A Note on the Numerical Solution for Fredholm Integral Equation of the Second Kind with Cauchy kernel

Transcription

1 Journal of Mathematcs an Statstcs 7 (): 68-7, ISS Scence Publcatons ote on the umercal Soluton for Freholm Integral Equaton of the Secon Kn wth Cauchy kernel M. bulkaw,.m.. k Long an Z.K. Eshkuvatov Department of Mathematcs an Insttute for Mathematcal Research, Unversty Putra Malaysa, 434 Serang, Selangor, Malaysa bstract: Problem statement: In ths stuy, numercal soluton for the Freholm ntegral equaton of the secon kn wth Cauchy sngular kernel s presente. pproach: The Chebyshev polynomals of the secon kn are use to appromate the unknown functon. Results: umercal results are gven to show the accuracy of the present numercal soluton. Concluson: The present numercal soluton to the Freholm ntegral equaton of the secon kn wth Cauchy kernel s accurate. Key wors: Integral equatons, Cauchy sngular kernel, Chebyshev polynomal appromatons ITRODUCTIO Conser the Freholm ntegral equaton of the secon kn wth Cauchy kernel of the form: ϕ(t) ϕ () +λ t = f (), ( λ s a constant) () wth specfe en contons: ϕ± ( )= () where f s assume to be real-value functons belong to the class of Holer on the nterval [-, ] an ϕ s the unknown functon to be etermne. The sngular ntegral n Eq. beng unerstoo n the sense of Cauchy prncpal value. Many researchers evote ther works on appromatng the sngular ntegrals of Cauchy type. Partcularly, Dagnno an Sant (99) obtane a prouct quarature rules, base on splne nterpolaton, for the numercal evaluaton of Cauchy sngular ntegrals. They also propose an error boun an obtane convergence results for functons f C k [-,], k=, or 3. Ors (99) prove the unform convergence of some quarature formulas base on splne appromaton for Cauchy prncpal value ntegral. They also presente some numercal applcatons. In partcular, they apple ther rules to the well-known Prantl s ntegral equaton. Rabnowtz (99) prove convergence results for prouct ntegraton rules base on appromatng splnes. These results are both for boune an unboune ntegrans. Pontwse an unform convergence results are prove for sequences of Cauchy prncple values of these appromatng sples..dagnno an Sant (99) consere the same rules as n Dagnno an Sant (99) an nvestgate ther convergence for a large class of functons f. They establshe an error boun an some unform convergence results n the case of equally space quarature noes, for functon f, satsfyng a Holer conton of orer μ on [,], < μ. Hasegawa an Tor (99) presente an automatc quarature for computng Cauchy prncpal value ntegrals for smooth functons f(t). They appromate the functon f(t) by a sum of Chebyshev polynomals whose coeffcents are compute usng the Fast Fourer Transform (FFT). Hasegawa an Tor (994) presente an automatc quarature for appromatng Haamar fnte-part (fp) ntegrals of a smooth functon, wth a ouble pole sngularty wthn the range of ntegraton. The quarature rule s rve from the fferentaton of an appromaton to a Cauchy prncpal value ntegral. Dethelm (99) nvestgate the numercal appromaton of the Cauchy prncpal value ntegral. He presente the quarature formula for appromatng the Cauchy prncpal value ntegral. He prove the convergence of the quarature formula an gave the estmaton for the errors. Dagnno an Lambert (996) evaluate the Cauchy prncpal value ntegral by applyng a local splne appromaton metho, efne for any functon f L [,]. They establshe convergence results wth error boun. Dethelm (997) consere the so-calle mofe quarature formulas,.e. formulas obtane by frst subtractng out the sngularty an then applyng a classcal quarature Corresponng uthor: Dr. Mohamma bulkaw, Department of Mathematcs an Insttute for Mathematcal Research, Unversty Putra Malaysa, 434 Serang, Selangor, Malaysa 68

