A New Algorithm for Solving Coupled. Schrödinger KdV Equation: An Application. of the Fourier Transform Adomian. Decomposition Method

Transcription

1 . Ses Theor. Phys. Vol. 8 o HIKRI.-hkar.o hp://.o.or/.988/asp..6 e lorh for Sol ople Shröer KV Eqao: pplao of he orer Trasfor oa Deoposo Meho reshr ha Sahareh Depare of Mehaal Eeer Soh Tehra rah Isla za Uersy Tehra Ira Paya Izay Depare of Mehaal Eeer Soh Tehra rah Isla za Uersy Tehra Ira rash Za * Depare of Mehaal Eeer Soh Tehra rah Isla za Uersy Tehra Ira * orrespo ahor opyrh reshr ha Sahareh Paya Izay a rash Za. Ths s a ope aess arle srbe er he reae oos rbo ese hh pers resre se srbo a reproo ay e proe he oral ork s properly e. bsra I hs sy a hybr of orer rasfor a oa Deoposo Meho TDM for f aalyal solos of he ople Shröer KV eqao s osere. The aalable aalyal solos of he ople Shröer KV eqao

2 58 reshr ha Sahareh Paya Izay a rash Za obae by lple rael ae eho are opare h TDM o eae he aray of he eho. opar he ehooloy h soe oher ko ehqes shos ha he prese approah s effee a poerfl. Moreoer TDM aes ha he ao of opaoal ork s h less ha he opaoal ork reqre for oher se-aalyal ehos. Keyors: orer rasfor a oa Deoposo Meho TDM; he ople Shröer KV eqao Iroo The solos of he olear eolo eqaos play a pora role he fel of olear ae pheoea. The ea solos falae he erfao of eral ehos he hey es. I he las fe eaes sbsaal proress has bee ae o researhes hs area a oes h hs reo. Mlple rael ae solos of olear eolo eqaos sh as he ople Shröer KV eqao [ ] hae bee sessflly apple o e ea solos by a []. The oel eqao for he ople Shröer KV eqao a be presee he follo for U U U 6 U Where. The TDM as frs propose by S.S.orazar [] a The TDM s sefl o oba ea a approae solos of lear a olear ffereal eqaos. I hs eho e se a hybr of orer rasfor a oa Deoposo Meho slaeosly []. The a oal of he prese sy s o f he aalyal solos of he ople Shröer KV eqao by he he hybr of orer rasfor a oa Deoposo Meho.[]. as Iea of orer Trasfor oa Deoposo eho TDM We oser he eeral fors of oe-esoal olear paral ffereal eqaos for llsra he bas ea of he TDM.. The operaor a eerally speak be e o o pars lear a olear as:

3 e alorh for sol ople Shröer KV eqao 59 Tak he orer rasfor fro boh se of Eq. e oba } } } o e hae o apply he oep of oa Deoposo Meho The oa eoposo eho osss of eopos he ko fo of ay eqao o a s of a fe ber of opoes efe by he eoposo seres: 5 here he opoes are o be eere a rerse aer. I s obos for lear par e a re so easly 6 for olear ers hae o be epresse by a fe seres of he so-alle oa polyoals a e he for... 7 o e are able o ealae for all fors of oleary by s Taylor seres.e. The oa polyoals for he olear er a be ealae by s he follo epresso ] [! λ λ λ 8 So by ak se 8 oa polyoals are e by!! 9 so o. fer hs eqao 5 a be re as follos } } } -a } } } -b Us -b a able s o roe he rerse relao as belo: } } } } k k k k Oe a rere he rerse relao as:

4 6 reshr ha Sahareh Paya Izay a rash Za } } } } } } so o. Hereafer by ak se Maple pakae he frs par of Eq. es he ale of } for e he aser of frs par s sffe o apply erse orer rasfor o } he o e hae he ale of herefore e a apply he seo par of Eq. a se he rerse relao he e apply he erse orer rasfor o all opoes hh e e obae by ak se rerse relao. pply TDM for Sol ople Shröer KV Eqao I hs seo o ffere solos of he ople Shröer KV eqao [] ll be eae by s he TDM. y s U oe a separae Eq. o real a aary pars. Therefore oe a e a -esoal rple syse he follo for 6 8 I orer o oba TDM solo of he Eq. 8 e apply orer rasfor o hese eqaos } } 6 } 9-ab y apply orer rasfor o seo hr forh ffh a sh er lef se he hr eqao of Eqs. 9 e obae: } - } -ab o e a re

5 e alorh for sol ople Shröer KV eqao 6 } } } -ab o e a able o roe he rerse relao as belo: } } } } } -ab Whh hese rerse relaos e a efe as belo: ab... o o eae he TDM for he ople Shröer KV eqao Jaob obly pero ae solo [] s se he follo seos.. Jaob pero solo o ople Shröer KV eqao The Jaob ea solos are e [] by os θ ξ b a θ ξ s b a ξ b a p -ab

6 6 reshr ha Sahareh Paya Izay a rash Za a here a b p are arbrary osas aξ p aθ p p p. or b sply a b p are se hs aalyss. The Eqs. -ab ake he follo for for. os / s / / 5-ab o e oba fro aboe eqaos al oos as: os s / 6-ab boary oos are: os / s / / 7-ab o e sar h orer rasfor of al oos 8-ab o f e sbse boary oos 7-a b a orer rasfor of al oos 8-a b he rerse relao -a b oe a easly re: s/ os / } os / os/ s/ s / } / / } 9-ab Hereafer by ak se Maple pakae for e erse orer rasfor e oba: os s s os os s os s os s -ab -ab

7 e alorh for sol ople Shröer KV eqao 6 so o.. a sho he oparso beee ea a TDM solos of a a e a see ha he resls of TDM are ery oo.. The oparso beee ea a TDM solos of a 5. olsos To p he sse o perspee TDM s a poerfl ool hh s apable of hael lear a olear paral ffereal eqaos. I hs sy he orer Trasfor Varaal Meho as se for he ople Shröer-KV eqao h al a boary oos. The TDM rees he ole of allaos a has rea aaaeos hh s apply boary oo ao o al oo. Referees [] E. a Mlple rael ae solos of olear eolo eqaos s a fe alebra eho J. Phys

8 6 reshr ha Sahareh Paya Izay a rash Za [] S.S. orazar. azar-golsha. Ylr a M. orazar O he Hybr of orer Trasfor a oa Deoposo Meho for he Solo of olear ahy Proble of he Reao-Dffso Eqao Z.arforsh. 67a 55-6 [] S. S. orazar M. Raezapor. Dooshose e lorh o Sole he Gas Dyas Eqao: pplao of he orer Trasfor oa Deoposo Meho pple Mahea See [] R. Hose M. Raezapor. Dooshose a E. Rezae e Meho for Sol ople Shröer KV Eqao pple Mahea See [5].. Rao olear ae olaos plasas Praaa J. Phys [6].M. Wazaz The ofe eoposo eho for aaly reae of ffereal eqaos ppl. Mah. op [7] G. oa Sol roer Probles of Physs: The Deoposo Meho Kler oso 99. [8] G. oa olear Sohas Operaor Eqaos ae Press Sa Deo 986. [9] M. Raezapor. Moazer P. Izay a. Dooshose e pproah o he Gas Dyas Eqao: pplao of he orer Trasfor Varaoal Ierao Meho pple Mahea Sees Reee: Deeber