NATURAL TRANSFORM AND SOLUTION OF INTEGRAL EQUATIONS FOR DISTRIBUTION SPACES

Transcription

1 Americ J o Mhemic d Sciece Vol 3 o - Jry 4 Copyrih Mid Reder Plicio ISS o: 5-3 ATURAL TRASFORM AD SOLUTIO OF ITERAL EQUATIOS FOR DISTRIBUTIO SPACES Deh Looker d P Berji Deprme o Mhemic Fcly o Sciece J V Uiveriy Jodhpr IDIA E-mil : dehp@yhoocom d erjipk@yhoocom Arc I hi pper ierl eqio ch Volerr covolio ype o ir d ecod ype d Ael ierl eqio re olved i rl rorm d rher he olio o oied re deied o ceri diriio pce eyword : rl rorm Forier rorm Lplce rorm diriio pce ierl eqio Mhemic Sjec Cliicio : 44A 44A35 44A99 45E 46F 46F 46F99 Irodcio To olve diereil d ierl eqio everl ierl rorm ch Forier Lplce Smd d my more re ed ] A ew ierl rorm he - rorm i died y h d h 9] d heir properie d pplicio re decried The ivere o he rorm d ddiiol properie o he me re ive y Belcem e l 3 4] Diriiol rl rorm i deied y Looker d Berji 3] Thi ecio o he pper del wih ic ermioloie d properie o he rl rorm I Secio he rl rorm i ed o oi olio o ierl eqio Secio 3 del wih he pplicio o he olio o oied i Secio o ceri diriio pce The rel cio d or i eciowie coio o epoeil order d deied i he e where M A y / j j ; A { : M Me i co o iie mer d The rl rorm R o he cio where ] R R e e my e iie or iiie or ll d d re ime vrile d i he reqecy vrile The dicree orm o rl rorm i 3 p Eq 4] } i ive y 3] 65

2 Deh Looker d P Berji! 3 ] R The ivere rl rorm i deied y 3 p Eq 7] d 4] R ] i ci c i e R d 4 The rl rorm i derived rom he Forier ierl ee 3] where we oice lo dliy relio ewee rl-lplce d rl-smd rorm oeher wih oher properie We meio cople o properie o he rl rorm 3 4] rl rorm o derivive : The derivive o o d wih repec o re repecively deied y wih repec o ] R R d h order derivive 5 k ] R R 6 k k k Covolio Theorem : I F d re rl rorm o he cio deied i e A he he covolio i ive y d ] F 7 3 Whe he Dirc del cio d whe or 4 Lieriy propery : I! he rl rorm ecome ] R 8 ; he rl rorm i ] re y co d d re cio he 9 ] F Ierl eqio occr i my ield o mechic d mhemicl phyic They lo rie repreeio orml or he olio o diereil eqio Ideed diereil eqio c e replced y ierl eqio which icorpore i odry codiio Ierl eqio lo orm oe o he mo el ool i my rche o pre lyi ee 6 7 ] A ierl eqio i which kow cio pper der oe or more ierl i The eqio d d where he cio i he kow cio while ll he oher cio re kow re ierl 66

3 ATURAL TRASFORM AD SOLUTIO eqio Thee cio my e comple vled cio o rel vrile d A ierl eqio d i clled lier i oly lier operiol perormed i i po he kow cio I c hee c e wrie L ] 3 where L i pproprie ierl operor The mo eerl ype o lier ierl eqio i h d 4 where he pper limi my e eiher vrile or ied The cio where i o e deermied; he kerel The pecil ce o eqio 4 re h d i o zero rel or comple prmeer The cio Fredholm ierl eqio : I hi he pper limi o ierl i Whe h he 4 will e d i kow Fredhlom ierl eqio o he ir kid ii Whe h he 4 ecome i ied re kow cio 5 d 6 i kow Fredhlom ierl eqio o he ecod kid iii Whe i 6 clled homoeo Fredhlom ierl eqio d 7 i clled Volerr Eqio : Volerr eqio o he ir homoeo d ecod kid re deied preciely epec h i he vrile pper limi o ierio 3 Covolio Ierl Eqio : The kerel i coidered cio o he dierece where k i ceri cio o oe vrile The ierl eqio i clled Fredlom eqio o he covolio ype 4 Ael Ierl Eqio i ive d i olio i ollow k d i d d 8 d 9 Solio o Ierl Eqio d rl Trorm I hi ecio he rl rorm i ivoked o oi olio o ome ierl eqio d oher pplicio c e ee i 6 7 ] Coider he Volerr ierl eqio o ir kid wih covolio ype kerel 67

4 Deh Looker d P Berji k d k where d i 7 we oi deped oly o he dierece Applyi he rl rorm o oh he ide F F By i iverio rl rorm i we oi he olio o F 3 Coider he Volerr ierl eqio o ecod kid wih covolio ype kerel k d 4 O pplyi he rl rorm o oh he ide d i covolio orml 7 4 ive F F 5 d he ivere rl rorm ive F 6 which i he reqired olio o 4 3 Coider he Ael ierl eqio i he orm d 7 which c e wrie where H d 8 ide d i he covolio 7 i 8 we hve Whe ] ] i Heviide i ep cio Applyi he rl rorm o oh he ] ] ] 9 Pi hi vle i 9 we e d pi 68 i hi or i 9 we oi ] ] 3 F ]

