Construction of Canonical Polynomial Basis Functions for Solving Special N th -Order Linear Integro-Differential Equations

Transcription

1 Aerc Jourl of Egeerg Reserch AJER Aerc Jourl of Egeerg Reserch AJER e-iss : -87 p-iss : -9 Volue- Issue-5 pp-- wwwerus Reserch per Ope Access Costructo of Cocl olol Bss Fuctos for Solg Specl th -Orer er Itegro-Dfferetl Equtos Two O A R M T Deprtet of Mthetcs Uerst of Ilor Deprtet of Mthetcs Sttstcs The ol I Astrct: The prole of solg specl th -orer ler tegro-fferetl equtos hs specl portce egeerg sceces tht costtutes goo oel for sstes rous fels I ths pper we costruct cocl polol fro the fferetl prts of specl th -orer tegro-fferetl equtos use t s our ss fucto for the uercl solutos of specl th -orer tegro-fferetl equtos The results ote ths etho re copre wth those ote Ao Decoposto etho It s lso osere tht the ew etho s effecte etho wth hgh ccurc Soe eples re ge to llustrte the etho Kewor: Itegro - fferetl equto cocl polol fferetl collocto etho I ITRODUCTIO Itegro - fferetl equto s equto whch olg oth fferetl tegrl equto Ths tpe of prole rse Scece Egeerg ecuse of coplet of ths prole we scoer tht orer to get ect or ltcl soluto of the proles uercl lst re ow to eelope terest ths re ths otte the resercher to stu ths clss of prole The Cocl olol estlshe o [-5] s thoroughl use reserchers to hle we ret of scetfc egeerg pplctos: ler oler hoogeeous hoogeeous s well It ws show uthors [ ] tht ths etho proes proeets oer estg uercl techques The etho ges rpl coerget seres soluto pproto of the ect soluto f such soluto ests Two [] otte ths resercher work ue to soe propertes of Cocl polols reporte the work tht: Cocl olol c e geerte oer ge terl of coserto; It c e esl progre; It c e geerte recursel Wthout loss of geerlt the resercher cosers specl th -orer ler tegro-fferetl equto IDE of the for: p f K t t t re poste tegers where re rel costts p f K t re ge sooth fuctos whle s to e etere Eq - occur rous res of egeerg echcs phscs chestr stroo ecoocs potetl theor electrosttcs etc M ethos re usull use to hle the hgh-orer IDE - such s the successe pprotos Ao ecoposto Hootop perturto etho Tlor collocto Hr Welet Tu Wlsh seres ethos Mote Crlo Metho Drect etho se o Fourer lockpulse fuctos etc [- -7-5] ut ue to the proles ecoutere soe of these uthors w w w e r u s ge

2 Aerc Jourl of Egeerg Reserch AJER tegrtg cople fuctos lke e cos e etc the the etho seres s tge Sgfct of the stu The costructo of ew ss clle Cocl polols pple to the ler o-ler proles Tht goo choce of ss pls portt role oth the ccurc effcec of collocto etho s well-kow the lterture The eteso of Cocl polols s ew ss for collocto etho s ee the followg oserto were ote Cocl polols proe soe coputtol tge og whch re the followg; The re geerte sgle recurse forul The re epeet of the terl of coserto The re epeet of the ssocte cotos The esure hghl stle etho A-stlt optol orer ccurc Oe or tge of the pproch s tht t s esl frel to error estto We ote tht o ler proles re sole usg the collocto etho ters of cocl polols for the sequece of lerze pprote proles The ewto s lerzto process s use whch gurtees qurtc coergece rte of the terto We lso ote tht the etho proes the soluto rpl coerget seres wth copoets tht re elegtl copute Wth ll these oserto the resercher coclue tht cocl polol pls portt role ter of ccurc effcet II COSTRUCTIO OF CAOICA OYOMIA Fro the geerl equtos stte - we efe D s follows; For the cse = we efe our opertor s: o w w w e r u s ge

3 Aerc Jourl of Egeerg Reserch AJER w w w e r u s ge ; For = : For =: For = : For = : For = : Thus fro equto we ot the followg etc For the cse = we efe our opertor s: o ; 5

