CAS Wavelet Function Method for Solving Abel Equations with Error Analysis

Transcription

1 It J Res Id Eg Vol 6 No Iteratoal Joural of Research Idustral Egeerg wwwrejouralco CAS Wavelet Fucto ethod for Solvg Abel Equatos wth Error Aalyss E Fathzadeh R Ezzat K aleejad Departet of atheatcs Karaj Brach Islac Azad Uversty Karaj Ira A B S R A C I ths paper we use a coputatoal ethod based o CAS wavelets for solvg olear fractoal order Volterra tegral equatos We solve partcularly Abel equatos A operatoal atrx of fractoal order tegrato for CAS wavelets s used Bloc Pulse Fuctos BPFs ad collocato ethod are eployed to derve a geeral procedure for forg ths atrx he error aalyss of proposed uercal schee s studed theoretcally Fally coparso of uercal results wth exact soluto are show Keywords: Abel tegral equatos CAS wavelet fractoal order volterra tegral equatos operatoal atrx error aalyss Artcle hstory: Receved: 9 October 7 Accepted: 9 Deceber 7 Itroducto Fractoal calculus s a feld of appled atheatcs that deals wth dervatves ad tegrals of arbtrary orders It s also ow as geeralzed tegral ad dfferetal calculus of arbtrary order Fractoal dfferetal equatos are geeralzed fro classcal teger-order oes whch are obtaed by replacg teger-order dervatves by fractoal oes I recet years fractoal calculus ad dfferetal equatos have foud eorous applcatos atheatcs physcs chestry ad egeerg because of the fact that a realstc odelg of a physcal pheoeo havg depedece ot oly at the te stat but also o the prevous te hstory ca be successfully acheved by usg fractoal calculus ay authors have deostrated the applcatos of the fractoal calculus For exaples t has bee appled to odel the olear oscllato of earthquaes flud dyac traffc frequecy depedet dapg behavor of ay vscoelastc aterals cotuu ad statstcal echacs colored ose sold echacs ecoocs sgal processg ad cotrol theory [-] A large class of dyacal systes appearg throughout the feld of egeerg ad appled atheatcs s descrbed by fractoal dfferetal equatos For reaso t s deed requred a relable ad effcet techques for the Correspodg author E-al: ezat@auacr DOI:/rej7387

2 3 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss soluto of fractoal dfferetal equatos he ost frequetly used ethods are Walsh fuctos [6] Laguerre polyoals [7] Fourer seres [8] Laplace trasfor ethod [9]the Jacob polyoals [] the Haar wavelets [-3] Legedre wavelets [4-6] Euler wavelet [7] ad the Chebyshev wavelets [8-] have bee developed to solve the fractoal dfferetal equatos Kroecer operatoal atrces have bee developed by Klca for soe applcatos of fractoal calculus [] he operatoal atrx of fractoal ad teger dervatves has bee detered for soe types of orthogoal polyoals such as flatlet oblque ultwavelets [ 3] B-sple cardal fuctos [4] Legedre polyoals Chebyshev polyoals ad CAS wavelets [8 9] Furtherore the CAS wavelets have bee used to approxate the soluto of Volterra tegral equatos of the secod d [3] tegrodfferetal equatos [3] ad optal cotrol systes by te-depedet coeffcets [3 33] he structure of the paper s as follows I Secto we troduce soe basc atheatcal preleres that we eed to costruct our ethod We recall the basc deftos fro bloc pulse fuctos ad fractoal calculus I Secto 3 we recall defto of CAS wavelet he a purpose of ths artcle s to use of a operatoal atx of fractoal tegrato to reduce the soluto of a fractoal order Volterra tegral equatos to the soluto of a olear algebrac equatos by usg CAS wavelets he Secto we dscuss o the covergece of the CAS wavelets ad the error aalyss for the preseted ethod ad Secto 6 we apply our ethod to solvg Abel tegral equatos Fally a cocluso of uercal results s preseted Preleres I ths secto we recall the basc deftos fro fractoal calculus ad soe propertes of tegral calculus whch we shall apply to forulate our approach he Rea-Louvlle fractoal tegral operator space L [ b] s gve by [34]: I of order o the usual Lebesgue x x s u s ds > I u u he Rea-Louvlle fractoal dervatve of order > s orally defed as: d D u I u < dx where s a teger he fractoal dervatve of order > the Caputo sese s gve by [34]:

3 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg x 3 D* u x s u s ds < where s a teger > ad u L [ b] he useful relato betwee the Rea- Louvlle operator ad Caputo operator s gve by the followg expresso: x I D u u u < *! where s a teger > ad u L [ b] A -set of Bloc Pulse Fuctos BPFs the rego of [ s defed as follows: h t < h b OW where wth postve teger values for ad h ad are arbtrary postve tegers here are soe propertes for BPFs eg dsjotess orthogoalty ad copleteess 4 he set of BPFs ay be wrtte as a -vector B: where t [ B [ b b ] 6 A fucto f L [ ay be expaded by the BPFs as: f f b F B B F where B s gve by 6 ad F s a -vector gve by: F [ f f ] 8 the bloc-pulse coeffcets f are obtaed as: h f f h 9 h he tegrato of the vector B defed 6 ay be obtaed as: t B d;b Where s called operatoal atrx of tegrato whch ca be represeted by: h Klca ad Al Zhour see [3] have gve the Bloc Pulse operatoal atrx of fractoal tegrato where F as follows: I B F B 7

4 33 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss F ad CAS Wavelets I ths secto frst we gve soe ecessary deftos ad atheatcal prelares of CAS wavelets he fucto approxato va CAS wavelets ad bloc pulse fuctos s troduced Wavelets cosst of a faly of fuctos costructed fro dlato ad traslato of a sgle fucto called the other wavelet Whe the dlato paraeter a ad the traslato paraeter b vary cotuously we have the followg faly of cotuous wavelets [ 6]: t b 4 a b a a br a a If we restrct the paraeters a ad b to dscrete values a a b b a a > > b where ad are postve tegers the we have the followg faly of dscrete wavelets: a at b where for a wavelet bass for L I partcular whe a the t b fors a orthooral bass [ 7] he CAS wavelets t have four arguets; s ay oegatve teger s ay teger ad t s the oralzed te he orthooral CAS wavelets are defed o the terval [ by [7 8]: / CAS t t < 6 OW where CAS cost st 7 ad {} It s clear that CAS wavelets have copact support e Supp { t : } [ ] t

5 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg We troduce the followg useful otato correspodg to CAS wavelets as follows: / CAS t < OW Ht For the CAS wavelets have the followg for: / / t < B OW Where terval [ { B } are a bass set that are called the Bloc Pulse Fuctos BPFs over the 3 Fucto Approxato wth CAS Wavelets he set of CAS wavelets fors a orthooral bass for L [ hs ples that ay fucto f defed over [ ca be expaded as: where c f c c Z C < f > f t ad < f g> s the er product of the fucto f ad g C ad are vectors gve by: C [ c [ c 8 9 c c c c c c ] Notato Fro ow we defe such that {} 3 Operatoal atrx of Fractoal Itegrato wth Hybrd Fucto ] Eq 7 ples that CAS wavelets ca be also expaded to a f b By usg the propertes of CAS wavelets ad Eq 9 we have: / / f / {s / s -ter BPFs as: 3

6 3 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss } cos cos } { / for ad otherwse f herefor we get [ / / t ] t B Where ad herefore: t B t 4 where Dag ad for s a atrx whch s troduces as: * where / / / / Ht * s pot wse product Notato by usg the propertes of CAS wavelets we ow that For exaple for the CAS operatoal atrx to BPFs ca be expressed as