2 J. Math. & Stat., 7 (): 68-7, formula, for the numercal appromaton of Cauchy prncpal value ntegrals. They have gven new bouns nvolvng the total varaton Var f (s) an L p -norms f (s) p of some ervatve of the ntegran functon. Eshkuvatov et al. (9) constructe a new quarature formulas for evaluatng the sngular ntegral of Cauchy type. The constructon of the quarature formulas s base on the mofcaton of screte vortces metho an lnear splne nterpolaton over the fnte nterval [, ]. They prove that the constructe quarature formula converges for any sngular pont not concng wth the en ponts of the nterval [, ]. They have gven error bouns n the classes of functons H α [, ] an C [, ] of orer O(h α ln h ), < α an O(h ln h ), respectvely. Eshkuvatov et al. () consere the sngular ntegral wth the Cauchy kernel. They constructe new quarature formulas base on the mofcaton of screte vorte metho to appromate the sngular ntegral. They have shown error bouns n the classes of functons H α [,] an C [,] for ether = t or = t where t,,..., are the noe ponts an t =(t + t + ) /. The ntegral equatons wth Cauchy kernel have been wely use n solvng problems assocate wth aeroynamc, hyroynamc an elastcty (Lfanov, 996; Laopoulos, ; bou an aser, 3; Mohankumar an ataraan, 8; Lara an Maragraza, ; Kasoz an Paulsen, a; Kasoz an Paulsen, b; Gan et al., 8; Thukral, ). In ths stuy, we present a numercal soluton for the Eq. wth contons (). MTERILS D METHODS The unknown functon ϕ of Eq. whch satsfes contons () can be represente as: ϕ()= ψ(), (3) where, ψ () s a well behave functon of on the nterval [-, ]. The functon ψ () n Eq. 3 s appromate usng the Chebyshev polynomals of the secon kn, U, as: From (3) an (4), we have: () au (), ϕ (6) Substtutng (6) nto (), yels: au () +λ a t U (t) t =f() (7) It s known that (Kythe an Schaferkotter, ; bulkaw et al., 9) t U (t) t = π T + () (8) where, T are the Chebyshev polynomals of the frst kn whch s efne by: T () = cos cos (), =,,, (9) Usng (8) nto Eq. 7, yels: a U () πλt + () =f() () Multplyng both ses of Eq. by U an ntegratng from - to, we obtan the followng system of lnear equatons; a =, =,,, () where: = B + C () π, = B = t U (t)u (t)t = (3), ψ () where: a U () (4) C = πλ T (t)u (t)t (4) an: + sn[ cos ()] U () =,,, () sn[ cos ()] 69 = f(t)u (t)t ()

3 J. Math. & Stat., 7 (): 68-7, Solvng the above system for the unknown coeffcents a, =,,, an substtutng the values of a nto the appromate soluton (6), we obtan the numercal soluton of the Eq. wth contons (). RESULTS D DISCUSSIO Eample : Let us conser the sngular ntegral equaton: ϕ(t) (6) ϕ () + t = π + ( π ) + +π wth the contons: ϕ± ( )= (7) It s not ffcult to see that the eact soluton of the Eq.6 s: φ() = ( + ) (8) Due to Eq. -4, we obtan: π = π, =, 3 4 π 4 = π, = π 3 6 π =, = π, =. (9) π a + πa = π+, π π a + a + πa = π+, π a+ a = π. () It s easy to see that the soluton to the above system s: a a a = (3) Substtutng (3) nto (6) for we obtan: ϕ() = (+ ) (4) whch s the eact soluton. Eample : Conser the followng equaton: ϕ(t) ϕ () + t = U () πt 6() () wth the contons: ϕ ( ± )= (6) It s clear that the eact soluton of the Eq. s: 3 ϕ() = ( ) (7) From Eq. -4, we obtan: From Eq. (), we have: ( π + ( π) ) ( t ) U (t)t = t t t U (t)t + +π whch gves: = π+ π, 3 4 = π+ π, 3 6 = π. () () Thus the system of lnear equatons () for = becomes: 7 =, 3 =, 3 π 4 4 =, 3 8 =, 3 4 =, 3 π 6 3 =, =, 3 =, 3 =, π =, =, π 4 4 =, =, =, =, 9 8 π =, 3 =, 4 =. 9 (8)

