Chebyshev Polynomials for Solving a Class of Singular Integral Equations

Transcription

1 Appled Mahemas, 4, 5, Publshed Ole Marh 4 SRes. hp:// hp://d.do.org/.46/am Chebyshev Polyomals for Solvg a Class of Sgular Iegral Equaos Samah M. Dardery, Mohamed M. Alla Deparme of Mahemas, Fauly of See, Zagazg Uversy, Zagazg, Egyp Deparme of Mahemas, Fauly of See ad Ars Al-Mhab, Qassm Uversy, Qassm, KSA Emal: sd.974@homal.om, allamm999@homal.om Reeved 8 November ; revsed 8 Deember ; aeped 6 Deember Copyrgh 4 by auhors ad Sef Researh Publshg I. Ths work s lesed uder he Creave Commos Arbuo Ieraoal Lese (CC BY). hp://reaveommos.org/leses/by/4./ Absra Ths paper s devoed o sudyg he appromae soluo of sgular egral equaos by meas of Chebyshev polyomals. Some eamples are preseed o llusrae he mehod. Keywords Sgular Iegral Equaos; Cauhy Kerel; Chebyshev Polyomals; Wegh Fuos. Iroduo Durg he las hree deades, he sgular egral equao mehods wh applaos o several bas felds of egeerg mehas, lke elasy, plasy, aerodyams ad fraure mehas have bee suded ad mproved by several sess (see []-[6]). Hee, s eresg o solve umerally hs ype of egral equaos (see [7] [8]). Chebyshev polyomals are of grea mporae may areas of mahemas parularly appromao heory (see [9] []). I hs paper we aalyze he umeral soluo of sgular egral equaos by usg Chebyshev polyomals of frs, seod, hrd ad fourh kd o oba sysems of lear algebra equaos, hese sysems are solved umerally. The mehodology of he prese work epeed o be useful for solvg sgular egral equaos of he frs kd, volvg parly sgular ad parly regular kerels. The sgulary s assumed o be of he Cauhy ype. The mehod s llusraed by osderg some eamples. Sgular egral equao of frs kd, wh a Cauhy ype sgular kerel, over a fe erval a be represeed by k (, ) ( ) d + L (, ) ( ) d f, < < (.) How o e hs paper: Dardery, S.M. ad Alla, M.M. (4) Chebyshev Polyomals for Solvg a Class of Sgular Iegral Equaos. Appled Mahemas, 5, hp://d.do.org/.46/am.4.547

2 S. M. Dardery, M. M. Alla where k (, ), L (, ) ad f oues fuos ad k( ) are gve real-valued ouous fuos belogg o he lass Holder of,. I Equao (.) he sgular kerel s erpreed as Cauhy prple value. Iegral equao of form (.) ad oher dffere forms have may applaos (see [] [] [6] [] []). The heory of hs equao s well kow ad s preseed [] [4]. A appromae mehod for solvg (.) usg a polyomal appromao of degree has bee proposed [7]. I s well kow ha he aalyal soluos of he smple sgular egral equao a k (, ) ad ( ) d f, < < (.) L,, for he followg four ases: ) The soluo s ubouded a boh ed-pos ±, ) The soluo s bouded a boh ed-pos ±, ) The soluo s bouded a ed, bu ubouded a ed +, 4) The soluo s ubouded a ed, bu bouded a ed +, are gve by [5]. I hs paper he used appromae mehod for solvg Equao (.) sems from ree work [] where a appromae mehod has bee developed o solve he smple Equao (.). The appromae mehod developed below appears o be que approprae for solvg he mos geeral ype Equao (.). Some eamples are preseed o llusrae he mehod.. The Appromae Soluo I hs seo we prese he mehod of he appromae soluo of Equao (.) four ases. Le he ukow fuo Equao (.) be appromaed by he polyomal fuo ( ) j j j W Ψ, j (.) (,,,4) ( j where ),,,,, are ukow oeffes, o be deermed, ad ase (I): Ψ T, ase ( (II): ) ( Ψ U, ase (III): ) ( 4 Ψ V ad ase (VI): Ψ ) W, where T, U, V ad W,,,,, are The Chebyshev polyomals of he frs, seod, hrd ad fourh kds respevely a be defed by he reurree relaos [9] [6]., T T T T T,, U U U U U,, V V V V V,, W W + W W W, ad W,,,,, are he orrespodg wegh fuos. Subsug he appromae soluo (.) for he ukow fuo o (.) yelds ( j ) ( j ) j k W, Ψ ( ) ( ) ( d, ) j j + LW Ψ ( ) d f, < < (.6) I above Equao (.6), we e use he followg Chebyshev appromao o he kerels (, ) (, ) L, gve by (for fed, f. [7]) (.) (.) (.4) (.5) k ad 754

