Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 agurr l ad is prograig B Sayaaraya Y Pragahi Kuar Asa Abdullah 3 3 Dpar o Mahaics Acharya Nagarjua Uivrsiy Adhra pradsh Idia Dpar o Mahaics Collg o Naural ad Copuaioal Scics Adigra Uivrsiy Adigra Ehiopia Absrac I his papr h auhor cosrucs w agurr l ucio wih is progra by usig MAAB progra Also h auhor drivaiv ad igraio wih is powrs i rs arics ar cosrucd h icicy o h abov ucios hrough h us o hs vrbs i h soluio o so xapls ha will show us h validiy o wha w hav said Morovr so o h hypohsis was provd as h hors o orhogoaliy ad Covrg Kywords- agurr ls MAAB progra opraioal arix o igraio opraioal arix o drivaiv ad powrs i rs I INRODUCION Firs w show ha h ors ha cu daa io dir rqucy copos ar ahaical ucios [ ] I is show hrough ay sourcs ha ach copo will b sudid wih a rsoluio proporioal o is rag whr hy hav radiioal Fourir hods i cas aalysis i dir Scic or xapl physics [9 ] Ovr h pas yars hr hav b xchags bw sciiic ilds ha iclud h dvlop o ors idpdly i h ilds o girig scic ad gology [3-5] Irchags bw hs ilds durig h pas yars hav ld o ay w l applicaios such as iag coprssio urbulc hua visio radar ad arhqua prdicio [6 3] Wavl aalysis is a powrul ahaical ool ha has b usd widly i iag digial procssig quau ild hory [5-7] urical aalysis ad ay ohr ilds i rc yars oday hr ar ay wors o ls hods or approxiaig h soluio o h probls [8 9] such as Haar ls hod [8] SAC ls hod Haroic ls hod irs ad scod Chbyshv ls [4] ad gdr ls hod [7] I h prs papr gav so ipora characrisics o agurr polyoials wih is will b giv icludig w propris Procssig iag is also discussd i his papr II AGUERRE POYNOMIAS AND IS PROPERIES agurr s dirial quaio: h dirial quaio o agurr s polypial giv by xy " x y ' y () whr =3 his quaio has polyoial soluios calld agurr polyoials is giv by x d x x x () dx Which is also rrrd o as Rodrigu s orula or h agurr polyoials h irs w agurr polyoials ar x x x 3 x x 4x x 6 8x 9x x (3) 3 whr x is a polyoial o dgr Prograig agurr polyoial: By usig Malab ucio =ag() i == =; ls Su = ; or = : su = su+(- )^*acorial()*acorial()*^/ (acorial()*acorial()*acorial(-)); d =su; Ed 3 So ipora propris o agurr polyoials: I h ollowig w lis so propris o h agurr polyoials Graig ucio: x (4)! Rcurrc orula: x x x (5) x x x (6) x x x x (7) ISSN: 3-5373 hp://wwwijjouralorg Pag 9
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 3 Orhogoaliy x x x (8) 4 Sris xpasios: A dx! I x A x! i i h x x dx x (9) 4 Miscllaous orhogoal polyoials ad hir propris: hr ar ay ohr xapls o orhogoal polyoials So o h or ipora os oghr wih hir propris ar giv i h ollowig lis Associad agurr Polyoials x d x x dx () ad saisyig h quaio x y x y y () I h x x w hav x x () x x! x dx p dx p! II AGUERRE WAVEES p (3) I his scio w cosrucd agurr l ro h aily ucio r sr s or sr R s s (4) whr h ls M M ar h basis ucios orhogoal o h [] 3 Cosrucd agurr Wavls: agurr l is dod by (ag) is h yp o ls usd or solvig dirial quaios igral quaios variaio probls dir scics ad girig probls as wll