On the convergence of the VHPM for the Zakharove-Kuznetsov equations

Transcription

1 IJST ( A (Specil isse-mheics: 5-58 Iri Jorl of Sciece & Techology hp://wwwshirzcir/e O he covergece of he VHPM for he Zhrove-Kzesov eqios M Mifr* M Ghsei d M Seidy Depre of Mheics Fcly of Scieces Mzdr Uiversiy PO Bo Bbolsr Ir E-ils: ifr@zcir ry78ghsei@yhooco & sidy@zcir Absrc I his pper he vriiol hooopy perrbio ehod (VHPM d is covergece is doped for he Zhrove-Kzesov eqios (ZK-eqios The i of his pper is o prese efficie d relible ree of he VHPM for he olier pril differeil eqios d show h his ehod is coverge The covergece of he pplied ehod is pproved sig he ehod of jors fro Cchy-Kowlevsy heore of differeil eqios wih lyicl vecor field Keywords: Vriiol hooopy perrbio ehod; covergece; Zhrove-Kzesov eqio Irodcio Trvelig wves re very ipor becse vrios pheoe i re sch s vibrio d selfreiforcig soliry wves re described by he So ivesigio of rvelig wve solio plys ipor role i olier sciece Rose d Hy [] irodced clss of pril differeil eqios (PDEs K (: ( ( >< ( which is geerlizio of he Koreweg-de Vries (KdV eqio sch h he heory of wer wves i shllow chels is described For ore iforio refer o [] I Eq( d For hese re Soliry wves or so-clled Copcos Recely Wzwz [] hs give he ew soliry pers for he olier dispersive K( eqios: ( ( > > ( The ew soliry wve specil solios wih copc sppor for he olier dispersive K( eqios: ( ( > > ( *Correspodig hor Received: 5 Noveber / Acceped: 9 Jry re preseed by Isil d Th [] d Wzwz [5] They se fiie differece ehod d fiie elee ehod o ivesige he pproie solios of K( d K( i Eq( The ZKeqio (shorly clled ZK( of he for yy ( b( c( ( where re iegers d b c re rbirry coss Eq( govers he behvior of wely olier io-cosic wves i pls coprisig cold ios d ho isoherl elecros i he presece of ifor geic field [6] Eq( is solved by differe ehod For isce i [7] ZK-eqio ws solved by he sie-cosie d he hyperbolic ge (h-fcio ehods I his pper he vriiol Hooopy Perrbio ehod sig He's polyoils is pplied o solve ZK-eqio d covergece of he cosidered echiqe is pproved Mehodology To irodce he VHPM i is ecessry o ow VIM d HPM Vriiol Ierio d Hooopy Perrbio Mehod To illsre he bsic coceps of he VIM d HPM firs cosider he followig olier differeil eqio L N g( (5