3 S. M. Dardery, M. M. Alla wh kow epressos for Kp ( ) ad Lq where wh ad m p q (, ) p, (, ) q (.7) k k L L p q. The (.6) gves s ( j) ( j ) f, < <, ( j,,, 4) (.8) m ( j ) ( j ) ( j) kp up, + Lq q, (.9) p q ( j ) ( j ) s p j W Ψ up, d, < <, ( j,,, 4) (.) ( j) q ( j ) j q, Ψ W d (.) ( j Le ) k, j,,,4, be he zeros of U, T+, W+ ad V +, respevely. Subsug he ( j olloao pos ), j,,,4 o (.8) we oba he followg sysems of lear equaos: where k ( j) ( j) ( j) j k f k, k,,, +, j,,,4 (.) ( ) m s ( j) ( j) j j j j j k p k p, k q k q, p q,(,,, ),(,,,4) (.) k u + L k + j ( j Solvg he sysem of Equao (.) for he ukow oeffes ), j,,,4, ad subsug he ( j) values of o (.) we oba he appromae soluos of Equao (.).. Numeral Eamples I hs seo, we osder some problems o llusrae he above mehod. All resuls were ompued usg FORTRAN ode. Eample Cosder he followg sgular egral equao ( + ) ( ) where 4 + ( + ) ( ) d, < < (..) 8 k, +, L, +, f. So, oe ges 8 4,,,, ( > ) k k k k p,,,,, ( > ) L L L L L q Hee we fd ha relao (.8) produes 8 j Thus (.9) gves ( j) ( j ) 4, < <, (,,, 4) j j j ( j ) ( j ) p q u, + u, +, +,, j,,, 4,,,, 755

4 S. M. Dardery, M. M. Alla Frsly, le us osder deal he ase (I), j, for. Ths resuls By applyg he followg relaos T T( ) d, d,, (..),.,. u u < < T( ) T( ), d,, d, (..) T ( ) d π U, d (..4) ( ) ( ) j T( ) Tj ( ) d π j (..5) π j I s easy o esmae he values u,, u,,, ad,. From (.) ad (..)-(..5) we ge ( ) By hoosg he olloao pos sysem of lear equaos: π + ; 7 π + + ; 8 π( + ); 4 π ; 8 k ( k ) π ( + ) os,,,,4 ( ) f, k,,,4 k k By solvg hs sysem for he ukow oeffes,,,, ha produes.898, , From (..7) we oba he appromae soluo of Equao (..) he form + π (..6), for, we oba he followg (..7) (..8) Whh odes wh he ea soluo. The error of appromae soluo (..8) of Equao (..) a s gve by Table. Seodly, le us osder deal he ase (II), j, for. Ths resuls By applyg he followg relaos U d, U d,, (..9),.,. u u < <,, U d, U ( ) d, (..) 756