as racioal dirial quaios agurr l hav our argus ordr or agurr polyoials ad oralizd is is ad i I w dilaio by parar s raslaio by parar r ad x us rasor x i (4) h w will g h ollowig quaio ohrwis (5) whr ~! or = 3 Prograig o agurr ls: By MAAB progra w ca g abov ucios or ls ucio o ordr by h ollowig algorihs Cas : ucios o h irval [ 5) ucio = ag() i = = *sqr() ls s = ; or = : s = s + (-) ^ *acorial()*acorial()* ( ddd*) ^/ (acorial()*acorial()*acorial(-)); d =(*sqr()/acorial())*s Ed Cas : ucios o h irval [ 5) ucio = ag() i == =*sqr() ls s = ; or = : s = s +(- )^*acorial()*acorial()*(ddd()) ^/ (acorial()*acorial()*acorial(-)); d =(*sqr()/acorial())*s Ed IV ORHOGONAIY OF AGUERRE WAVEES Fro scio (3) ad quaio (8) w ow x has orhogoaliy wih rspc o h wigh ucio w h s o lagurr ls ar h orhogoal wih rspc o wigh ucio o h irval w I is a sp ucio aig valus ls o 5 ad 5 rspcivly whr is ow ha ay coiuous ucio approxiad uiorly by agurr ucio W will did by usig h agurr ls Dilaios ad ISSN: 3-5373 hp://wwwijjouralorg Pag 3
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 raslaios o h ucio di a R orhogoal basis i h spac o all squar igrabl ucios his as ha ay l i R ay b rprsd as a liar cobiaio (possibly i ii) o hs basis ucios asy hor : h orhogoal o is o chc I is appar ha d (6) Whvr ad is o saisid siulaously I o zro valus o h l (say ) h ar coaid i h s whr h l coais agurr ucio h ha as igral qual o zro I bu h a las o acor i h produc is zro hus h ucio is orhogoal d d i = 8! i (7) V FUNCION APPROXIMAION A ucio approxiaio xpadd as ay b A whr A (8) I quaio (8) do h ir produc wih wigh ucio w o h Hilbr Spac I h iii sris i abov quaio is rucad h quaio (7) ca b wri as (9) M A A Whr A ad giv by ar M arics A A A A A A A M M () M M M () VI SHIFED AGUERRE WAVEES Shiig h agurr ls by usig polyoials h quaio (5) will bco * i ohrwis () Whr or M! M should o i dalig wih agurr ls h wigh ucio ad raslad w w hav o dilad a ucio ovr ca b xpadd as A whr A did (3) I iii sris i (3) is rucad h i ca b wri as M A A (4) o ar M whr A ad arics giv by A A A A MA A M (5) A A M M ISSN: 3-5373 hp://wwwijjouralorg Pag 3
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 (6) hor : M M A ucio N wih liid scod drivaiv say b liid scod say N ca b widd as a uliid aggrga o agurr ls ad h sris covrgs uiorly o ls) or = ad M= by diriaio quaio () 4 3 4 4 4 Ad 5 44 6 44 45 h Eploy Marix o Drivaiv or agurr Wavl ha is N A A N 3 4 (7) 4 4 Proo o h hor: D ag() w hav A w d ad 4 4 4 A w d O Dag() 4 i by subsiuig O x i yilds ad O x A x x d (8) x A x x dx 3 x whvr x by copl igraio N () is A N 3 his copls h proo o h hor VII EMPOYMEN MARICES FOR AGUERRE WAVEE 7 h Eploy Marix o Drivaiv or agurr l: I his scio w us shid agurr ls i ploy arix o drivaiv or agurr ls Firs w cosruc 6 6 arix ad i dod by D ag() (drivaiv or agurr Fig (h Eploy Marix o Drivaiv or agurr Wavl) hor 3: b h agurr ls vcor did i quaio (6) h drivaiv o his vcor ca b xprssd as d D d ag whr ISSN: 3-5373 hp://wwwijjouralorg Pag 3