7 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg Let: where atrx I P P s called the CAS wavelet operatoal atrx of fractoal tegrato Usg Eqs 4 ad we have: I I B I B F B 6 By Eqs ad 6 we get: P ; F B herefore the CAS wavelet operatoal atrx of fractoal tegrato [8]: P s gve by see P F 7 For exaple for the CAS operatoal atrx to BPFs ca be expressed as 3/4 P Ipleetato of the ethod Cosder the geeralzed Abel tegral equato of the frst ad secod ds respectvely as [36]: x y f < < x 8 ad x y y f < < x 9

8 37 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss where f ad y are dfferetable fuctos Here we cosder Abel tegral equato as a fractoal tegral equato ad we use fractoal calculus propertes for solvg these sgular tegral equatos By usg oe ca wrte: x y 3 I y x By replacg 3 equatos 8 ad 9 we have: f I y 3 y f I y 3 Let: y U 33 by substtutg 33 equatos 3 ad 3 we obta: f I U 34 3 U f I U Fally by usg equatos 34 ad 3 we obta the fractoal for of Abel tegral equato of the frst ad secod d respectvely as follows: f U P 36 f U U P 37 Now by collocatg the equatos 36 ad 37 at collocato pots } { x where x are the CAS wavelet pots of degree we obta the followg syste of algebrac equatos: f x U P x 38 f x U U P x 39 Clearly by solvg ths syste ad deterg approxate soluto of the equatos 36 ad 37 as y U Error Aalyss t U [ u u u ] we obta the I ths secto we provde a theoretcal error ad covergece aalyss of the proposed ethod for solvg Abel tegral equatos At frst we dcate that the CAS wavelet expaso of a fucto f wth bouded secod dervatve coverges uforly to f But before that for ease referece we preset the followg theore: heore If the CAS wavelet expaso of a cotuous fucto f coverges uforly the the CAS wavelet expaso coverges to the fucto f

9 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg Proof Let: g c where c < f > ultplyg both sdes of 4 by whch p ad q are fxed ad the tegratg ter wse justfed by uforly covergece o [] we have: 4 g dx c dx pq c pq dx c hus < g > c for ad Cosequetly f g have sae Fourer expasos wth the CAS wavelet bass ad therefore f g ; x [37] heore see [38] A fucto f L [] wth bouded secod dervatve say f ca be expaded as a fte su of the CAS wavelets ad the seres coverges uforly to f that s f c Z Furthereore we have: [] 43 P f f x Proof Fro Eq t follows that : 44 < > c f x x f x x dx f CAS x dx By substtutg x t Eq44 yelds: t c f CAS t s cos f d t st cost t f d f s cos 3 t CAS f d 3 t t f CAS f CAS 3 hus we get: pq pq

10 39 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss t c f CAS t f CAS N CAS t fro orthooralty of CAS wavelets we ow that have: Hece the seres c 4 c Z c CAS Sce s absolutely coverget O the other had we have: Z c Z c Accordgly utlzg heore the seres c uforly oreover we coclude that: P f f c 4 Now fro 3 we obta: P hs copletes the proof f f c we 3 4 coverges to f 6 Now we proceed by dscussg the covergece of the preseted ethod heore 3 Suppose that f L [] wth bouded secod dervatve f ad P f the trucated expato of CAS wavelet for f by he we have the error boud as follows: e P f f E E 7 where 8 e e dx E Proof Proof easly results fro theore Now by creasg e whe the error fucto approaches to zero If e x s suffcetly large eough the the error decreases

11 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg Nuercal Exaples o show the effcecy of the proposed ethod we wll apply our ethod to obta the approxate soluto of the followg exaples All of the coputatos have bee perfored usg ALAB 78 Exaple Cosder the frst d Abel tegral equato of the for: x y x 4 x s 4 he exact soluto s y x 4 Let y U l ad X for l are collocato pots Now fro Equatos 4 ad we have: x y 6 I y 4 x I U U P B So we obta algebrac equatos for of Exaple as follows: 63 X U P By solvg ths syste ad deterg U we obta the approxate soluto of equato 9 as y U Plot of error that s resulted by ethod for s llustrated Fgure 9 Exaple Cosder the secod d Abel tegral equato of the for: y x he exact soluto s 6 x x y x y x 64 Let y U l ad X for x y I y x I U l By Equatos 4 ad we have: 6 66

