Irregular Boundary Area Computation. by Quantic Hermite Polynomial

Transcription

1 It. J. Cotmp. Mat. Sccs, Vol. 6,, o., - Irrgular Boudar Ara Computato b Quatc Hrmt Polomal J. Karwa Hama Faraj, H.-S. Faradu Kadr ad A. Jamal Muamad Uvrst of Sulama-Collg of Scc Dpartmt of Matmatcs, Sualma, Iraq jwamr97@gmal.com Uvrst of Sulama-Collg of Scc Educato Dpartmt of Matmatcs, Sulama, Iraq karzma@aoo.com Sulama Tccal Isttut, Dpartmt of Egrg Drawg, Sulama, Iraq jamalmuamadam@aoo.com Abstract I ts stud, a w mtod s drvd for computg rrgular boudar ara b quatc rmt polomal. Fall, som umrcal ampls prstd to sow t ffcc of t tortcal rsults. Kwords: Irrgular Ara, Quatc Hrmt Polomals, Talor s Mtod. INTRODUCTION T trm ara t cott of survg rfrs to t ara of a tract of a lad projctd upo t orzotal pla, ad ot to t actual ara of a lad surfac [.It s oft cssar to comput t ara of a tract of a lad wc ma b rgular or rrgular sap Fg..Lad s ordarl bougt ad sold o t bass of cost pr ut ara [5. Som ruls wr usd rctl b a survors ad grs for computato of rrgular boudar of a tract, suc as trapzod ad Smpso s o trd rul. Tr s o lmtato trapzodal rul, ts rul ca b appld for a umbr of ordats ad, t ca b appld wl boudars btw t ds of ordats ar assumd to b stragt, ara b ts rul s qual

2 J. Karwa Hama Faraj t al to t product of t commo trval ad t sum of trmdat ordats plus avrag of t frst ad last ordats. I Smpso s rul,t boudar btw t ds of ordats ar assumd to form a arc of a parabola, c Smpso s rul s som tms calld t parabolc rul, ts rul ca b applcabl ol w t umbr of dvsos s v t umbr of ordat s odd,ad t boudar btw t ordats s cosdrd to b a arc of a parabola [. Smpso s o trd ruls wr tdd for uqual trvals b [6.Prstd a mtod tat mplod dffrt polomal fuctos usg salt boudar pots [. A w tral do r ts papr for drvg a w formula for calculatg rrgular boudar ara b usg quatc rmt polomal, t w sowd umrcall tat our formula rsults bttr ta t usd mtods suc as Smpso s ad Trapzodal ruls.. QUANTIC HERMITE POLYNOMAIL Cosdr a rrgular boudar for wc t offsts,,..., ar masurd at,,..., wr s t umbr of trvals Fg..T corrspodg trvals ar,,,,...,.t actual boudar for ac trval s appromatd b a QH polomal,wc s dfd class of Hrmt polomals [,. T QH polomal s dfd blow, followd b ara for a sgl QH, t compost QH formula for uqual trvals, ad t spcal cas of qual trvals. Cosdr t trval, Fg.. A quatc Hrmt polomal for ts trval s a fv-dgr polomal passg troug t two pots, ad, ad satsfg t frst ad scod drvatvs at ad.t QH polomal s: 5 Q a b c d f Ts satsfs t s codtos k k Q, Q, Q Ad k k Q,,,,..., ad k,. Clarl, dfto of Q w av s ukows wc ar a, b, c, d, ad f To fd ac of ts, w must us all codtos as: at, Q a 5 at, Q a b c d f at, Q b 5 at, 5 Q b c d f 6 at, Q c 7

3 Irrgular boudar ara computato 5 at 6, f d c Q 8 Solvg -8 ad w gt d { } 6 f Esq., 5, 7, 9, ad ad gv t s ukows.. SINGLE QUANTIC HERMIT AREA T ara udr t sgl quatc polomal btw ad s obtad b tgratg [ d c b a d Q A f d c b a d f Substtutg for d c b a,,,, ad f from, 5, 9, ad to ad tgratg, t A. COMPOSITE QH AREA FOR UNEQUAL INTERVALS T total ara from to, u A s t sum of t sgl quatc aras gv b. Tat s u A A Substtutg for A from ad collctg smlar trms, t [ u A 5 Eq. 5 s t quatc rmt formula for uqual trvals.ts formula, wc volvs ol a mor modfcato to t trapzodal rul s kow as t corrctd trapzodal rul. T frst trm of 5 s t trapzodal rul. T ukow frst ad scod drvatvs ad

4 6 J. Karwa Hama Faraj t al for,...,, ar stmatd umrcall usg kow offsts.emplog Talor s torm of ordr two ac of ts obtad as follows: 6,,..., `, 7 8, [ 9,,...,, [ ad [ A algortm for ara computato usg 5- s Appd I,II, ad III. 5. SPECIAL CASE OF EQUAL INTERVALS T prcdg rlatosps ca obvousl b appld to uqual or qual trvals. For qual trvals, owvr, t rlatosps ca b smplfd ad a sgl formula for computg t ara ca b obtad. Ara computato wt suc formula would b mor ffct.assumg tat t trval qual, t aftr all smplfs.. usg 6- to 5 Equ.5 bcoms 9 9 [ A 6. Vrfcato Obtad quato tstd o som fuctos, t rsults tabulatd tabl ad tabl, as follow: 6.: Algbrac fucto:

5 Irrgular boudar ara computato 7 Tabl : comparso btw act tgrato valus ad valu b Quatc formula S Fucto Quatc valu Eact valu MATLAB Algortm Yf Sms 5 7 d Yt ^5 * ^ ^ * ^ 7,, Yf 5 7 d Sms Yt ^5 ^ 7*,, Yf Sms 5 7 d Yt * ^5 * ^ 7* ^,, 6.: Trgoomtrc fucto: Tabl : comparso btw act tgrato valus ad valu b Quatc formula S Fucto Quatc valu Eact Valu MATLAB Algortm π Yf s d π Yf Cos d π Sms Its,,p Sms Itcos,.5*p,.5*p

6 8 J. Karwa Hama Faraj t al 6.: Practcal fld data from survg tt book [: A comparso was do btw dffrt mtods b usg stg data [, p.5, ad t rsults wr tabulatd tabl. as follow: Tabl : Comparso btw dffrt mtods S Mtods Ara Error % Eact** Trapzodal rul % Smpso o trd rul % Quatc Hrmt 9.5.% * Ol qual trvals cas tstd r bcaus drvd quatos for uqual trvals s lttl compl for applg b survor, ad vr ca b us wtout programmd adld calculator.it s wrt som tm outsd boudar codto d to coos uqual trvals to obtad a bttr rsult but ts cas ca b tratd b slcto offsts at rgular trval ad at closr dstac btw offsts ** Eact ara was calculatd b MATLAB, troug trg t data to EXCEL sprad st, drawg bst curv wc ca pass t offsts d, t calculatg t coffcts for Quatc Hrmt polomal, ad calculatg t act ara troug takg tgrato s algortm for MATLAB appd II. 7. CONCLUTION I ts papr,w coclud tat t Quatc Hrmt polomal troducd ts papr prforms bttr rsults ta t Trapzod ad Smpso s mtod for calculatg rrgular boudar ara w APPENDIX I, II AND III: APPENDIX I. Algortm for computg rrgular boudar ara usg quatc rmt mtod. * Stp : Iput a Numbr of trvals, 6 ; b t offsts,,,,,..., ; ad c t trvals,,,,,..., or f t trvals ar qual.st ad S * Stp : f t trvals ar qual, go to stp. Otrws, cotu.

7 Irrgular boudar ara computato 9 * Stp : Comput t frst drvatvs at t frst ad last offsts, ad, * Stp : Comput t frst drvatv at offsts, 5 *Stp 5: Comput t scod drvatvs at t frst ad last offsts, ad, [ 6 [ 7 * Stp 6: Comput t scod drvatv at offsts, [ 8 *Stp 7: Comput t trm sd t summato quato 5, S S 9 St t cumulatv trm, S, qual to t currt valu plus S ; S S. *Stp 8: St. f,go to stp 9.Otrws, go to stp *Stp 9: Comput t rqurd ara, u A,ad go to stp S A u *Stp : Comput t rqurd ara for qual trvals, A 9 9 [ A *Stp : Output u A or A APPENDIX II. Algortm for calculato of act ara for t practcal fld data b usg EXCEL & MATLAB Stp : Etr data to EXCEL sprad st Stp : Startmatlab start MATLAB putmatr sd data to MATLAB

8 J. Karwa Hama Faraj t al Stp : Df varabl am MATLAB dfs & varabls Stp: Evalstrg cut t MATLAB commad cftool op bst curv ft wdow Stp 5: Fttg Draw w ft b slctg 5 t dgr polomals draw bst curv wc ca pass offsts d Stp 6: Aalss Calculat ara udr t draw bst curv b takg tgrato APPENDIX III. MATLAB output rlatg bst curv quato ad ts coffcts lar modl Pol5: f p*^5 p*^ p*^ p*^ p5* p6 Coffcts wt 95% cofdc bouds: p ,.-6 p ,.876 p ,.78 p ,.6 p ,.6 p , 5. Goodss of ft: SSE:.76 R-squar:.998 Adjustd R-squar:.957 RMSE:.78 Rfrcs [ F. A. Amad, Ara Computato Usg Salt Boudar Pots, Joural of survg grg, 998. [ J.H-F.Karwa K.G.Rada, Gralzato of, lacuar trpolato b quatc spl, Joural of matmatcs ad statstcs, Nw work, 6,7-78. [ N.N.Basak,Survg ad lvlg, Tata McGraw- Hll, 99, 6-6. [ L.R.Burd,D.L.Fars,ad C.A.Rolds,Numrcal Aalss.Prdl,Wbr & Scmdt,Bosto,Mass,978.

9 Irrgular boudar ara computato [5 S.K.Ro,Fudamtals of Survg,Prtc-Hall of Ida Pravt lmtd, 5, [6 S. M. Easa Smoot Boudar Appromato for drctl computg Ara, Joural of survg grg,999. Notatos: T followg smbols ar usd ts papr a,b lmt of tgrato total umbr of trvals lgt of o qual trval A Ara of rgo for trval A total ara offsts at pots f o, f,f scod drvatvs at pots,, rspctv Q ordat of 5t dgr Quatc polomal at pot

10 J. Karwa Hama Faraj t al Rcvd: Sptmbr,