4 Due to Eq., we have: π 6 = ( t U (t) T (t))u (t)t (9) whch gves: =, =, =, =. 9 J. Math. & Stat., 7 (): 68-7, (3) Thus the corresponng system of lnear equatons for = s: a =, =,,, (3) where,,, =,,, are efne n (8) an, =,,, are efne n (3). It s not ffcult to see that the soluton to the above system s: a =,a =,a =,a =,a =,a = (3) 3 4 Substtutng the values of a, =,,, nto (6) where =, we obtan the numercal soluton of Eq. whch s entcal to the eact soluton (7). COCLUSIO The truncate seres nvolvng the Chebyshev polynomal appromaton of the secon kn s use to appromate the unknown functon for solvng the Freholm ntegral equaton of the secon kn wth Cauchy kernel an constant coeffcents. The metho of appromaton llustrate here gves a goo way for obtanng the numercal soluton avong the complcate ntegratons. umercal results show the accuracy of the metho presente whch, for some functons f (), gves the eact soluton. CKOWLEDGEMET Ths stuy was supporte by Unversty Putra Malaysa uner Postoctoral Fellowshp an proect o RU. REFERECES bou, M.,. an.. aser, 3. On the numercal treatment of the sngular ntegral equaton of the secon kn. pple Math. Computaton, 46: DOI:.6/S96-33() bulkaw, M., Z.K. Eshkuvatov an.m... Long, 9. numercal treatment for solvng Cauchy type sngular ntegral equaton. FJM., 3: 7-8. Dagnno, C. an E. Sant, 99. Splne prouct quarature rules for cauchy sngular ntegrals. J. Comput. pple Math., 33: DOI:.6/377-47(9)9363- Dagnno, C. an E. Sant, 99. On the convergence of splne prouct quaratures for Cauchy prncpal value ntegrals. J. Comput. pple Math., 36: DOI:.6/377-47(9)9-F Dagnno, C. an P. Lambert, 996. umercal evaluaton of Cauchy prncpal value ntegrals base on local splne appromaton operators. J. Comput. pple Math., 76: DOI:.6/S377-47(96)-7 Dethelm, K., 99. Gaussan quarature formulae of the thr kn for Cauchy prncpal value ntegrals: Basc propertes an error estmates. J. Comput. pple Math., 6: DOI:.6/377-47(9)3-4 Dethelm, K., 997. ew error bouns for mofe quarature formulas for Cauchy prncpal value ntegrals. J. Comput. pple Mathe., 8: DOI:.6/S377-47(97)4-9 Eshkuvatov, Z.K.,.M... Long an M. bulkaw, 9. Quarature formula for appromatng the sngular ntegral of Cauchy type wth unboune weght functon on the eges. J. Comput. pple Math., 33: DOI:.6/.cam Eshkuvatov, Z.K.,.M... Long an M. bulkaw,. umercal evaluaton for Cauchy type sngular ntegrals on the nterval. J. Comput. pple Math., 33: 99-. DOI:.6/.cam Gan, D.D., M. mn an. Kolahooz, 8. nalytcal nvestgaton of Hyperbolc equatons va he s methos. m. J. Eng. pple Sc., : DOI:.3844/aeassp Hasegawa, T. an T. Tor, 99. n automatc quarature for Cauchy prncpal value ntegrals. Mathe. Comput., 6: Hasegawa, T. an T. Tor, 994. Hlbert an haamar transforms by generalze chebyshev epanson. J. Comput. pple Math., : DOI:.6/377-47(9)3-R Kasoz, J. an J. Paulsen, a. Flow of vens uner a constant force of nterest. m. J. pple Sc., : DOI:.3844/aassp Kasoz, J. an J. Paulsen, b. umercal ultmate run probabltes uner nterest force. J. Math. Stat., : 46-. DOI:.3844/mssp..46.

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