Quadrature Rules for Evaluation of Hyper Singular Integrals

Transcription

1 Applied Mthemticl Sciences, Vol., 01, no. 117, HIKARI Ltd, Qudrture Rules or Evlution o Hyper Singulr Integrls Prsnt Kumr Mohnty Deprtment o Mthemtics School o Applied Sciences KIIT University, Bhubneswr Odish, Indi-7510 Mnoj Kumr Hot Deprtment o Mthemtics School o Applied Sciences KIIT University, Bhubneswr Odish, Indi-7510 Copyright c 01 Prsnt Kumr Mohnty nd Mnoj Kumr Hot. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct Some Newton-Cote s type o qudrture rules hve been ormulted or the numericl evlution o hyper singulr integrls, which re interpreted s o Hdmrd inite prt type. Rules hve been tested numericlly by some stndrd test integrls nd their respective error bounds hve been determined. Mthemtics Subject Clssiiction: 65D30; 65D3 Keywords: Singulrity; Hilbert trnsorm; Cuchy principl vlue; Hdmrd inite prt integrl; residue; symptotic error estimte; error bound

2 50 Prsnt Kumr Mohnty nd Mnoj Kumr Hot 1 Introduction The inite Hilbert trnsorm 9] o the type: I, c = P d ; < c < b 1 c where is suiciently dierentible in, b] led to uncontrolled instbility when stndrd qudrture rules either Newton-Cote s type or Guss type re to be pplied or their pproimte evlution, due to the presence o the singulrity = c in the rnge o integrtion. The Cuchy principl vlue o this integrl is deined s: I, c = lim ε 0 + c ε c d + c+ε ] c d provided the limit eists. In cse this limit eists, the limiting vlue is known s the Cuchy-principl vlue CPV nd the integrl is denoted s by eqution1. Singulr integrls o the type 1 occur requently in mny brnches o physics, in the theory o erodynmics nd scttering theory etc.however, the qudrture rules ment or the numericl integrtion o the integrl 1 behve very much unstble when these rules re pplied or the pproimtion o the integrl: J, c = d, α > 1; < c < b c α due to the presence o higher order singulrity t = c. divergent integrl: Further, the c ε ] d = lim c ε 0 + c d + c c+ε d ; 3 is very much epressed s: c d = H c d + lim. c ε 0 + ε The integrl present in the right side o the bove eqution is clled s the Hdmrd inite-prt integrl9]. Rmm nd Vn der Sluis 1], Groetsch], Criscuolo6], Pget], Elliott3] nd mny more s vilble in literture hve been contributed their work or the pproimte evlution o this integrl. However the bsic purpose o this pper is to ormulte some qudrture rules which re uniormly convergent to the Cuchy principl vlue o the integrl o the type 1; nd the sme rule hs been employed or the pproimte evlution o the inite prt integrl by reducing the order o singulrity. For this, we consider the integrls o the type eqution1 nd the integrls o Hdmrd type s given in eqution, or α = in the intervl, ].

3 Qudrture rules 51. Formultion o Qudrture Rules This section hs two subsections: subsection-.1 nd subsection-. s given below..1 Rules or the Approimte evlution o Rel CPV Integrls The n 1point rule is generted by decomposing the intervl o integrtion, ] into n equl prts by the points: 0, ± n, ± n, ± 3 n is denoted by R n nd deined s: n 1,... ± ; n R n = w n0 0 + n 1 k=1 w nk k k ] n n 5 Since the nodes re preied, thus it is only remin to determine the coeicients w n0 nd w nk ; or k = 11n 1 ssocited with 0 nd with the block k n k n] respectively. It is to be noted here tht in such rules the coeicient o 0 i.e. w n0 is zero, or ll n. The coeicients w nk o the rule R n or ll n ssocited with the bove block re the solutions o the ollowing set o moment equtions AW =B in coeicients w nk ; where 1... n-1 w n n-1 3 w n A = n-1 5 ; W = n 1 n 1... n-1 n 1 w nn 1 The rules corresponding to n = 1,, 3 nd re noted below. R 1 = R = 13 5 R 3 = R = ] ] 06 5 ] ] ] ] ] ;B = ] ] n/1 n 3 /3.. n n ] ; 7 ] ] ; nd ] ] ] 5 7 ] 3 3 It is pertinent to note here tht or ll n, the rule R n is n open type rule since both the end points nd o the intervl o integrtion, ] re ecluded in the set on 1 nodes... Scheme or the Numericl Approimtion o Hdmrd Finite Prt Integrls o the Type J = H d. 10 For the construction o the scheme, here we ssume tht z is the nlytic continution o in the disc: Ω = {z C: z r ; r > }. 9

