Numerical KDV equation by the Adomian decomposition method

Transcription

1 America Joral o oder Physics ; () : -5 Pblished olie ay (hp://wwwsciecepblishiggropcom/j/ajmp) doi: 648/jajmp merical KDV eqaio by he Adomia decomposiio mehod Adi B Sedra Uiversié Ib Toail Faclé des Scieces Dépareme de Physiqe LHESI Kéira orocco address: msedra@icpi ( B Sedra) To cie his aricle: Adi B Sedra merical KDV Eqaio by he Adomia Decomposiio ehod America Joral o oder Physics Vol o pp -5 doi: 648/jajmp Absrac: Usig he Adomia decomposiio mehod (AD) we prese i his paper a merical approximaio o he solio o he oliear KDV eqaio The pricipal as cocers esseially he compaio o he Adomia polyomials or his ype o eqaio ad hereaer deermiig a sigiica crierio o esre he codiios or covergece o he mehod Keywords: KDV Eqaio merical Approach Adomia Decomposiio AD ehod or The KDV Eqaio Cosider he ollowig ormlaio o he KdV eqaio: x x ( x ) x ( ) which ca be rewrie as ollows: () 6 ( ) ) ( x) () where / x represes he liear operaor o he eqaio; ad ( ) / is he o-liear cio Accordig o he Adomia decomposiio he solio is expressed as: ad he o-liear par by: wih: ) ) () A ( ) (4) d i A λ i por! dλ Ν (5) i λ By iegraig wih respec o ime ad sig he iiial codiios we have: So: + [ L 6 ( ) ] ) ( x) ds (6) ) ( x) L s) + 6 A ds (7) For he KdV eqaio he Adomia polyomials ca be expressed as ollows: A i i i This allows s o dedce ) amely: ) ( x) + ) [ + A ] wih he ollowig iiials codiios: ds (8) (9)

2 Adi e al: merical KdV eqaio by he Adomia decomposiio mehod ) sech We proceed i he ollowig o comp ) or [ ] approximae solio ~ ) ollowig ormla: x ad o be able o deermie he e ( x ) A () ad p he h order sig he ~ ) i ( x) i ) L i i s) + 6 A i ds i () Ths he approximae solio is : ~ ( x ) i ) merical esls i We will develop i he ollowig a compariso o he graphical preseaios bewee diere ieraios o he approximae solio o a give order ad he exac solio i a rage o space [ ] ad wo diere iervals o 5 5 as show below : ime [ ] ad [ ] () Fcio Graphic Preseaio Graphic Preseaio Based o graphical preseaios preseed above i shold be oed he ollowig observaios: The asympoic resls o he approximae solio rom abo 5 h order divergig a he growig ed o he ime

3 America Joral o oder Physics () : -5 parameer; Good agreeme bewee he exac solio ad he approximae solio i he redced ierval ime [ 5] To emphasize he secod remar we proceed o he graphical preseaio o wo cios: ( ) sech ɶ ( ) () 6 8 ( )! The Taylor series expasio o ( ) ( ) which coverges or all have:! or : ( ) ( ) (5) (6) ad more we ( )!! ρ (7) We deie he cio: g ( ρ) ρ (8) where: The las preseaio clearly shows ha i he eighborhood o he redced ierval ime here are almos coicidece bewee he wo graphs ad ha beyod i he approximae solio diverges i a ssaied maer compared wih he exac solio which ca esimae he rae bewee he maximm vales o he wo cases o 4 imes We propose o calclae a coveie ime vale desigaed by τ rom which he approximae solio ~ ) begis o diverge Covergece Aalysis o he AD ehod or he KDV Eqaio We cosider ha : is a real aalyic cio i a disc B ( ) whose radis is ad L: veriies E ( ) C where C ad edow o he orm ( x) ( ) sp The ormlaio o Adomia mehod eables s o wrie: ( ) E( s) A ( ( s) ( s) ) ds (4) + Accordig o he Cachy esimaio here exiss e sch ha: Ths we have: ρ (9) ( ) ( g ) ( ) Usig he ormlaio o Adomia mehod we id: ( C ) () + () ( s) A ds C E A ds () Cg ( ) Cρ ( ) ( s) A ds C E A ds () C g ( ) g( ) C ρ ( ) + C + ( + ( ) ) + ρ ( ) (4)! ( + ) This allows s o dedce ha he Adomia series a are boded by he coverge series o powers based o

4 4 Adi e al: merical KdV eqaio by he Adomia decomposiio mehod ρ ( ) C or all [ τ ] τ i or case sdy C So we have : ( ) ( x) ) sech ad sig he Cachy esimae made earlier: ( )! We explici or he case : ad we ae: Le he solio ( ) ( )! x ( ) where (5) (6) (7) 6 (8) E be he operaor associaed wih he liear Cachy problem as ollows: sch as ( ) have: ( ) v E v Lv v( ) ad based o he above we ( ) C (9) E () Ths by ideiyig ad ollowig he expressio o he Forier rasorm o he solio ( x ) ˆ iξ ha: E( ) e v ca be dedced O he oher had sig he Parseval eqaliy we have: E + ( ) ( + ) ξ π π π + + ( + ξ ) ( + ξ ) Thereby veriyig he ieqaliy: wih : C i his case We have also : Eˆ( ) ˆ Eˆ( ) ˆ ˆ ( ) C () E (4) or by deiiio we ca wrie: + E( ) + ( C + ) ( C ) (5) + (6) The deermiaio o he hreshold covergece i ime eqal o τ he expressed by: C τ 6 (7) I he case o he KdV eqaio he liear operaor is: L ad as previosly meioed sig he Forier rasorm or he resolio o he liear problem we id: v + π iξ + iξx ) e ˆ ( ξ ) ) where: () ˆ + π iξx ( ξ ) e ( x) dx ξ () Aer calclaio we id ha:

5 America Joral o oder Physics () : -5 5 τ 5 Perormig he graphical compariso bewee he exac solio ad he approximaed oe a he TH order we id ha he deachme bewee he wo crves pracically sars rom he hreshold vale calclaed ad illsraed i he ollowig graphic: Throgh his wor i was possible o apply he AD mehod or he resolio o he KdV eqaio sig a polyomial o ime o TH order Solio ha has prove eecive i a give ime ierval Ths we have compleed or sdy wih a aalysis which allowed s o deermie he hreshold o covergece o he said solio eereces [] K Abbaoi ad Y Cherral Covergece o adomia s mehod applied o diereial eqaios Comp ah Appl (994) [] K Abbaoi ad Y Cherral Covergece o Adomia s mehod applied o oliear eqaios ah Comper odel (994) [] Himo K Abbaoi ad Y Cherral ew resls o covergece o Adomia s mehod Kyberees (999) [4] Y Zh Q Chag ad S W A ew algorihm or calclaig Adomia polyomials Appl ah Comp (5) 4 Coclsio [5] Ic O merical solio o parial diereial eqaio by he decomposiio mehod Kragjevac J ah (4)