2 IJST ( A (Specil isse-mheics: - 5 where L is lier operor N is olier operor d g ( is ihoogeeos er Accordig o he VIM [89] we c cosrc correcio fciol s follows: ~ ( ( ( { L N g( } d (6 where ( is geerl Lgrge liplier ( c be ideified opilly vi he vriiol heory he sbscrip deoes he h-order pproiio d ~ is cosidered s resriced vriio ie ~ The esseil ide of his ehod is o irodce hooopy preer sy p which es he vles fro o Whe p he syse of eqios is i sfficiely siplified for which orlly dis rher siple solio As p grdlly icreses o he syse goes hrogh seqece of ``deforio'' he solio of ech sge is ``close'' o h he previos sge of ``deforio'' Evelly p he syse es he origil for of eqio d he fil sge of ``deforio'' gives he desired solio To illsre he bsic cocep of hooopy perrbio ehod cosider he followig olier syse of differeil eqios A(U f ( r r (7 wih bodry codiios U B U r where A is differeil operor B is bodry operor f (r is ow lyicl fcio d is he bodry of he doi Geerlly speig he operor A c be divided io wo prs L d N where L is lier d N is olier operor Therefore Eq(7 c be rewrie s follows: L( U N ( U f ( r We cosrc hooopy V ( r p : [] R which sisfies H(V p ( p[l(v L(U ] p[ A(V f ( r ] p [] r or eqivlely H (V p L(V L(U pl(u p[ N (V f ( r ] (8 where U is iiil pproiio of Eq(7 I his ehod sig he hooopy preer p we hve he followig power series preseio for V V V pv p V The pproie solio c be obied by seig p ie U U U U Vriiol Hooopy Perrbio Mehod sig He's polyoils To illsre he bsic ide of he VHPM cosider he followig geerl differeil eqio L N g( (9 where L is lier operor N is olier operor d g ( is ihoogeeos er Accordig o he VIM s illsred previosly we c cosrc correcio fciol s follows: ~ ( ( ( ( L N g( d ( where ( is geerl Lgrge liplier Applyig he hooopy perrbio ehod ( g( d p ( p ( N p d ( which is he coplig of VIM d He's polyoils d is clled he odified vriiol ierio ehod (MVIM The copriso of siilr powers of p gives solios of vrios orders So c be obied s d where f ( ( g( d ( ( H d

3 5 IJST ( A (Specil isse-mheics: - d H N( p p! dp For ler ericl copio we le he epressio i ( y i o deoe he - er pproiio o ( For ore iforio bo he VHPM refer o [] The VIM d VHPM for ZK-eqio The VIM for ZK-eqio I order o solve ZK-eqio ( b( c( yy by he VIM correcio fciol c be cosrced s follows: ~ ( ( ( { ( y ( y b ~ ( y c~ ( y d ( yy where ~ is cosidered s resriced vriios ie ~ To fid he opil vle of ( we hve ~ ( ( ( { ( y ( y yy ~ b ( y c~ ( y d ( or ( y ( y ( d (5 which resls i ( y ( y ( ( y ( ( y d (6 Therefore he siory codiios re obied i he followig for ( ( (7 which resls i ( Sbsiig his vle of he Lgrge liplier io Eq( gives ( y ( y { ( y ( y yy b ( y c ( y d (8 The ierio forl will give severl pproiios d he ec solio is obied he lii of he reslig sccessive pproiios The VHPM for ZK-eqio Usig he vle of Lgrge liplier h ws clcled i he previos secio d pplyig he VHPM gives: p p f ( p (( p p bp (( p p d d cp (( p p yyd (9 The copriso of siilr powers of p gives solios of vrios orders d he copoe which cosies ( y is wrie s ( y i i ( y Covergece of VHPM for ZK eqios Cosider he iiil vle proble L N > ( wih iiil codiio ( f ( where do deoes differeil i ie Asspio Le L: X Y for coios seigrop E( ep( L for R N( : X X be lyic er f d X be Bch lgebr wih he propery fg f g f g X ( Noe h for ZK- eqio so E ( I By Dhel s priciple proble ( c be reforled s iegrl eqio L