4 5 Prsnt Kumr Mohnty nd Mnoj Kumr Hot As result, z = ; ll, ]. The integrl given in 10 cn be trnsormed to: J = d = K + L;where K = = K1 g R ng; nd L = { } d; K 1 = =0 d; g = K1 K 1 d. Now since the integrl K is singulr integrl o the type 1, thus the rules R 1 to R s given rom equtions 6 to 9 ment or the numericl integrtion o the rel CPV integrls my be pplied or its numericl pproimtions. Further, though the integrl L is lso singulr integrl but s its primitive 1 eists, thus it cn be evluted by using the Fundmentl Theorem o Integrl Clculus nd thus As result the integrl: L = K 1 d = K J = g = d Rng K 1 ; where K1 ; nd is the one o the end points o the intervl o integrtion, ]. Net we consider: 3. Error Anlysis The error bounds o the trunction error E n ssocited with the qudrture rules R n or n = 1,, 3 nd or the numericl evlutions o rel Cuchy principl vlue o integrls Equtions 6 to 9 hve been determined by ollowing the techniques due to Lether5] nd is given in Theorem-1.Since derivtion o ech prts o the Theorem-1 re similr we hve derived only the prt-i to void repetition. Theorem-1. I z is n nlytic continution o deined in the closed disc:ω = {z C: z r ; r > } ;then i E 1 Me 1 r; ii E 1 Me r; iii E 3 Me 3 r; iv E Me r; wherem = m z r z ; e 1 r = ln r+1 r 1 r r 1 ; e r = ln r+1 r 1 r r 1 06 r r 9] ; r+1 e 3 r = ln r r 55 36r r 337 9r r 50 r r+1 e r = ln r r 9 6r r 63 16r r 71 6r r 3 6r 5 11 r r r 671 6r 9 ; or = 1; echowhich 0 s r. Lether5]. Proo-i: Let 60r 36r 5 ; nd r r 1 The quntity e k r is deined s error constnt due to z = ; or z, ]. Now by epnding z by Tylor s theorem bout z = 0 we hve z = k=0 c k k where c k = k 0 k! ; or k = 0, 1,..., z, ] re the Tylor s coeicients. Now since E 1 being liner opertor,we obtin E 1 c µ+1 E 1 µ+1. 1 µ=1

5 Qudrture rules 53 r e 1 r e r e 3 r e r Tble 1: Vlues o Error Constnts e nr or r > 1 nd = 1 Rules First Leding Term o Error Epression R R R R Tble : Asymptotic Error Bounds or norn = 1,, 3nd Thus, by Cuchy Inequlity] E 1 M µ=1 1 r µ+1 E 1 µ nd then by using the technique due to Lether5] we obtin E 1 Me 1 r; where e 1 r = E 1 1 ] 1 r + 1 = ln r r r 1 r 1 which 0 s r. This proos the prt-i o Theorem-1 nd hence the theorem is estblished. Comprtive Study o the Error Constnts For = 1, the error constnts e n r corresponding to the rules R n or n = 1,, 3 nd hve been evluted or vlues o r > 1 nd the results o computtion re given in Tble-1. It is observed rom the Tble-1 o vlues o error constnts nd the corresponding grphsfig.-1drwn bsed on the tble tht e r < e 3 r < e r < e 1 r. Also it is evident rom Tble- tht the degree o precision o the rules re, 6, 10 nd 1 respectively. In generl the degree o precision o the ruler n is n.net we consider:. Numericl Eperiments This rticle consists o two prts: Prt-I to Prt-II. Prt-I. Approimte Evlution o Rel CPV Integrl with Singulrity t Origin The integrl considered here is:

6 5 Prsnt Kumr Mohnty nd Mnoj Kumr Hot Figure 1: Grph o Error Constnts or Dierent Vlues o r > 1 Rules Appro. o I Abs.Err R R R R Tble 3: Evlution o Rel CPV Integrl with Singulrity t Origin I = 1 e 1 d= ,re. Longmn7] nd the result o numericl pproimtions re given in Tble-3. Prt-II. Appro.o Hdmrd Finite Prt Integrl J = H 1 e 1 d;= The result o numericl integrtions o this integrl is given in Tble-. 5. Conclusion From the Tbles o numericl results it is observed tht the vlues obtined by the sequence o rules o incresing precision converges to vlue i.e. equl to the ect vlue o the respective integrls correct up to t lest ten igures ter the deciml point in cse o rel CPV integrls s well s integrls o the type Hdmrd inite prt. Also, it is not require to evlute derivtive o the integrnd t ny o its nodes, which is positive dvntge over the eisting rules ound in the literture. Rules Appro. o J Abs.Err R R R R Tble : Eperiments on Hdmrd Finite Prt Integrl

7 Qudrture rules 55 Reerences 1] A.G. Rmm nd A. Vn der Sluis: Clculting singulr integrls s n ill-posed problem, Numer. Mth.,57,1990, ] C.W. Groetsch: Regulrized product integrtion or Hdmrd inite prt integrls, Computers, Mth. Applic., vol. 30, no. 3-6,1995, pp ] D. Elliott: Three lgorithms or Hdmrd inite prt integrls nd rctionl derivtives, J. Comput. Appl. Mth., 6,1995, ] D.F. Pget: The numericl evlution o Hdmrd inite prt integrls, Numerische Mthemtik, 36,191, pp ] F.G. Lether: Error bounds or ully symmetric qudrture rules, SIAM J. Numer. Anl., 11,197, ] G. Criscuolo: A new lgorithm or Cuchy principl vlue nd Hdmrd inite prt integrls, J. Comput. Appl. Mth., 7,1997, ] I.M. Longmn: On the numericl evlution o Cuchy Principl vlues o integrls, MTAC1,1950, ] J. B. Conwy: Functions o one comple vrible,nd edition,nros publishing house, New York,190. 9] P.J. Dvis nd P. Rbinowitz: Methods o numericl integrtion, nd edition, Acdemic press, New York, 19. Received: July 9, 01