4 Aerc Jourl of Egeerg Reserch AJER w w w e r u s ge For = : For =: For = : For = : For = : ] 8 [ 5 Thus fro equto 5 we ot the followg etc et e the ect soluto of the tegro-fferetl equto f t t t D wth c c 7 where f t re ge cotuous fuctos c c soe ge costts III MATRIX RERESETATIO FOR THE DIFFERET ARTS et : V e polol ss : X V V where V s o-sgulr lower trgulr tr egree for Also for tr VV ow we coert the Eq 7 to the correspog ler lgerc equtos three prts; c Mtr represetto for D :

5 Aerc Jourl of Egeerg Reserch AJER Ortz Sr propose [8] lterte for the Tu techque whch the clle the opertol pproch s t reuces fferetl proles to ler lgerc proles The effect of fferetto shftg tegrto o the coeffcets ector ~ : ~ ~ ~ u ~ X Of polol s the se s tht of post-ultplcto of respectel u ~ X u ~ X u t t ~ X where We recll ow the followg theore ge Ortz Sr [8] ~ the trces Mtr represetto for the tegrl ter: et us ssue tht t t V The we c wrte t t t where M Wth t o t for l c o t t t MV Mtr represetto for the suppleetr cotos: 8 9 Replcg the left h se of 7 t c e wrtte s c c c c where for = c c c c B c c B w w w e r u s ge 5

6 Aerc Jourl of Egeerg Reserch AJER We refer to B s the tr represetto of the suppleetr cotos The followg reltos for coputg the eleets of the tr B c e euce fro : c c for c c for k B s ts th colu We trouce the ector tht cots rght h ses of cotos The the suppleetr cotos tke the for B It follows fro 8 9 tht D t t M V 5 et M : M M sts for ts th colu let f f f V wth f f f The the coeffcet of ect soluto V of prole 7 stsfes the followg fte lgerc sste: M f ; M ; B ; settg G B B M M g f f We c wrte ste of G g Rerk: For G 7 Eq s trsfore to Frehol tegrl equto of seco k for t s trsfore to fferetl equto IV DESCRITIO OF THE METHOD For the purpose of our scusso we ssue pprote soluto of the for Where re costts to e etere re the cocl polols costructe oe We wrte equto the for: D f V So tht D or I V D ; 9 K tt t 8 w w w e r u s ge

7 Aerc Jourl of Egeerg Reserch AJER The puttg equto 8 to equto we ot Q I f V Q f k t Q t t re kow fuctos to e supple re ukow costts to e etere; re cocl polol geerte secto together wth the followg cotos: ' I equto the tegrl prt hs to e elute fter whch the left oer re the collocte t pot to ot k where k k k f k kk t t t k k ; k Thus equto ge rse to - lgerc ler sste of equtos + ukow costts The reg equtos re ote usg the our cotos stte equto whch re the susttute to These equtos re the sole to ot the ukow costts equto 8 to ot our pprote soluto Rerk: ll these proceure scusse oe he ee trslte the etre process s utote the use of solc lgerc progr MATAB 79 o ul coputto s requre e where V ERROR I ths secto we he efe our error s s the ect soluto s the pprote soluto copute for rous lues of VI UMERICA EXAMES I ths secto we coser soe eples of thr fourth orer ler tegro-fferetl equtos Reso: Becuse of frequet occurrece of prole flu cs ologcl oel scece egeerg we ece to pck soe prole whch re cool use copre the result ote ltc soluto result lle Mthetcl oellg of rel lfe phscs egeerg proles usull results these clsses M thetcl forultos of phscl pheoe cots tegro-fferetl equto these equtos rse flu cs ologcl oels checl ketcs Itegro-fferetl equtos re usull ffcult to sole ltcll so t s requre to ot effcet pprote soluto Therefore the ee of ths stu lso to scuss the estece uqueess of the solutos for these clsses of proles The preset work s otte the esre to ot ltcl uercl solutos to our lue proles for hgh-orer tegro-fferetl equtos Eple : Coser the thr orer ler tegro fferetl equto w w w e r u s ge 7

8 Aerc Jourl of Egeerg Reserch AJER s t tt t t wth the cotos ect soluto s cos We use the solute error to esure the fferece etwee the uercl ect solutos I tle result ote for = 7 8 re ge wth the ect soluto t Eple : Coser the thr orer ler tegro fferetl equto cos t tt t t t wth the cotos ect soluto s s t w w w e r u s ge 8

9 Aerc Jourl of Egeerg Reserch AJER Eple : Coser the ler our lue prole for the fourth-orer tegro fferetl equto e e t t wth the cotos e e The ect soluto of the oe our lue prole s e w w w e r u s ge 9