5 S. M. Dardery, M. M. Alla ( ) I s easy o esmae he values u,, u,,, ad,. From he relaos (.) ad (..9)-(..) we ge ( ) By hoosg he olloao pos sysem of lear equaos : U d πt + (..) U( U ) j ( ) d π j (..) j π ( + ) π 5 4 π π ( k ) π k ( + ) os,,,,4 k k f, k,,,4 By solvg hs sysem for he ukow oeffes,,,, ha produes.66697, , From (..4) we oba he appromae soluo of Equao (..) he form ( ) (..), for, we oba he followg (..4) (..5) π Whh odes wh he ea soluo. The error of appromae soluo (..5) of Equao (..) a s gve by Table. Thrdly, le us osder deal he ase (III), j, for. Ths resuls By applyg he followg relaos I s easy o esmae he values + V + V( ) d, d,, (..6),.,. u u < < + + ( ) ( ),, V d, V ( ) d, (..7) + j V ( V ) j ( ) d π j (..8) ( ) ( ) ( ) u,, u,,, ad + V d πw (..9) ( ),. 757

6 S. M. Dardery, M. M. Alla Table. Illusraes errors of appromae soluos of Equao (..) Cases (I)-(IV) a. error (j ) error (j ) error (j ) error (j 4) 9.5E-.E+.E E E-8 9.E-.E+.E+.99E-7.99E-7 7.E-.E+.E+.99E-7.99E-7 5.E E E E-7.E-.E E E E-7.E E E-8.99E-7.99E-7.E E E-8.99E-7.99E-7.E-.99E E E E-8.E E E E E-8 5.E E E E E-8 7.E-.48E E-8.496E-8.496E-8 9.E- 9.6E E E-8.85E-8 9.5E E-8.658E E E-9 From he relaos (.4) ad (..6)-(..9) we ge ( ) ( ) π π 4 π π 8 5 (..) By hoosg he olloao pos sysem of lear equaos : ( ) kπ k os,,,,4 ( + ) ( ) ( ) ( ) k k By solvg hs sysem for he ukow oeffes f, k,,,4 ( ),,,, ha produes ( ) ( ).898, , ( ) ( ).59549, 8.9 From (..) we oba he appromae soluo of Equao (..) he form of + π ( + ), for, we oba he followg (..) (..) Whh odes wh he ea soluo. The error of appromae soluo (..) of Equao (..) a s gve by Table. Fourhly, I ase (IV), j 4, for. Ths resuls 758

7 S. M. Dardery, M. M. Alla By applyg he relaos W W( ) d, d,, (..) 4 4,.,. u u < < + + ( 4) ( 4),, + + ( 4) ( 4) ( 4) ( 4) I s easy o esmae he values u,, u,,, ad,. From he relaos (.5) ad (..)-(..6) we ge ( 4 ) By hoosg he olloao pos sysem of lear equaos : W d, W ( ) d, (..4) j W ( W ) j ( ) d + π j (..5) W d πv (..6) + 7π ; π ; 4 π 4 ; 5 4 π ; 4 π os,,,,4 ( k ) k ( + ) ( 4) ( 4) ( 4) 4 k k By solvg hs sysem for he ukow oeffes f, k,,,4 ( 4 ),,,, ha produes ( 4) ( 4).898, ( 4) ( 4).59549, From (..8) we oba he appromae soluo of Equao (..) he form of π + ( ) (..7), for, we oba he followg (..8) (..9) whh odes wh he ea soluo. The error of appromae soluo (..9) of Equao (..) a s gve by Table. Eample. Cosder he followg sgular egral equao ( ) d+ ( + ) ( ) d + (..) whh orrespods wh k (, ) ad L(, ) k k ( p > ) +. So oe ges,, p,,, ( > ) L L L L q q 759

8 S. M. Dardery, M. M. Alla Hee we fd ha relao (.8) produes Thus (.9) gves ( j) ( j ) +, < <, ( j,,, 4) j j ( j ) ( j ) u, +, +,, j,,, 4,,,, Frsly, le us osder deal he ase (I), j, for. Ths resuls T ( ) From he relaos (..)-(..5) ad (..) we oba By hoosg he olloao pos sysem of lear equaos : ( ) k, d, (..) π + π 9π 4 π( 4 ) ( k ) π ( + ) os,,,,4 ( ) f, k,,,4 k k By solvg hs sysem for he ukow oeffes,,,, ha produes 8.898, ,.8649 From (..4) we oba he appromae soluo of Equao (..) he form of + 9π ( 7 4 ) (..), for, we oba he followg (..4) (..5) whh odes wh he ea soluo. The error of appromae soluo (..5) of equao (..) a s gve by Table. Seodly, le us osder deal he ase (II), j, for. Ths resuls, By applyg he relaos (..9)-(..) ad (..6) we ge By hoosg he olloao pos ( ) U d, (..6) π 7 4 π( ) 5 π π( ) ( k ) π k ( + ) os,,,,4 (..7), for, we oba he followg 76