12 36 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss U P B he we obta algebrac equatos for of exaple as follows: 6 U X X U P Fgure shows error Plot for Exaple usg preseted ethod wth Exaple 3 Now cosder aother secod d Abel tegral equato of the for: x y y x x x he exact soluto s y e erfc Where x erfc e t 7 Fgure 3 shows Error plot for Exaple 3 wth Fgure Error plot for Exaple wth

13 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg Fgure Error plot for Exaple wth Fgure 3 Error plot for Exaple 3 wth 8 7 Coclusos I ths paper we have preseted a uercal schee for solvg Abel tegral equatos of the frst ad secod ds he ethod whch s eployed s based o the CAS wavelet By cosderg Abel tegral equatos of the frst ad secod ds as a fractoal tegral equato we use fractoal calculus propertes for solvg these sgular tegral equatos Error aalyss s provded for the ew ethod Fgure -3 show the error of preseted ethod ad the exact

14 363 CAS wavelet fucto ethod for solvg Abel equatos wth error aalyss soluto of the Abel equatos proposed exaples -3 respectvely he obtaed results shows that the used techque ca solve the fractoal order Volterra tegral equatos effectvely Refereces [] He J H 998 Nolear oscllato wth fractoal dervatve ad ts applcatos Iteratoal Proceedgs of teratoal coferece o vbratg egeerg 88-9 [] aard F Fractoal calculus: soe basc probles cotuu ad statstcal echacs arxv preprt arxv:863 [3] Rossh Y A & Shtova V 997 Applcatos of fractoal calculus to dyac probles of lear ad olear hereary echacs of solds Appled echacs revews -67 [4] Balle R 996 Log eory processes ad fractoal tegrato ecooetrcs Joural of ecooetrcs 73-9 [] Evas R Katugapola U N & Edwards D A 7 Applcatos of fractoal calculus solvg Abel-type tegral equatos: Surface volue reacto proble Coputers & atheatcs wth applcatos [6] Che C F & Hsao C H 97 Desg of pecewse costat gas for optal cotrol va Walsh fuctos IEEE trasactos o autoatc cotrol [7] [7] D S Shh F C Kug C Chao 986 Laguerre seres approach to the aalyss of a lear cotrol syste corporato observers Iteratoal Joural of Cotrol vol pp3-8 [8] Parasevopoulos P N Spars P D & ouroutsos S G 98 he Fourer seres operatoal atrx of tegrato Iteratoal joural of systes scece [9] Podluby I 997 he Laplace trasfor ethod for lear dfferetal equatos of the fractoal order arxv preprt fuct-a/97 [] Sadr K A A & Cheg C 8 A ew operatoal ethod to solve Abel s ad geeralzed Abel s tegral equatos Appled atheatcs ad coputato [] L Y & Zhao W Haar wavelet operatoal atrx of fractoal order tegrato ad ts applcatos solvg the fractoal order dfferetal equatos Appled atheatcs ad coputato [] Babaaghae A & aleejad K 7 Nuercal solutos of olear two-desoal partal Volterra tegro-dfferetal equatos by Haar wavelet Joural of coputatoal ad appled atheatcs [3] Fathzadeh E Ezzat R & aleejad K 7 Hybrd ratoal haar wavelet ad bloc pulse fuctos ethod for solvg populato growth odel ad abel tegral equatos atheatcal probles egeerg 7 [4] Ur Reha & Kha R A he Legedre wavelet ethod for solvg fractoal dfferetal equatos Coucatos olear scece ad uercal sulato [] Heydar H Hooshadasl R Gha F & Feredou F 3 wodesoal Legedre wavelets for solvg fractoal Posso equato wth Drchlet boudary coos Egeerg aalyss wth boudary eleets [6] Heydar H Hooshadasl R Catta C & L 3 Legedre wavelets ethod for solvg fractoal populato growth odel a closed syste atheatcal probles egeerg [7] Wag Y & Zhu L 7 Solvg olear Volterra tegro-dfferetal equatos of fractoal order by usg Euler wavelet ethod Advaces dfferece equatos 7 7

