On the General Solution of First-Kind Hypersingular Integral Equations

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1 Amerca Joural of Egeerg ad Appled Sceces Orgal Research Paper O the Geeral Soluto of Frst-Kd Hypersgular Itegral Equatos Suza J. Obays, Z. Abbas, N.M.A. Nk Log, A.F. Ahmad, A. Ahmedov ad 3 Hader Khaleel Raad Departmet of Physcs, Uverst Putra Malaysa, Serdag, Malaysa Isttute for Mathematcal Research, Uverst Putra Malaysa, Serdag, Malaysa 3 Departmet of Physcs, Xaver Uversty, Ccat, Oho, USA Artcle hstory Receved: --5 Revsed: 3--5 Accepted: --6 Correspodg Author: Suza J. Obays Departmet of Physcs, Uverst Putra Malaysa, Serdag, Malaysa Emal: suza_ye@yahoo.com Abstract: A ew algorthm s preseted to provde a geeral soluto for a frst type Hyper sgular Itegral Equato (HSIE). The sgular tegral has bee coverted to a regular form by cacellg the sgularty ad the trasformg t to a system of algebrac equato based orthogoal polyomals. We appled the covergece method preseted by (Obays, 3) to coform the umercal soluto of the regularzed Hadamard equato, whch has show a absolute agreemet wth the eact soluto wth small values of. The proposed method has bee eamed by varous HSIEs where the dsplacemet fucto satsfes the Hӧlder-cotuous frst dervatve. Such equatos are partcularly mportat may physcs ad egeerg problems, such as fracture mechacs, acoustcs ad elastcty. Keywords: HSIE, Sgular Itegrals, Fredholm Itegral Equato, Chebyshev Polyomals Itroducto Basc Cocepts Oe of the most frequetly used umercal applcatos s the tegral equato methods due to ther ablty to resolve the strog sgulartes that arse stress felds where the boudary codtos chage type (Lfaov et al., 3; Iovae et al., 3; Obays et al., a). We preset a ew geeral soluto for the frst type HSIEs, whereas ths approach depeds o the aalytcal evaluato of the sgular tegral after the reducto for the hgher order sgularty. Itegral equatos arse may physcal problems, such as elastcty, acoustcs ad fracture mechacs whch requre the aalytcal soluto of Fredholm tegral equato (Mart ad Rzzo, 989; Mart, 99; Obays et al., 3): λ ( t, ) g( t) dt = f ( ), L () L where, λ s a square tegrable kerel of (t, ) that has a logarthmc sgularty o the dagoal t = : λ ( t, ) dtd = B < () L L If L s a pecewse smooth cotour that cludes the terval [-,], (.e.: Here we deal wth a equato wth Hlbert kerel). The regular part of the kerel λ ca be separated as sgular ad osgular parts by usg the decomposg Fourer trasform. Thus, by cosderg the segmet [-,]as a specal case of L, Equato becomes: K( t, ) g( t) dt + G( t, ) g( t) dt = f ( ), [,] (3) where, G(t,) s a regular kerel of t ad whle the frst kerel K(t,) has the form: h( ) K( t, ) =, α ( t ) α () Whch s called the hyper sgular kerel ad K(,) ad h(t) s ukow fucto to be determed. Furthermore α s the order of the tegral whch classfes ts sgularty. If α =, the Equato 3 becomes: 6 Suza J. Obays, Z. Abbas, N.M.A. Nk Log, A.F. Ahmad, A. Ahmedov ad Hader Khaleel Raad. Ths ope access artcle s dstrbuted uder a Creatve Commos Attrbuto (CC-BY) 3. lcese.