9 S. M. Dardery, M. M. Alla Table. Illusraes errors of appromae soluos of Equao (..) Case (I), Case (II) ad Case (IV) respevely a. Error (j ) Error (j ) Error (j ) 9.5E-.E+.E+.E+ 9.E E E-8.E+ 7.E E-8.99E E-8 5.E E-8.99E-7.99E-7.E E E-7.99E-7.E-.99E E E-7.E+.48E E E-7.E-.99E E-7.99E-7.E E E E-8 5.E E-8.99E-7.99E-7 7.E E-8.99E-7.E+ 9.E E E E-8 9.5E-.E+.E+.E+ sysem of lear equaos: k k By solvg hs sysem for he ukow oeffes f, k,,,4 ( ),,,, ha produes.7559, , From (..8) we oba he appromae soluo of Equao (..) he form of (..8) π (..9) whh odes wh he ea soluo. The error of appromae soluo (..9) of Equao (..) a s gve by Table. Thrdly, I ase (IV), j 4, for. Ths resuls W ( ) d, (..) ( 4), + By applyg he relaos (..)-(..6) ad (..) we ge ( 4 ) + 9 π π( ) π π 4 (..) 76

10 S. M. Dardery, M. M. Alla By hoosg he olloao pos sysem of lear equaos: 4 π os,,,,4 ( k ) k ( + ) ( 4) ( 4) ( 4) 4 k k By solvg hs sysem for he ukow oeffes f, k,,,4 ( 4) ( 4) ( 4) ( 4) ( 4 ),,,, ha produes, for, we oba he followg ,.9487 (..) From (..) we oba he appromae soluo of Equao (..) he form of ( + )( ) (..) π + whh odes wh he ea soluo. The error of appromae soluo (..) of Equao (..) a s gve by Table. Smlarly, dog he same operaos as we dd for Case (I), Case (II) ad Case (IV), oe a solve for Case (III). Eample. Cosder he followg sgular egral equao ( ) d+ ( + ) ( ) d +, (..) whh orrespods wh k (, ) ad L (, ) k k ( p > ) Hee he relao (.8) produes where (.9) gves +. So, oe ges, p,,,, ( > ) L L L L q q ( j) ( j ) +, < <, j,,, 4 (..) j j ( j ) ( j ) u, +, +,, j,,, 4,,,, Frsly, le us osder deal he ase (II), j, for. From (..9)-(..) ad (..6) we ge π ( ) 8 ( ) π (..) π ( 4 ) 8 4 π( ) By solvg he sysem (..), a he olloao pos oeffes ( ),,,, we oba ( k ) π k ( + ).66697, ,.8757 So he appromae soluo of Equao (..) s gve by ( 4 ) os,( k,,,4), for he ukow (..4), (..5) π 76

11 S. M. Dardery, M. M. Alla whh odes wh he ea soluo, he error of he appromae soluo (..5) of Equao (..) a s gve by Table. Seodly, ase (III), j, for. Ths resuls From (..6)-(..9) ad (..6) we ge ( ) + V ( ) d, (..6) ( ), π + 5 π π( ) By solvg he sysem (..), a he olloao pos oeffes ( ),,,, we oba π 4 ( ) ( ).899,.58987, ( ) ( ).97887, Hee, he appromae soluo of Equao (..) s gve by + π ( 5+ 4) (..7) ( ) kπ k os,( k,,,4), for he ukow ( + ) (..8) (..9) whh odes wh he ea soluo, he error of he appromae soluo (..9) of Equao (..) a s gve by Table. Table. Illusraes errors of appromae soluos of Equao (..) Case (II) ad Case (III) a. Error (j ) Error () 9.5E-.98E-8.98E-8 9.E-.98E E-8 7.E-.E E-8 5.E-.E+.99E-7.E-.E+.99E-7.E E-8.99E-7.E E-8.99E-7.E E-8.99E-7.E-.99E-7.99E-7 5.E-.99E E-8 7.E-.99E-7.E+ 9.E E E-7 9.5E E E-7 76

