P.O. Bo: 7 AlKarak - Jorda Absrac: The Adomia decomposiio mehod (ADM) is implemeed for aalyic-meric reame for cerai class of oliear parial differeial sysems called sysem of coservaio laws. We prove he covergece of ADM applied o hese ypes of sysems. The aalysis will be illsraed by ivesigaig several eamples o idicae reliabiliy accracy ad covergece of he mehod. Key words: Hyperbolic sysem of coservaio laws Adomia decomposiio mehod Adomia polyomials Covergece INTRODUCTION deermiisic ad sochasic differeial eqaios [8-7] ad refereces herei. This mehod is well-sied o Noliear copled parial differeial eqaios are physical problems sice i makes he ecessary very impora i a variey of scieific fields. They arise liearizaio perrbaio problem beig solved. i large mber of mahemaical ad egieerig I his sdy he ADM is eeded o problems especially i flid mechaics solid sae aalyic-meric simlaio for sricly hyperbolic sysem physics plasma physics plasma waves capillary-graviy of coservaio laws. The covergece aalysis for he waves ad chemical physics []. The sysem of ADM applied o sch sysems is discssed. coservaio laws is oe of he mos famos eamples of copled oliear parial differeial eqaios whose The Adomia Decomposiio Mehod: To smmarize he form is. basic solio procedre of he ADM. Cosider he oe dimesioal eqaio of coservaio law i operaor U( ) + F' ( U) U( ) = () form. where F (U) is he Jacobia mari of he fl vecor F []. L ( ) + Lf( ( )) = ( ) = () The sysem i Eq.() is sricly hyperbolic if he eigevales of F (U) are posiive ad real which will be where he oaios L ad L symbolize he liear he case sdy i his work. differeial operaors wih respec o ad respecively. May ahors paid aeio o sdy solios of L hyperbolic ellipic ad mied hyperbolic-ellipic ype of Applyig he iverse operaor (). = (). d o Eq.() coservaio laws eqaios. Amog hese aemps are he fiie differece mehod [] he Sic-Galerki yields. mehod [] he redced differeial rasform mehod [5-6] variaioal ieraio mehod [7-9] homoopy ( ) = L ( Lf( ) ). () aalysis mehod [9-] homoopy perrbaio mehod [-] ad he Adomia decomposiio mehod (ADM) The ADM [6-7] assmes ha he solio ( ) [-5]. ca be wrie i erms of a ifiie series of he form. The ADM was irodced by Adomia [6-7]. I ivesigaed o solve effecively ad easily a large class ( ) = ( ) of liear oliear ordiary parial fracioal () Correspodig Ahor: Emad A. Az-Zo'bi Deparme of Mahemaics ad Saisics Mah Uiversiy Mah P.O. Bo: 7 AlKarak - Jorda. 75

2 Middle-Eas J. Sci. Res. (): ad he oliear erm f() ca be represeed by he ifiie series of he form. coverges owards a pariclar solios wiho iiial ad bodary codiios. f = A d k A = f k.! d k = (5) Proof: We verify he codiios H ad H of covergece. Sarig wih H for he operaor L Eq.(7) ca be wrie as. where A =... are he Adomia polyomials which are give by. L ( ) = - f (( )) () (6) The for all v H we have: ( L Lv v) = ( Lf( ) + g+ Lf( v) g v) = L ( f ( ) f ( v) ) v Sbsiig Eqs. (5) ad (6) io Eq.() ad assmig () ( ha ( ) = () oe ca geerae he res of he ) erms of he series solio sig he recrsive relaio: Accordig o Cachy-Schwarz ieqaliy we ge. [ ] = = L L A. + ( ( ) ) ( ) (7) L f f v v L f f v v. () The eac solio ( ) will be. ad sice L is differeial operaor i Hilber space H here is some cosa sch ha. ( ) = lim k ( ). (8) k = L f f v v f f v v. (5) : H wih s dsd < (9) Sbsiig he ieqaliies (7)-(9) io (6) we ( ) [ T ] ca dedce. ad he orm L f f v v L f f v v = ( s ) dsd. ( ) [ T ] ( ( ) ) Covergece Aalysis: I his secio we discss he B f saisfies Lipchiz codiio wih Lipchiz covergece aalysis [8] of he ADM for he cosa L (say) he we ge. coservaio laws i Eq. (7). Le s cosider he Hilber space H = L (( ) [ T]) defied by. f f v L v (6) () The ADM is covergece if he followig wo hypoheses are saisfied: Theorem.: If f() is Lipschizia fcio i Hilber space H he ADM applied o he sysem ( ( ) ) ( ) f ( ) f ( v) v L v = C v So hypohesis H is valid. Now we verify he covergece hypohesis H for he operaor L which is H: ( L Lv v) C v C > v H for every K > here is a cosa C (K) > sch ha for v H wih K v K we have (L - Lvw) C (K) H: Whaever maybe K > C ( K ) > sch ha for v - v w for every w H. To do ha sig he mea- H wih v K we have ( L Lvw ) C ( K ) v w for vale heorem ad Cachy-Schwarz ieqaliy as every w H. meioed above we have. ( ( ) ) = f w f ( K) w L f f v w f f v w L + Lf () =. () (7). 76