2 Suza J. Obays et al. / Amerca Joural of Egeerg ad Appled Sceces 6, 9 (): 95. DOI:.38/ajeassp h( ) g( t) dt ( t ) + G( t, ) g( t) dt = f ( ), [,] (5) The choce of g(t) Equato s sgfcat ad drectly affects the sgularty order where f g(t) represets the slope fucto, the Equato s a CPVI; whereas f t s chose as a dsplacemet fucto the a HSIE s performed. A deep lked eus of the dfferet ordered sgulartes ca be performed by the followg echageablty relatoshp (Mart ad Rzzo, 989): g( t) g( t) d g( t) dt= dt = dt ( t ) ( t ) α d ( t ) (6) α+ α α For the secod order sgularty where α=, the above equato becomes: g( t) d g( t) dt=, (,) ( ) dt t d t (7) The rght had sde of Equato 7 s called Cauchy Prcple Value Itegral (CPVI), whch represets the base for ourappromato ad defed by (Iovae et al., 3; Obays, 3; Obays et al., 3). Defto If a fucto g(t), t [-,], be ubouded at some pot (-,) ad Rema tegrable over [-, -ε) [-,) ad ( + ε,) (,], ε>. The CPVI of g over [-,] s defed by: = (8) Provded that ths lmt ests. The CPVI Equato 8 s a well-defed tegral wheever g s a mproper Rema tegrable o [-,], whlst a commo CPVI s the Hlbert trasform. The pot s called a weak sgularty of g (Prem ad Mchael, 5). It s mportat to kow that oce the CPVI has bee appromated, the HSIEs ca be obtaed successfully by the cosderato of the relatoshp Equato 6 ad t s defed as: Defto Lfaov et al. (3) suppose that g() s a real fucto defed o [-, ]. The: Prelmares Numerous epaso of more advaced ad effcet methods for the umercal soluto of tegral equatos have bee coducted (Madal ad Bhattacharya, 7; Helsg, ). The HSIE of the form Equato 5 where the regular kerel G(t,) =, h() = ad the dsplacemet fucto g satsfy the Hӧlder-cotuous frst dervatve, becomes: g( t) ( t ) dt= f ( ), () Obvously g represets the gap the velocty potetal of the flow across the plate. To esure the cotuty of the velocty at both edpots = ±, the followg boudary codto s mportat: g ( ± ) = () Let the ukow fucto g Equato be wrtte as: g( t) = t ϕ( t) () where, φ() s a well-defed fucto of t ad by substtutg Equato to Equato wth the use of the relatoshps Equato 7, yeld: ϕ ( ) ( t) d ϕ( t) f ( ) = t dt = t dt t d t To regularze the above CPVI, we add ad subtract the value of φ(), whch yelds: Sce: d ϕ( t) ϕ( ) ϕ( ) f ( ) = t + dt d t t g( t) g( t) g( ) dt= lm + dt ( t ) ( t ) ε ε (9) ε + ε ( / ( )) t t dt = Wth estg ad bouded lmts of tegrato. Results: 96

3 Suza J. Obays et al. / Amerca Joural of Egeerg ad Appled Sceces 6, 9 (): 95. DOI:.38/ajeassp d ϕ( t) ϕ( ) f ( ) = t dt ϕ( ) d t ' ϕ t + ϕ t ϕ ' ϕ ( t ) ( ) ( ) ( ) = t dt ( ) ϕ( ) ( ) ζ (, ) F = t t dt Where: ' ϕ ( t ) + ϕ( t) ϕ( ) ζ ( t, ) = ( t ) Ad: F f ' ( ) = ( ) + ϕ ( ) + ϕ( ) (3) ( ) ζ ( ) g t C (6) Ad the codto: g ( ) t dt = (7) Is desrable for the uque soluto of g whle C, =,,,,, are the ukow coeffcets to be determed, the Equato 5 becomes: = f ( ) t dt ζ ( t) h( ) t dt ( t ) C + t G( t, ) ζ ( ) (8) The above tegral Equato 3 s a regularzato of Equato after subtractg ts sgularty where: Both hyper sgular ad regular kerels Equato 8 are appromated as follows: '' ζ ( t, ) ϕ ( )/ whe t = where, ζ s a real fucto belog to the class