12 S. M. Dardery, M. M. Alla Smlarly, dog he same operaos as we dd for Case (II) ad Case (III), oe a solve for Case (I) ad Case (IV). 4. Coluso Numeral resuls (Tables -) show ha he errors of appromae soluos of Eamples - dffere Cases wh small value of are very small. These show ha he mehods developed are very aurae ad fa for a lear fuo gve he ea soluo. Referees [] Chakrabar, A. (989) Soluo of Two Sgular Iegral Equaos Arsg Waer Wave Problems. ZAMM, 69, hp://d.do.org/./zamm [] Ladopoulous, E.G. () Sgular Iegral Equaos Lear ad No-Lear Theory ad Is Applaos See ad Egeerg. Sprger, Berl. [] Ladopoulous, E.G. (987) O he Soluo of he Two-Dmesoal Problem of a Plae Crak of Arbrary Shape a Asorop Maeral. Egeerg Fraure Mehas, 8, hp://d.do.org/.6/-7944(87)9-8 [4] Zabreyko, P.P. (975) Iegral Equaos A Referee Te. Noordhoff, Leyde. hp://d.do.org/.7/ [5] Prossdorf, S. (977) O Appromae Mehods for he Soluo of Oe-Dmesoal Sgular Iegral Equaos. Applable Aalyss, 7, [6] Zss, V.A. ad Ladopoulos, E.G. (989) Sgular Iegral Appromaos Hlber Spaes for Elas Sress Aalyss a Crular Rg wh Curvlear Craks. Idus. Mah., 9, -4. [7] Chakrabar, A. ad Berghe, V.G. (4) Appromae Soluo of Sgular Iegral Equaos. Appled Mahemas Leers, 7, hp://d.do.org/.6/s (4)95-5 [8] Abdou, M.A. ad Naser, A.A. () O he Numeral Treame of he Sgular Iegral Equao of he Seod Kd. Appled Mahemas ad Compuao, 46, 7-8. hp://d.do.org/.6/s96-()587-8 [9] Abdulkaw, M., Eshkuvaov, Z.K. ad Nk Log, N.M.A. (9) A Noe o he Numeral Soluo of Sgular Iegral Equaos of Cauhy Type. Ieraoal Joural of Appled Mahemas ad Compuer See, 5, 9-9. [] Eshkuvaov, Z.K., Nk Log, N.M.A. ad Abdulkaw, M. (9) Appromae Soluo of Sgular Iegral Equaos of he Frs Kd wh Cauhy Kerel. Appled Mahemas Leers,, hp://d.do.org/.6/j.aml.8.8. [] Gakhov, F.D. (966) Boudary Value Problems. Addso-Wesley, Boso. [] Mar, P.A. ad Rzzo, F.J. (989) O Boudary Iegral Equaos for Crak Problems. Proeedgs of he Royal Soey A, 4, hp://d.do.org/.98/rspa [] Sheshko, M. () Sgular Iegral Equaos wh Cauhy ad Hlber Kerels ad Ther Appromaed Soluos. The Leared Soey of he Cahol Uversy of Lubl, Lubl. ( Russa) [4] Muskhelshvl, N.I. (977) Sgular Iegral Equaos. Noordhoff Ieraoal Publshg, Leyde. hp://d.do.org/.7/ [5] Lfaov, I.K. (996) Sgular Iegral Equao ad Dsree Vores. VSP, Lede. [6] Kyh, K.P. ad Shaferkoer, R.M. (5) Hadbook of Compuaoal Mehods for Iegrao. Chapma & Hall/ CRC Press, Lodo. 764