3 ( ) = ( ) C ( K) w L Lv v L f f v w Eample.: Firs we es he iiial bodary vale problem ( sysem wih fl f ( ) = ) Middle-Eas J. Sci. Res. (): for some cosa ad v < < (i he sese of orm) Sbsiig he decomposiio series () for ( ) wih v K. Leig C (K) = f (K) we ge. io Eq.() irodces he recrsive relaio. wih C (K) = f (K) H is hold. The proof is complee. (8) Illsraed Eamples: I his secio he ADM has bee sccessflly sed o sdy hree models of hyperbolic sysem of coservaio laws. Choosig eamples wih kow solios allows for a more complee error aalysis as show i he firs wo eamples. For prposes of illsraio he comper applicaio program Mahemaica is sed. + = > > (9) ( ) = + = A d. () I view of Eq.() we obai. 5 6 = = 8 5 = 6 5 = = 5 6 = Sbjec o he codiios. = > = >. The eac solio is give by. 5 ( ) = = + + () Applyig he operaor L defied i secio o Eq.(9) gives. ( ) = f ( ) d where he fcio f ( ) = = A = = + = = A A A () () A s are he Adomia polyomials which ca derived sig formla (6). The few erms of A s are. A () k k = The series solio ( ) = lim ( ) ca be easily obaied ad coicide wih he eac solio (). Eample.: Cosider he oliear iiial vale sysem i () wih fles F( U) = v v Sbjec o iiial codiios. T = e v = e. The sysem is sricly hyperbolic wih eigevales. = ±. v f v A v = = ( ) ad g( ) = = B ( ). J-S. Da [9- (5) (6) - - The eac solio for his problem is U( ) = (e e T ). I his eample he oliear operaors are ] proposed a simple recrrece echiqe of calclaig Adomia polyomials for sigle ad mlivariable oliear operaors. The firs few Adomia polyomials for f are give by. 77

4 Middle-Eas J. Sci. Res. (): = = A f v v = = A v f v vv A= vf ( v) + vf ( v) = vv + vv! A vf ( v ) vvf ( v ) vf! ( v ) vv vvv vv = + + = + ad i he same way for B s. Followig same procedre i Eample we formlae he recrsive relaios as. ( ) = ( ) = e A d + ( ) = ( ) = + v e v B d = e = e! ( ) = e! ( ) = e! v = e v = e! v ( ) = e! v ( ) = e! T U( ) = e + + e !!!!!! = ( + ) F U v v. T (7) We fid he firs few ieraes of he recrsive scheme (7) o be. ad Ths he series solio is. As we ge he eac solio. Eample.: The sysem i () wih fles. (8) is sricly hyperbolic wih eigevales. = ± + v. I a operaor form L + L + vlv = L + Lv + vl = sbjec o iiial codiios =.5cos v =.5si. Operaig L (). = (). d o boh sides of (9) gives. = A + B v d v = v C v + Dv d A = A B = B Cv = C Dv = D. The firs few compoes of he Adomia polyomials of A ad C are give by. A = A= + () A= + + A = C= v C= v + v C= v + v + v C = v + v + v + v ad i he same way for B ad C. Also he kow fcios ( ) ad v ( ) ca be epressed by a ifiie series of he form. (9) () () where he fcios A() = B(v) = vv C( v) = v ad D( v) = v are relaed o he oliear erms ad ca be epressed i erms of Adomia polyomials as followig. () () 78

5 Middle-Eas J. Sci. Res. (): ad v ( ) = si = ( ) v si cos 6 = ( ) v si si 5cos 8 v ( ) = cos cos si si + si. 9 Fig. : The behavior of fcio () verss for. I his eample oly si compoes of he decomposiio series were obaied. We obaied he approimae solios = ( ) = ( ) + ( ) + + ( ). v v v v 6 The mai goal of his work is o fid a approimae solio for he oliear sricly hyperbolic sysem of coservaio laws sig he Adomia decomposiio mehod. A proof of covergece for sch sysems is preseed. The Adomia decomposiio mehod provides he solio i sccessive compoes ha will be added o ge he series solio; i appears o be very promisig Fig. : The behavior of fcio () verss for wih greaer sabiliy for solvig oliear sysems.. REFERENCES ( ) = ( ) The behaviors of he solios obaied by he ADM for differe vales of are show i Figs. ad respecively. CONCLUSION. Ablowiz M.J. ad P.A. Clarkso 99. Noliear (5) Evolio Eqaios ad Iverse Scaerig Cambridge Uiv. Press. v( ) = v ( ).. Az-Zo bi E.A. ad K. Al-Khaled. A New Covergece Proof of The Adomia Decomposiio The firs few erms of he decomposiio series (5) are Mehod for a Mied Hyperbolic Ellipic Sysem of Coservaio Laws Applied Mahemaics ad ( ) = cos Compaio 7: Sod G.A A Srvey of Several Fiie Differece ( ) = Mehods for Sysems of Noliear Hyperbolic ( ) = cos ( cos 5si ) Coservaio Laws J. Comp. Phys. 7: Alqra M.T. ad K.M. Al-Khaled. Nmerical Compariso of Mehods for Solvig Sysems of ( ) = cossi( cos si ) 8 Coservaio Laws of Mied Type I. J. Mah. Aal. 5:

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