of Hӧlder o the set [-,] [-,]. It s kow that HSIE Equato has the followg eact soluto (Madal ad Bhattacharya, 7): t f( t)l dt, f f ( ), g( ) = t ( )( t ) () f f ( ) =. Appromato I computg tegrals, there are several approprate choces to select from. We developed a geeral appromatg method for the bouded soluto of ay HSIE of the form Equato 5, where h() = ad G(t,). Whereas the hyper sgular kerel K(t,), the desty fucto g ad the regular kerel G(t,) are supposed to be real fuctos belog to Hӧlder class o the sets [-,] [-,],[-,] ad [-,] [-,], respectvely. The fucto φ() Equato appromated by a fte sum of a approprate polyomal of degree : ϕ( ) Cζ ( ) (5) Whch meas: ( t ) p s ρr ζ r q ζ q r= q= ( ) ( t), G( t, ) k ( ) ( t) (9) where, ρ r () ad k q () are kow epressos of ad by substtute Equato 9 to Equato 8, yelds: CQ( j) = f ( j), [,] () = Where: p h( ) ρ ( ) ζ ( t) r r+ r= Q( ) = t dt s + kq( ) ζ q+ ( t) q= () Ths approach reduces the tegral equato problem Equato 5 to a fte lear algebrac system of + lear equatos wth + ukow coeffcets C of the form: CQ( j) = f ( j), j =,,..., () = where, j to be chose as the root of the polyomal j o [-,],.e. = j ad the coeffcets { } (Obays, 3): C = satsfy Kg + Gg f, gk( ) =, k =,,,... (3) 97

4 Suza J. Obays et al. / Amerca Joural of Egeerg ad Appled Sceces 6, 9 (): 95. DOI:.38/ajeassp where, K ad G are the sgular ad regular kerels defed Equato 3 respectvely ad the calculato of C edorses the evaluato of g (t) Equato 6. For the error estmate of the HSIE of the frst kd Equato 5, t s prove (Obays, 3) that f the l l fuctos f C ([,]) ad K( t, ) C ([,] [,]), the: l g g = C, l (), w Wth w=. If l ca be chose to be ay large postve umber, the the error Equato decreases very quckly ad the covergece s very fast to the eact soluto. The umercal eamples perform that for ay sgular pot [-,] the sequece {g } coverge uformly L,w orm to {g} as creases (Obays, 3). Numercal Eamples Eample If we cosder h() = ad G(t,) = ad f() =, the Equato 5 takes the form: g( t) dt, (5) = ( t ) Whch have the eact soluto Equato. The fucto ζ() s calculated umercally from the relatoshp Equato 6 ad g() s appromately obtaed. We ca easly show the proof by cosderg ζ () Equato as Chebyshev polyomal of the secod kd over the terval [-,]. It s kow that orthogoal polyomals have a great varety ad wealth of propertes whch playa great role ad mportace for quadrature methods as well as for the soluto of mathematcal ad physcal problems wth a very effcet terpolato formula. The zeros of these polyomals are usually costat real, dstct ad belog to a partcular terval of [-,] whch coform wth our research terest. These systems of zeros are used as odes of the quadrature rules, whch possess addtoal propertes, lke that of postvty ormmalty of quadrature error. Oe ca also obta polyomals very close to the optmal oe by epadg the gve fucto terms of Chebyshev polyomals ad the cuttg off the epaso at the desred degree. The secod kd Chebyshev polyomal s defed as (Maso ad Hadscomb, 3; Obays et al., b): s( + ) θ U( ) =, cosθ = (6) sθ Where: U = U = U = (7) ( ), ( ), ( ) Whereas the rest of the terms satsfy the followg recurrece