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 D is h ag M 4 D ag i which is is M M D ag arix ad i s h l is did as ollows (M ) ohrwis Proo: By usig shid agurr polyoials i h h l o vcor ca b wri as! ohrwis (3) M M ad (9) Diriaio quaio (9) d d! ohrwis ha is M M M So is agurr ls xpasio has h ollowig ro d d i a (3) his iplis ha h ploy D is arix ag() D ag a bloc arix as did i (7) orovr x d d d d M i i or M Cosquly h irs row o arix is zro Now by usig h ollowig quaio g whr (3) C l 3 l i Cli l l Subsiuig quaio (3) ad (3) i quaio (3) d Cli d! i (33) (34) Choos 5 d d M D quaio ag holds 7 Eploy Marix o Igraio or agurr Wavls: I his scio Igraio or agurr ls ar discussd For his ploy arix o igraio or agurr ls P Now id 6 6 arix P I quaio (5) or M=3 h six basis ucios ar giv by: 4 5 6 4 7 ad 4 4 5 6 4 3 (35) By igraio h abov six ucios ro o ad usig quaio (8) w obai d 4 3 d 5 8 4 4 3 7 d 4 4 ISSN: 3-5373 hp://wwwijjouralorg Pag 33
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 d 4 3 d 5 8 4 4 3 d 4 4 6 d P6 6 6 whr hus 6 By usig abov quaios h opraioal arix o igraio P is Eploy Marix o Igraio or agurr Wavls 4 3 8 4 4 3 7 4 4 P ag 4 3 8 4 4 3 4 4 A33 O3 3 P ag O3 3 A 33 Fig (h Eploy Marix o Igraio or agurr Wavl) 73 Powrs i rs o agurr Wavls: I his scio w will driv powrs i rs o agurr Wavls or M 3 Ad is h oralizd i will driv h powrs I rs o agurr Wavls which hlp o solv Probls M 3 3 basis Fucios ar giv by: I I arix or h powrs o ca b rwri as ollows Z W 5 4 5 3 6 8 8 ad 5 4 7 5 6 8 8 Whr Z W 5 Z ad W 5 Powrs i rs agurr Wavls: 4 5 3 6 8 8 6 6 8 7 5 6 8 8 Cosrucig h opraio arix o igraio ad diriaio ha ca b usd i solvig ay probls which ar illusrad i h ollowig xapls VIII APPICAION OF MARICES D ag AND P ag FOR SOVING CACUUS OF VARIAIONA PROBEMS I ordr o solv liar or oliar dirial quaios by usig h ploy arics D ag ad P ag so urical xapls illusra h Procdur [8] I ordr o solv liar or oliar ISSN: 3-5373 hp://wwwijjouralorg Pag 34
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 X Exac soluio Approxia soluio =M=3 Approxia soluio =M=4 87779 8 87 4 9458 945544 94544 6 7548 586756 786756 8 554898 5433957 5533957 99999999 dirial quaio by usig h abov opraioal arics so urical xapls illusra h procdur which will bgi wih x x h xac soluio U x x abl Shows h urical rsuls or his xapl wih 4 M 3 wih rror M 4 rror 5 ar copard wih xac soluio graphically i ig3 abl shows so urical rsuls or xapl () i y A i i i y A d y A P y i i i i i3 y A P y y y Exapl () Cosidr h ollowig variaio probl [8] i y y 8y Wih h boudary codiios d y y h corrspodig Eulr agrag quaio is y y h h xac soluio or his Probl is y wih h abov boudary codiios o solv his probl assuig y A P ha ag Fid y y o go y A D ag y A D ag will copl his xapl s [8] jus rplac old arics by ()s' arics o igraio ad drivaiv rachd h xac soluio Exapl (): Cosidr h ollowig Volrra igro dirial quaio (VIDE) [4] [5] Fig3 (3D o rsuls o xapl ()) U U x x x x U U d wih ISSN: 3-5373 hp://wwwijjouralorg Pag 35