15 E Fathzadeh R Ezzat K aleejad/ It J Res Id Eg [8] Wag Y & Zhu L 7 Solvg olear Volterra tegro-dfferetal equatos of fractoal order by usg Euler wavelet ethod Advaces dfferece equatos 7 7 [9] Wag Y & Fa Q he secod d Chebyshev wavelet ethod for solvg fractoal dfferetal equatos Appled atheatcs ad coputato [] Fathzadeh E Ezzat R & aleejad K 7 he costructo of operatoal atrx of fractoal tegrato usg the fractoal chebyshev polyoals Iteratoal joural of appled ad coputatoal atheatcs -3 [] Klca A & Al Zhour Z A A 7 Kroecer operatoal atrces for fractoal calculus ad soe applcatos Appled atheatcs ad coputato 87-6 [] Dara R A Adb H & Laesta Nuercal soluto of tegro-dfferetal equatos usg flatlet oblque ultwavelets Dyacs of cotuous dscrete & pulsve systes seres A 7-74 [3] Dara A Adb H Agarwal R P & Saadat R 8 Flatlet oblque ultwavelet for solvg tegro-dfferetal equatos Dyacs of cotuous dscrete ad pulsve systes seres A: ateatcal aalyss [4] Laesta Dehgha & Iradoust-Pach S he costructo of operatoal atrx of fractoal dervatves usg B-sple fuctos Coucatos olear scece ad uercal sulato [] Daubeches I 99 e lectures o wavelets Socety for dustral ad appled atheatcs [6] Keert F 3 Wavelets ad ultwavelets CRC Press [7] Yousef S & Bafate A 6 Nuercal soluto of Fredhol tegral equatos by usg CAS wavelets Appled atheatcs ad coputato [8] Saeed H oghada ollahasa N & Chuev G N A CAS wavelet ethod for solvg olear Fredhol tegro-dfferetal equatos of fractoal order Coucatos olear scece ad uercal sulato [9] aleejad K & Ebrahzadeh A 4 Optal cotrol of volterra tegro-dfferetal systes based o Legedre wavelets ad collocato ethod World acadey of scece egeerg ad techology teratoal joural of atheatcal coputatoal physcal electrcal ad coputer egeerg [3] Adb H & Assar P 9 Usg CAS wavelets for uercal soluto of Volterra tegral equatos of the secod d Dyacs of cotuous dscrete ad pulsve systes seres A: atheatcal aalyss [3] Dafu H & Xufeg S 7 Nuercal soluto of tegro-dfferetal equatos by usg CAS wavelet operatoal atrx of tegrato Appled atheatcs ad coputato [3] Abualrub Sade I & Abuhaled 9 Optal cotrol systes by te-depedet coeffcets usg cas wavelets Joural of appled atheatcs 9 [33] Y & Huag J CAS wavelet ethod for solvg the fractoal tegro-dfferetal equato wth a wealy sgular erel Iteratoal joural of coputer atheatcs [34] Podluby I 998 Fractoal dfferetal equatos: a troducto to fractoal dervatves fractoal dfferetal equatos to ethods of ther soluto ad soe of ther applcatos Vol 98 Acadec press [3] Klca A & Al Zhour Z A A 7 Kroecer operatoal atrces for fractoal calculus ad soe applcatos Appled atheatcs ad coputato 87-6 [36] Wazwaz A 997 A frst course tegral equatos World Scetfc Publshg [37] Follad G B 3 Real aalyss: oder techques ad ther applcatos Joh Wley & Sos [38] Barzar A Assar P & ehrpouya A Applcato of the cas wavelet solvg fredhol-haerste tegral equatos of the secod d wth error aalyss World appled sceces joural /doswasj8467