ISSN: [Ramalakshmi* et al., 6(1): January, 2017] Impact Factor: 4.116

Transcription

1 IJESRT ITERATIOAL JOURAL OF EGIEERIG SCIECES & RESEARCH TECHOLOGY AALYSE THE OPTIMAL ERROR AD SMOOTHESS VALUE OF CHEBYSHEV WAVELET V.Raalaksh, B.Raesh Kua, T.Balasubaanan (Depatent o Matheatcs, atonal Engneeng College,Kovlpatt, Taladu, Inda) (Depatent o Matheatcs, See Sowdabka College o Engneeng, Auppukotta, Taladu, Inda) (Depatent o Matheatcs, Kaaaj College, Thoothukud, Taladu, Inda) DOI: 0.58/zenodo.7 ABSTRACT In ths Chapte, we obtaned Wavelet eo analss o optal contol n nonlnea deental equaton b appl the Chebshev Wavelet and to analze the Resdual o Wavelet Soothness. Ths ethod tansos the nonlnea sstes o deental equaton to nonlnea sstes o statstcal equatons. The convegences o the nuecal ethod ae gven and the applcablt s llustated wth soe eaples. KEYWORDS: Wavelet, Chebshev Polnoal, Shted Chebshev Polnoal. ITRODUCTIO The wavelet ethods ae ve eectve and ecent tool o solvng atheatcal pobles. It has an useul popetes. Wavelets can be used o algebac opeatons n the sste o equatons obtaned whch lead to bette condton nube o the esultng sste. In wavelet based ethods, thee ae two potant was o povng the appoaton o the solutons: Inceasng the ode o the wavelet al and the nceasng the esoluton level o the wavelet. Thee s a gowng nteest n usng vaous wavelets to stud pobles, o geate coputatonal coplet. Aong the wavelet tanso ales the Haa, Legende and Chebshev wavelets deseve uch attenton. The basc dea o Chebshev wavelet ethod (CWM) s to convet the deental equatons to a sste o algebac equatons b the opeatonal atces o ntegal o devatve. The an advantage o these ethods les n the accuac o a gven nube o unknowns. In contast, nte deence and nte-eleent ethods eld onl algebac convegence ates. In [9], a ethod o nuecal soluton o nonlnea equatons utlzng postve dente unctons s poposed b Alpanah and Dehghan. [7] Intoduced a nuecal ethod o solvng nonlnea. In [], a Chebshev nte deence ethod has been poposed n ode to solve lnea and nonlnea second-ode Fedhol ntego-deental equatons. Fst te, Chebshev wavelet as anothe polnoal wavelet was dscussed n [, ]. Usng o chebshev wavelet to solve ntegal equatons contnued n soe othe papes [9, 0]. Man eseaches [,,,, ] deved the popetes o chebshev polnoal n wavelet undaentals. The an pupose o ths atcle usng the popetes o Chebshev wavelet o solvng Eq. () and Eq. () and obtan the eo optal value o Wavelet uncton and to analze the esdual analze o DE. The nteval o the sste les between 0 < t <. Shted Chebshev polnoals The ange [0, ] s useul to the ange o the nteval [-, ]. Soete the vaable s ndependent n [0, ] that te vaable tanso to s n [-, ]. The tansoaton dened as s= - o =0.5(s+). Ths tansoaton shted to the chebshev polnoal T ( n ) o nth degee s gven b T ( s) T ( n n n. Thus we have the polnoal s T ( n s ( ) T0 ( and T ae ntal condtons T In geneal ecuence elaton o the ) T n, hee ), T, T n = (-) ( ) T n - [70]

2 Popetes o the Shted chebshev polnoal: [5,8] ae deved the ecuence oula o the chebshev polnoals s n Tn, ( ) Tn, Tn, ( ),,,... n Hee s the hghest degee o polnoal T (! ( ( )!()! n n, ( 0 ), The othogonal condton o the ntegal equaton s whee hee a0 and a n and hk a / The k th devatve o the th ode shted polnoal dened as: Poposed Algoth: Consde the onlnea Deental Equaton: ( P) T (0) ) n, ( and n Tn, Tn, j w( ) 0 p (,, ) () Whee p =, n and a < < b j Addtonal condton o the DE s ( p) p T T n, ( n). d hk Whee j w( ),, n (! n, ( s k ( )! ( k ( k ( Ck ( ) Dk k n k k () whee s an analtc uncton, C and D ae constant polnoals. In ode to solvng equaton () b usng copason ethod, we obtaned the appoate () dened as: b,,..., > 0, s the unknown shted chebshev coecents. b (), whee 0 k p Fo () and () we have P (, b,... b ( T ) ) () j 0 k Substtutng equaton nube () n equaton (), we obtaned the k th polnoal equaton as ollows: p ( k k k ) ( ) Ck b T ( ) Dk b T ( ) (5) k k Fo Equaton ( and 5), we povde the ( nonlnea nuecal equatons. Solvng equaton (5) we nd the unknown coecents o b. Theeoe, we usng ORIGI 0.6 n equaton (), easl obtan the appoate value o vaous soluton and analze the equenc o the chebshev wavelet. Poposed Algoth o Tuncated Eo Analss: Assue that () s a uncton on [0, ], () s the ntepolatng polnoal to at, =,,.n ae the chebshev ntepolatng ponts, then we ve ( ) 0 ( (! k ( ) [7]