4 IJST ( A (Specil isse-mheics: - 5 ( E( f E( N (( d > ( If N( is lyic er f i sisfies locl Lipschiz codiio i he bll B ( f of rdis > ceered f ie here is cos K > sch h N( N( f f ( I he followig he covergece of he VHPM is pproved for Eq( Theore (Picrd-Ko Le L d N ( sisfy sspio d f X There eiss T > d iqe solio ( of he iiil-vle proble Eq( o [T] sch h ( C([ T ] X C ([ T ] X ( d ( f Moreover he solio ( depeds coiosly o he iiil d f See [] for he proof of Picrd-Ko heore Usig his heore locl well-posedess of solios of he iiil-vle proble ( wih Lipschiz vecor field N( c be proved for sll ie iervls Theore (Cchy-Kowlevsy Le sspio be sisfied wih X Y d ( be iqe solio of Eq( i C ([ T ] X where T > is he il eisece ie The here eis (T sch h :[ ] X is lso rel lyic fcio For frher deils d he proof of Cchy- Kowlevsy heore see [] By Cchy esies here eis coss b > sch h:! b N( (5 where N( deoes operor i he sese of Freche derivive eg N( N ( is he Jcobi operor The Tylor series of N ( f coverges for y oreover we obi f < d N(! b : g( where fcio g (! N( f ( f b f (6 f < Fro he jor i is cler h! N( f g( (7 Now cosider he jor proble d g( d ( R > (8 where The jor proble hs eplici solio ( b which is lyicl fcio of o ( Usig hese relios he covergece b of he VHPM c be pproved for ZK- eqios Covergece heore Le he sspio be sisfied d ( be iqe solio of Eq( i C ([ T X where T > is he il eisece ie As eioed before ( is defied s ( H d For ZK-eqios ( So H d (9 There eis (T sch h he h pril s coverges o he solio ( i C([ ] X 5 Rer S whe

5 55 IJST ( A (Specil isse-mheics: - Eisece d iqeess of he solio ( i C ([ T X is pproved i Picrd-Ko Theore 6 proof Fro Picrd-Ko heore for y > here eis ( sch h T sp [ ] f ( < Choosig where is he rdis of lyiciy of N ( er f The Cchy esies Eq(5 c be geerlized s N(! b g ( ( N( f!! ( ( (!!! f f ( where ( sisfy he jor proble Eq(8 for [ The ( d ll is derivives b wih respec o re icresig fcios of d so re g( ( d ll is derivives wih respec o By sig seigrop propery d Eq(9 H d N( ( d g( ( d g( ( ( H d N ( ( y d ( g (( ( g( ( d g( ( g( (! (! By idcio we sse h (! ( d prove h he se relio reis re : ( (! [ b As ( is lyic i for ll [ for b y sll p > here eiss C fcio ~ p ( o [ sch h b ( ~ p p p ( (( p! (! p if p he p p p p (( p By defiiio of H ~ p ( (! H we hve d (! N( p p! dp d p N( p (! so h dp d g( (( p d p dp H N( p! dp! d g( ( d! (! p( g( (! p ( where p is polyoil of g d is derivives p o he h order wih posiive coefficies (he se s i he proof of Cchy-Kowlevsy heore Usig he ierive forl Eq (9 we obi

6 IJST ( A (Specil isse-mheics: - 56 Therefore he power series ( H d ( (! ( is jor i X by (! b which coverges for ll < Recll he b cosri [ T ] i bod Eq( By he Weiersrss M-es ( covergece i C([ ] X for y ( where i{ } o he iqe solio ( of b or eqio 7 Corollry There eiss cos C sch h he error of he VHPM is boded by E sp S > b C ( where ( b re defied i covergece heore 8 proof Fro covergece heore we hve sp E (! I follows fro he eplici for for ( h (!! b ( b ( / As resl we obi ( sp! (!! b b (! b b b ( b (!! b ( (5 (! b s (!! (! we obi b b E b ( ( b b b If ( < he ( d we b b hve E C C b ( b b b ( ( b b ( b b (6 where b C Whe b b E So wih pproprie choice for d b ZK-eqios will be coverge for < b Now we wold lie o choose wo specil eqios ely ZK( d ZK( wih specific iiil codiios Firs cosider he ZK( eqio: ( ( ( yy (7 8 8 wih specific iiil codiios ( f ( ( where is rbirry cos Asse d proceedig s before sig VIM he lgrge liplier is deeried s Bsed o he VHPM we hve: p p f (