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 wih boudary codiios o solv his probl by usig P ag will solv his probl ad rachd h xac soluio Fig5 (copar h rsuls wih xac soluio o xapl ()) IX CONCUSION h agurr ls opraioal arics o igraios wih h aid o spcral ad collocaio hods ar applid o solv ay probls h ls hod auhorizs h ood o vry as algorihs wh copard o h algorihs app roach usd (agurr Polyoials) Nurical rsuls wih coparisos ar giv o coir h rliabiliy o h proposd hod or solvig ay probls Exapl (3): Cosidr h ollowig variaio probl[8] y y y d Mi wih h boudary codiios y y h corrspodig Eulr agrag quaio is is y y h xac soluio or his probl y xac soluio Firs Chbyshv ls Scod ls Chbyshv agurr ls 57584 4883 85337 846984 85553 85337 7345 79744 757 7345 3 5984 94446 5933944 5984 4 349566 35374 3497464 349566 5 4434944 448784 4476674 4434944 6 54747 54374 5435565 54747 7 645496 64596 64483 645496 8 7557548 756358 7547746 7557548 9 8734869 874757 874554 8734869 999345 997456 ISSN: 3-5373 hp://wwwijjouralorg Pag 36
Iraioal Joural o Mahaics rds ad chology (IJM) Volu 49 Nubr Spbr 7 X REFERENCES AA Asa Nurical soluio o Opial probls usig w hird id Chbyshv Wavls Opraioal arix o igraio Eg & c Joural 3():45-56 4 AA Asa Dirc hod or Solvig Noliar Variaioal Probls by Usig Hri Wavls Baghdad Scic Joural Vol() 5 3 AA Asa A Algorih or h Ordr Igro-Dirial Equaios by Usig Hri Wavls Fucios Baghdad Scic Joural Vol(3) 4 4 AA Asa l collocaio hod or solvig igrodirial quaio IOSR Joural o Egirig Vol 5(3) PP -7 5 5 B Asady M Kajai AH Vchh A Hydari Solvig Scod Kid Igral Equaios wih Hybrid Fourir ad Blocpuls Fucios Appl Mah Copu 6 57-5 5 6 CF Ch CH Hsiao A Walsh Sris Dirc Mhod or Solvig Variaioal Probls J Frali Isi 3 65-8 975 7 RY Chag M Wag Shid gdr Dirc Mhod or Variaioal Probls J Opi hory Appl 39 99-37 983 8 IR Horg JH Chou Shid Chbyshv Sris Dirc Mhod or Solvig Variaioal Probls I J Sys Sci 6 855-86 985 9 C Hwag YP Shih Opial Corol o Dlay Syss via Bloc-puls Fucios J Opi hory Appl 45-985 C Hwag YP Shih agurr Sris Dirc Mhod or Variaioal Probls J Opi hory Appl 39 43-49 983 C Hwag YP Shih Soluio o Igral Equaios via agurr Polyoials J Copu Elc Egi 9 3-9 98 CH Hsiao Haar Wavl Dirc Mhod or Solvig Variaioal Probls Mah Copu Siul 64 569-585 4 3 H Jddu Dirc Soluio o Noliar Opial Corol Probls Usig Quasiliarizaio ad Chbyshv Polyoials J Frali Is 339 479-498 4 M Razzaghi Fourir Sris Dirc Mhod or Variaioal Probls I J Corol 48 887-895 988 5 S N Shihab AA Asaa Nurical Soluio o Calculus o Variaios by usig h Scod Chbyshv Wavls Eg & ch Joural 3(8) 39-39 6 S N Shihab AA Asa So Nw Rlaioships Bw h Drivaivs o Firs ad Scod Chbyshv Wavls Iraioal Joural o Egirig Busiss ad Erpris Applicaios (IJEBEA) () 64-68 7 S S Najb & M A Sarha Covrgc Aalysis o shid Fourh id Chbyshv Wavls IOSR Joural o Mahaics Volu () 54-58 4 8 S N Shihab AAAsa So Approxia Algoihs For Variaioal Probls(Boo) 9 SN Shihab AAAsa Solvig Opial Corol iar Syss by Usig Nw hird id Chbyshv Wavls Opraioal Marix o Drivaiv Baghdad Scic Joural Vol() 4 ISSN: 3-5373 hp://wwwijjouralorg Pag 37