10 Aerc Jourl of Egeerg Reserch AJER VII DISCUSSIO AD COCUSIO I ths pper Cocl polol hs ee successfull use s ss fucto for the uercl soluto of specl th -orer tegro-fferetl equtos The soluto ote es of the cocl polol s fte power seres for pproprte cotos whch c e tur epresse close for The results ote here re copre wth result of Sezer Gulsu [] reele tht Cocl polol s powerful thetcl tool for the uercl solutos of specl th-orer ler tegro fferetl equtos ters of ccurc chee REFERECES [] As S 8 Solto soluto for the Ftzhugh-guo equto wth the hootop lss etho pple thetcl oellg 7-7 o: /chos7 [] As B A K M T 5 Drect etho for solg tegro fferetl equtos usg hr fourer lock-pulse fuctos Iter J Coputer Mth 8 o [] Dfu H A Xufeg S 7 uercl soluto of tegro-fferetl equtos usg CS welet opertol tr of tegrto Appl Mth Coput 9 - [] Gol A A J M 7 Applcto of He s hootop perturto etho for the th-orer tegro-fferetl equtos Appl Mth Coput [5] Ht T A S M 7 O ltc soluto for th fl flow of fourth gre flu ow ertcl cler hscl etters A - [] Hosse S A Shhor S A tr forulto of the tu etho Volterr ler tegrofferetl equtos Kore J Coput Appl Mth [7] Ic M 8 Applcto of hootop lss etho for f effecec of coecte strght fs wth teperture-epeet therl couctt Mth Copt Sul Do:/tco79 [8] Ic M 7 O ect soluto of plce equto wth Drchlet eu our cotos the hootop lss etho hscs etters A 5-5 [9] Ic M 8 O uercl soluto of Burgers equto hootop lss etho hscs etters A 5- [] Krete A Sezer M A Tlor collocto etho for the soluto of ler tegro-fferetl equtos Iter J Coputer Mth [] o S J Bo erturto: Itroucto to the Hootop Alss Metho Chp Hll/CRC ress Boc Rto [] o S J O the hootop lss etho for oler proles Appl Mth Coput [] o S J 8 otes o the hootop lss etho: soe eftos theores couctos oler Scece uercl sulto Do:/css8 [] o S J A T Y 7 A geerl pproch to ot seres solutos to oler fferetl equtos Stues Apple Mthetcs [5] o S J A eplct lct soluto to the Thos-Fer equto Appl Mth Coput 95-5 [] Mleke K A Mrzee F uercl soluto of tegro-fferetl equtos usg rtolze Hr fuctos etho K-eretes It J Sst Mth [7] s S Ylcs S A Sezer M A Tlor polol pproch for solg hgh-orer ler Frehol tegrofferetl equtos It J Mth Euc Sc Techol 9-5 [8] Ortz E Sr H 98A opertol pproch to the Tu etho for the uercl soluto of o-ler fferetl equtos Coputg [9] Rh-Arl M Y Shhor S 7 Iterte uercl soluto of o-ler tegro-fferetl equtos the Tu etho Appl Mth Coput [] Rsh J A Zre M 7 The uercl soluto of tegro-fferetl equto es of the Sc etho Appl Mth Coput 88 - [] Sezer M A Gulsu M 5 olol soluto of the ost geerl ler Frehol Itergrofferetl fferece equtos es of Tlor tr etho Cople Vrles [] T A As S 8 Hootop lss etho for qurtc Rcct fferetl equto Couctos oler Scece uercl Sulto 59-5 [] Two O A 99 Collocto pproto for sple perture our prole hd Thess Upulshe Uerst of Ilor Ilor [] Tssol K M Ghse M A Bol E uercl soluto of ler egro-fferetl equto usg se-cose welets Appl Mth Coput [5] Wg W A lgorth for solg the hgher-orer oler Volterr-Frehol tegro-fferetl equto wth echzto Appl Mth Coput 7 - [] Wg s A He J 7 Vrtol terto etho for solg tegro-fferetl equtos hscs etter A [7] Ylcs S A Sezer M The pprote soluto of hgh-orer ler Volterr-Frehol tegrofferetl equtos ters of Tlor polols Appl Mth Coput 9-8 [8] Zho J A Corless R M Copct fte fferece etho for tegro-fferetl equtos Appl Mth Coput w w w e r u s ge