relatoshp: U ( ) = U ( ) U ( ) (8) Ad by takg the followg cases for ρ r () ad for ρ q () Equato 35 defed below, as follows: ρ ( ), ( ) ( ) = ρr = r >, & Gq = q The C = ad C =, > ad by substtutg to Equato for ay value of gves: g( ) = Whch performs Equato. Table shows that the appromate solutos cocde wth the eact values. Eample Cosder the followg HSIE: g( t) dt cos( ) ( ) + ( t ) = 5(6 + ) t g t dt cos( ) 3 (9) Wth the codto g(±) =. The eact soluto of Equato 9 s: g( ) = (6 + ) The ukow fucto g Equato 9 s appromated by usg the fte sum of Chebyshev polyomal of the secod kd defed by (6), gves: ( ) ( ) g t = t CU t (3) Table. The eact ad appromate solutos of Equato 5 = Eact Appromate ±.... ± ±..9.9 ± ±

5 Suza J. Obays et al. / Amerca Joural of Egeerg ad Appled Sceces 6, 9 (): 95. DOI:.38/ajeassp Substtute Equato 3 to Equato 7 gves: C t U( t) dt = (3) It s ot dffcult to show that (Maso ad Hadscomb, 3): t U( t) dt s( ) θ sθdθ ; = (33) = + = ; Ad by usg Equato 33 to Equato 3, we obta the value of the frst coeffcet: C = (3) The hyper sgular ad regular kerels Equato 9 are appromated as follows (Obays et al., b): ( t ) ( r + ) U ( ) U ( t), s r= G( t, ) k ( ) U ( t) q q= p q r r (35) where, Ur( ) ad kq( ) are kow epressos of ad by substtutg Equato 35 to Equato 9 ad usg the orthogoal property, yelds: C ( + ) U( ) + k( ) = cos( ) =5( 6 + ) 3 Ad: ( ) = (, ) ( ) k (36) t G t U t dt (37) The lear system Equato 37 ca be evaluated aalytcally or umercally usg a quadrature formula. By choosg the roots of T ( ) + as the collocato pots j alog the terval [-,], whch are: j j = cos, j =,,..., ( + ) For =, we get: k ( ) = cos( )/8, k ( ) =, k ( ) = 3cos( )/6 k ( ) =, k ( ) = cos( )/6, k ( ) = ; 5 3 (38) (39) Substtutg the values of Equato 37 to the system of Equato 36 for =, gves: C ( + ) U( j) + k( j) = cos( ) j =5( 6j j + ) 3 () By solvg the system Equato for the ukow coeffcets C; =,,..., ad substtutg the values of C to Equato 3, we obta the umercal soluto of Equato 9, whch s detcal to the eact soluto. MATLAB codes are developed to obta all the umercal results of Equato 9 where Table presets the umercal epermet whch perfectly agrees wth the theoretcal results. Eample 3 Cosder the umercal soluto of Fredholm tegral equatos of the frst kd wth a double pole sgularty of the followg form: g( t) dt + tg( t) dt = ( t ) () , The eact soluto of Equato s: g( ) = + 3 () We use the same steps eplaed Eample where the roots are defed Equato 38 ad = : C ( + ) U( ) + k( ) = (3) Table. The error of the umercal soluto of Equato 9 = = 6 -.e + -.e e e -.6.8e e -..6e e -..e 7 -..e..8e 7..33e. 3.86e e..85e 7..59e.6.8e 7.6.9e e 7.8.6e.e +.e + 99

6 Suza J. Obays et al. / Amerca Joural of Egeerg ad Appled Sceces 6, 9 (): 95. DOI:.38/ajeassp Table 3. The error of the umercal soluto of Equato 9 = = 3 -.e + -.e e e e e e e e e 3..89e 5..e e e 3..85e e e e 3.8.8e e 3.e +.e + The errors of the umercal solutos preseted Table 3 are computed as the absolute value of the dfferece betwee the eact ad umercal solutos. Here, the smple soluto of the lear system of equatos provded a more effcet appromato ad faster algorthm by creasg the