7 57 IJST ( A (Specil isse-mheics: - p (( /8p (( p p p p d d /8p (( p p d (8 Coprig he coefficie of siilr powers of p or sig Eq(9 we hve: p : ( y yy p : ( ( d ( d ( d 8 8 ( ( sih( 9 ( y sih ( y yy p : ( ( d ( ( d ( d ( ( sih ( ( sih 9 ( yy 6 ( sih ( d 8 ( yy d p : ( ( d ( sih( ( 656 sih ( 5 ( sih ( ( sih ( ( i i 6 ( d ( y ( y By sbsiig i Eq(7 d sdyig he resl of h i Fig i is cler h or pproiio sisfies he eqio wih high level of ccrcy So will be coverge o he ec solio To Coie cosider he ZK( eqio ( ( ( (9 yy wih specific iiil codiios ( f ( 5 sih(( /6 s eioed i he previos eple we sse By sig VIM he lgrge liplier is deeried s Applyig he VHPM gives: p : ( y 5sih(( /6 yy p : ( y ( d ( d ( d ( sih (( / 6(( / 6 (( y / 6 8 p : ( ( d ( d ( yy d ( sih (( /6 sih (( /6 (( / sih(( /6 (( /6 6 p : ( ( d ( d ( yyd ( sih (( /6 (( / sih (( /6 (( / sih (( /6 5 (( / (( /6 56 ( d proceedig s before ( y ( y i i Sbsiig i Eq(9 d sdyig Fig shows h he VHPM gives he solio of Eq (9 which hs ecelle greee wih he ec oe So or pplied ehod is coverge for his eple oo Fig The resl of sbsiig i Eq(7 y

8 IJST ( A (Specil isse-mheics: - 58 Fig The resl of sbsiig i Eq(9 y 5 Coclsio I his wor he vriiol hooopy perrbio ehod sig He's polyoils is sed o solve Zhrov-Kzesove eqios As show his ehod is effecive d srighforwrd echiqe Oe ipor objec of or reserch is he eiio of he covergece of he vriiol hooopy perrbio ehod sig He's polyoils Covergece heores re give i geerl for pril differeil eqios d he resl is eied o Zhrov-Kzesove eqios s specil cses The resls show h he VHPM is coverge ehod d c be sed o solve oher lier d olier eqios [8] He J H ( A coplig ehod of hooopy echiqe d perbrio echiqe for o-lier probles I J of No Mech 5( 7- [9] He J H ( Hooopy perbrio ehod: ew olier lyicl echiqe Appl Mh d Cop 5( 7-79 [] He J H ( Vriiol ierio ehod for ooos ordiry differeil syses Appl Mh d Cop ( 5- [] Mifr M Mhdvi M & Reisy Z ( The vriiol hooopy perrbio ehod for solvig lyic ree of he lier d olier ordiry differeil eqios J Appl Mh d Iforics 8( [] Mifr M & Ghsei M ( Vriiol Hooopy Perrbio Mehod for he Zhrove- Kzesov Eqios J of Mh d Sisics 6( 5- [] Abdelrzec A & Peliovsy D ( Covergece of he Adoi Decoposiio Mehod for Iiil-Vle Probles J N Mehods for Pril Differeil Eqios 7( Refereces [] Rose D & Hy J M (99 Copcos Solios wih fiie wveleghs Phys Rev Le 7( [] S J C (979 Qelqes géérlisios de l'éqio de Koreweg-de Vries J Mh Pres Appl 58-6 [] Wzwz A M ( Ec specil solios wih soliry pers for he olier dispersive ( eqios Cho Soli Frc ( 6-7 [] Isil M S & Th T R ( 998 A ericl sdy of Copcos Mh Cop Sil 7( [5] Wzwz A M ( New soliry-wve specil solio wih copc sppor for he olier dispersive ( eqios Cho Soli Frc ( - [6] Zh Y ( Ec specil solios wih soliry pers for Bossiesq-Lie B( eqios wih flly olier dispersio Cho Soli Frc ( - [7] Y Z ( New filies of solios wih copc sppor for Bossiesq-Lie B( eqios wih flly olier dispersio Cho Soli Frc (8 5-58