3 Fo [, 0, 6], we deved the th ( polnoal as ollows: ( ) --- (6) Theoe: Consde the uncton () s contnuous and satsng the condton on [-, ], then the sees epanson o chebshev polnoal s unol convegent. Poo: Let the uncton () ples the sae condton o ( ) (cos ). ( ) ( ) = (cos( ) (cos ) ( (cos( ) cos( )) ( ) Snce, t s easl shown that (cos( ) (cos ), ( ) s an nceasng uncton o Theoe: Suppose the uncton (+) tes contnuousl deental on [0,] and the eact soluton o the chebshev polnoal s Poo: Let b. The appoate soluton s obtaned () 0 Theeoe, we can wte the oula 8 s th ode degee o polnoal and () s the appoate soluton o ( ) ( ) ( ) (7) Usng equaton (6) n (7), we obtan ( ) ( ) ( ) ( ) d 0 0 (! / ( 0 0 ( b b ) T d d / / ( (! ( b b ) d / (9) (8) d Theeoe, addng the equaton 8+9, we obtaned the appoate soluton, Fo equaton (), the uncton (), =0,,., Usng the popetes o Chebshev wavelets the tuncated eo estated as ( P) p (,, ( p) Theeoe E p (,, ) uecal Eaple: j j ) (0) '' ' Consde the LDE ) cos 0 and ( '' ' 0, the bounda condton s ( 0) ( ) (0) () 0 / (). [7]

4 sn 5 6 cos (5 )cos( and Whee ) 6 sn( ) ( 5) sn ( 5) cos The appoate solutons ae b T 5 and ( 5)sn( ), Usng algoth, we have = ( ) b T ( ) b T ( ) bt ( ) 0 0 q () b T ( ) b T ( ) b T ( ) ( ) () Whee q= 0,,, ae oots o the shted Chebshev polnoal (0) ( q b 0 q0 q T (), () b () (0) ( q b 0 q (5), () b (6) q0 q0, necessa condton as ollows: Fo equaton ( to6), we obtaned the 8 nonlnea equatons wth the eght unknown coecents. Solvng the above equaton and substtutng the equaton (), we get the estated soluton o = 5, (7) Usng ou poposed algoth and teatve ethod (IM), the esult on eact soluton o X ( ) (=) ) Table: Eo Value Table ( (=) () IM Eo e o =., [7]

5 Fg () Analze the Wavelet Soothness o () e.0j 0.5j.0j 0.j 5.0j C WT Sooth Y j -5.0j -0.5j -.0j -.0j Fg () Analze the Wavelet Denosed o () e.0j 0.5j.0j 0.j 5.0j C B Denosed Y -0.j -5.0j -0.5j -.0j -.0j Fg () Analze the Wavelet soothness and Denosed o () e.0j 0.5j.0j 0.j 5.0j C B Denosed Y Soothed Y -0.j -5.0j -0.5j -.0j -.0j [7]

6 WT Sooth Table.: Standad Eo Value o VALUE SE Intecept IM Table : DF o : WT Sooth ube o Ponts 0 ube o Ponts 7 Degees o Feedo 0.09 Degees o Feedo Intecept Intecept Table : Sua o the statstcs: IM IM Statstcs Value Eo Value Eo Value Eo Adj. R- Squae Table 5: Usng Anova Value: DF Su o Squaes Mean Squae F Value Pob>F WT Sooth Model E-8 Model Eo Table 6: Adjusted Resdual o Polnoal Value: Value SE WT Sooth WT Sooth IM Fg : Resdual vs Independent Vaable: Resdual o WT Sooth Y Resdual o WT Sooth Y Independent Vaable [75]

7 Table: 7 Calculated Value o Independent Vaable Resdual o WT Sooth COCLUSIO In ths pape, poposed a new algoth to obtaned Wavelet eo analss o optal contol n nonlnea deental equaton b appl the Chebshev Wavelet and to analze the Resdual o Wavelet Soothness and ou poposed ethodolog s copaed wth the teatve ethod.. Ths ethod s ve useul to quckl dent the eo peoance. REFERECES. Awadeh F, Adaw.A and Z. Mustaa (009), Solutons o the SIR odels o epdecs usng HAM, Chaos, Soltons and Factals, Baza.J, Ile.M, and Khoshkena.A(005), A new appoach to the soluton o the pe and pedato poble and copason o the esults wth the Adoan ethod, Appled Matheatcs and Coputaton 7 () Baza.J and Montaze.R(005), A coputatonal ethod o soluton o the pe and pedato poble, Appled Matheatcs and Coputaton 6 (), Baza.J (006), Soluton o the epdec odel b Adoan decoposton ethod, Appled Matheatcs and Coputaton 7, Bod.J.P (000), Chebshev and oue spectal ethods, Unvest o Mchgan, ew Yok Busenbeg.S and Van den Dessche.P (990), Analss o a dsease tanssson odel n a populaton wth vang sze, J. Math. Bol. 8, Chnvast.S and Chnvast.W(00), uecal odellng o an SIR epdec odel wth duson, Appled Matheatcs and Coputaton 6, Fo.L and, I.B. Pake, Chebshev Polnoals n uecal Analss, Ood Unvest Pess, London, (968). 9. Gulsu.M and Oztuk.Y (0), uecal appoach o the soluton o hpe sngula ntego deental equatons, Appled Matheatcs and Coputaton 0, Mason.J.C and Handscob.D.C(00), Chebshev polnoals, Chapan and Hall/CRC, ew Yok.. Y.Oztuk and Gulsu.M (0), An opeatonal at ethod o solvng Lane Eden equatons asng n astophscs, Math. Meth. Appl. Sc. 7, A.M.Wazwaz(00), A elable algoth o solvng bounda value pobles o hghe-ode ntego-deental equatons, Appl. Math. Coput. 8, 7-.. Yousse.C.B, Goa.G and. Dchaa.A.O(005), Knetc odellng o Lactobacllus case ssp. hanosus gowth and lactc acd poducton n batch cultues unde vaous edu condtons, Botecnolog lettes 7, [76]