values of. Cocluso A geeral soluto for ay HSIE problem of the form Equato 5 s preseted. We reformulated the ma tegral problem as a set of lear algebrac equatos that ca be solved by applyg the usual collocato method. Moreover, ths umercal techque provded a effcet appromato algorthm that coverge very fast to the eact soluto eve for small values of. It ca also be see from the results that the error values based o Equato decrease very fast for ay sgular pot [,] by creasg the values of ad choosg a approprate weght fucto. MATLAB codes are developed to obta all the umercal results for dfferet kerel fuctos ad force fuctos f() where the umercal epermets agree wth the theoretcal results. Ackowledgemet The frst author would lke to thak the Uverst Putra Malaysa (UPM) for ther support ad provdg faclty accomplsh ths work. Author s Cotrbutos Suza J. Obays: Aalyzg, dervg ad wrtg of the mauscrpt. Z. Abbas: Revewg ad aalysg the mauscrpt. N.M.A. Nk Log: Checkg ad aalyzg the derve. A.F. Ahmad: Desg ad data aalyss. A. Ahmedov: Aalyze methematcal derve. Hader Khaleel Raad: Gve the fal revew ad approval for the mauscrpt to be submtted. Ethcs Ths artcle s orgal ad cotas upublshed materal. The correspodg author cofrms that all of the other authors have read ad approved the mauscrpt ad o ethcal ssues volved. Refereces Helsg, J.,. A fast stable solver for sgular tegral equatos o pecewse smooth curves. SIAM J. Sc. Comput., 33: DOI:.37/97798 Iovae, G., I.K. Lfaov ad M.A. Sumbatya, 3. O drect umercal treatmet of hypersgular tegral equatos arsg mechacs ad acoustcs. Actamechaca, 6: 99-. DOI:.7/s Lfaov, I.K., L.N. Poltavsk ad M.M. Vakko, 3. Hypersgular Itegral Equatos ad ther Applcatos. st Ed., CRC Press, Lodo, ISBN-: 36, pp: 8. Madal, B.N. ad S. Bhattacharya, 7. Numercal soluto of some classes of tegral equatos usg Berste polyomals. Appled Math. Comput., 9: DOI:.6/j.amc Mart, P.A., 99. Eact Soluto of a smple hypersgular tegral equato. J. Itegral Equat. Applc., : 97-. DOI:.6/jea/87568 Mart, P.A. ad F.J. Rzzo, 989. O boudary tegral equatos for crack problems. Proceedgs of the Royal Socety of Lodo A: Mathematcal, Physcal ad Egeerg Sceces, (PES 89), The Royal Socety, pp: Maso, J.C. ad D.C. Hadscomb, 3. Chebyshev Polyomals. st Ed., CRC Prress, ISBN-: 36, pp: 36. Obays, S.J., 3. O the covergece problem of oedmesoal hypersgular tegral equatos. Math. Problems Eg., 3: DOI:.55/3/9775 Obays, S.J., Z. Eshkuvatov ad N.N. Log, 3. O error estmato of automatc quadrature scheme for the evaluato of Hadamard tegral of secod order sgularty. UPB Scetfc Bull. Seres A: Appled Math. Phys., 75: Obays, S.J., Z. Eskuvatov ad N.N. Log, a. A Chebyshev polyomal based quadrature for hypersgular tegrals. AIP Cof. Proc., 5: 39-. DOI:.63/.73

7 Suza J. Obays et al. / Amerca Joural of Egeerg ad Appled Sceces 6, 9 (): 95. DOI:.38/ajeassp Obays, S.J., Z. Eskuvatov, N.N. Log ad M.A. Jamalud, b. Galerkmethod for the umercal soluto of hypersgular tegralequatos based Chebyshev polyomals. It. J. Math. Aal., 6: Prem, K. ad R. Mchael, 5. Hadbook of Computatoal Methods for Itegrato. st Ed., Chapma ad Hall, CRC Press, Boca Rato, ISBN-: 58888, pp: 6.