5 ATURAL TRASFORM AD SOLUTIO 69 i ; F i F i 9] ; i i7] ; ] i F ] i i d ] i ] 3 where d From 5 ] ] R which whe ivoked i 3 we e ] i ] d d i ] Thereore he complee olio o 7 i ive y i d d d 3 I wh ollow re ome illrive emple which ppor he e o he rl rorm o olve ierl eqio Emple : Fid he cio which iie he eqio d i 33 Solio : Ivoki he rl rorm o oh he ide d 7 i 33 we oi 4

6 Deh Looker d P Berji 3 4 Tki ivere rl rorm we e 3 3! 34 which i he reqired olio Emple : Solve d co 35 Solio : Ui he rl rorm o oh he ide we hve Tki ivere rl rorm o he ove eqio we e which i he reqired olio co 36 3 Diriiol Solio o Ierl Eqio I hi ecio we epli h he olio oied i Secio o he ierl eqio c e deied o he diriio pce I order o deie ierl eqio o diriio pce we eed o peciy he pce or pce o eerlized cio The we eed o ive ierpreio o he eqio i erm o operor deied i h pce o diriio Thi ierpreio hold e ch h whe pplied o ordiry cio ierl eqio c e recovered Oe i he pce which i kow mied diriio pce 4 c 6]] h c e ideiied wih he pce o diriio R whoe ppor i Aoher diriio pce i which c e ideiied wih he pce S R empered diriio 43 pce whoe ppor i coied i The ierpreio o ierl eqio c e chieved y i he cocep o covolio o diriio I oh d hve ppor oded o he le he v i lwy deied Aclly i pp d pp v he pp v Th he covolio c e coidered v ilier operio I d : v re loclly ierle cio he we hve 4 v v d 37 Whe d v we hve v Th he covolio wih 4 operor o he pce which i ive y v deie 7

7 ATURAL TRASFORM AD SOLUTIO v v v d 38 where d re loclly ierle cio The ierl eqio d i olio c e ierpreed i he diriiol ee Moreover we deie he ierl rorm o diriiol pce d rher i he diriiol ierl rorm o oi he olio o ierl eqio c e ierpreed diriiol ee For hi oe my reer 3] The rl rorm R o he cio 7 c e wrie 3] ] R e 39 Few properie ch covolio Prevl eqio o rl rorm o diriio pce i lo deied Ui he cocep o covolio o diriio or ierl eqio d diriiol rl rorm deied ove we c coclde h he olio oied i Secio or he ierl eqio i rl rorm c e coidered i he diriio ee The kow d he kow cio o he ierl eqio olved i diriiol rl rorm re chrcerized or he pce o diriio ch 4 43 d y oher diriio pce I imilr mer oher orm o ierl eqio c e olved i rl rorm d c lo e ivoked or diriio pce Coclio : The ierl rorm mely rl rorm revied rorm i ed o olve Volerr covolio ype o ir kid d ecod ype d Ael ierl eqio Few illrio re propoed o clriy he lyi icorpored i he pree pper The diriio pce re deied or he olio o ierl eqio oied i clicl d diriiol rl rorm Ackowledeme Thi work i prilly ppored y he UC Po Docorl Fellowhip or Wome Idi o F5-34/ SA- II cioed o he ir hor DL d he DST-USERS Scheme o HR/UR/49/ cioed o he ecod hor PB Reerece ] Air M A Smd rorm d he olio o ierl eqio o covolio ype Ieriol Jorl o Mhemicl Edcio i Sciece d Techoloy 3 6 pp 96-9 ] Belcem F B M 7 Applicio o he Smd rorm o ideiie periodic prolic eqio Proc 6h I Coerece o Mhemicl Prolem d Aeropce Sciece pp 5-6 3] Belcem F B M d Silmr R Theory o rl Trorm Mh E Sci Aeropce MESA 3 pp ] Elzki T M d Elzki S M O he olio o Iero-diereil eqio yem i Elzki rorm lol J Mh Sci 3 pp 3-3 5] Elzki T M d Ezki S M Solio o iero-diereil eqio yem y i Elzki rorm lol J Mh Sci 3 pp - 6] Erd R d wl R P Silr Ierl Eqio Birkhä er Boo Bel Berli 7] wl R P 97 Lier Ierl Eqio Acdemic Pre ew York Lodo 8] eh Q D d Belcem F B M Applicio o he Smd rorm o rciol diereil eqio olier Sdie 8 pp -5 9] h Zr H d h Wqr A 8 - Trorm - Properie d Applicio UST J E Sci 7-33 ] Looker Deh d Berji P O he olio o diriiol Ael ierl eqio y diriiol Smd rorm Ier J Mh Mh Sci pp -8 Aricle ID 4858 ] Looker Deh d Berji P Frciol Ierl d Derivive or Smd rorm o diriio pce Appl Appl Mh Ier J 7 pp 88-

8 Deh Looker d P Berji ] Looker Deh d Berji P O he diriiol Ael ierl eqio or diriiol Elzki rorm J Idi Mh Soc To Apper 3] Looker Deh d Berji P 3 rl rorm or diriio d Boehmi pce Mh E Sci Aeropce To Apper 4] Silmr R d Belcem F B M Applicio o he rl rorm o Mwell' eqio Pro Elecromeic Reerch Sympoim Proc Szho Chi pp