SHIFTED JACOBI COLLOCATION METHOD BASED ON OPERATIONAL MATRIX FOR SOLVING THE SYSTEMS OF FREDHOLM AND VOLTERRA INTEGRAL EQUATIONS

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1 Mathematcal and Computatonal Applcatons, Vol., o., pp , 5 SHIFED JACOBI COLLOCAIO MEHOD BASED O OPERAIOAL MARIX FOR SOLVIG HE SYSEMS OF FREDHOLM AD VOLERRA IEGRAL EQUAIOS Abdollah Borhanfar and Khadeh Sadr Department of Appled Mathematcs, Faculty of Mathematcal Scences Mohaghegh Ardabl Unversty, Ardabl, Iran borhan@uma.ac.r, kh.sadr@uma.ac.r Abstract- hs paper ams to construct a general formulaton for the shfted Jacob operatonal matrces of ntegraton and product. he man am s to generalze the Jacob ntegral and product operatonal matrces to the solvng system of Fredholm and Volterra equatons. hese matrces together wth the collocaton method are appled to reduce the soluton of these problems to the soluton of a system of algebrac equatons. he method s appled to solve system of lnear and nonlnear Fredholm and Volterra equatons. Illustratve eamples are ncluded to demonstrate the valdty and effcency of the presented method. Also, several theorems, whch are related to the convergence of the proposed method, wll be presented. Key ords- Collocaton Method, Shfted Jacob Polynomals, System of Fredholm and Volterra Integral Equatons; Convergence; Operatonal Integral and Product Matrces. IRODUCIO Fndng the analytcal solutons of functonal equatons has been devoted of attenton of mathematcans nterest n recent years. Several methods are proposed to acheve ths purpose, such as [-]. Mathematcal modelng for many problems n dfferent felds, such as engneerng, chemstry, physcs and bology, leads to ntegral equatons or system of ntegral equatons. Several methods have been proposed to solve these problems. For eample, Varatonal teraton method [], dfferental transform method [], ystrom method [], Haar functons method [], Homotopy perturbaton method [3], Chebyshev wavelet method [4] and many others. Between of present methods, spectral methods have been used to solve dfferent functonal equatons, because of ther hgh accuracy and easy applyng. Specfc types of spectral methods that more applcable and wdely used, are the Galerkn, collocaton, and tau methods [5-9]. Saadatmand and Dehghan ntroduced shfted Legendre operatonal matr for fractonal dfferental equatons [3], Doha derved a new eplct formula for shfted Chebyshev polynomals for fractonal dfferental equatons [3], Bhrawy used a quadrature shfted Legendre tau method for fractonal dfferental equatons [3]. Recently, Doha ntroduced shfted Chebyshev operatonal matr and appled t wth spectral methods for solvng problems to ntal and boundary condtons [33]. he mportance of Sturm-Louvlle problems for spectral methods les n the fact that the spectral appromaton of the soluton of a functonal equaton s usually regarded as a fnte epanson of egenfunctons of a sutable Sturm-Louvlle problem.

2 Shfted Jacob Collocaton Method Based on Operatonal Matr 77 he Jacob polynomalsp ( )(,, ) play mportant roles n mathematcal analyss and ts applcatons [34]. It s proven that Jacob polynomals are precsely the only polynomals arsng as egenfunctons of a sngular Sturm-Louvlle problem [35-36]. hs class of polynomals comprses all the polynomal soluton to sngular Sturm-Louvlle problems on [,]. Chebyshev, Legendre, and ultrasphercal polynomals are partcular cases of the Jacob polynomals. In ths paper, the shfted Jacob operatonal matrces of ntegraton and product s ntroduced, whch s based on Jacob collocaton method for solvng numercally the systems of the lnear and nonlnear Fredholm and Volterra ntegral equatons on the nterval [,], to fnd the appromate soluton u ( ). he each of equaton of the systems resulted are collocated at ( ) nodes of the shfted Jacob- Gauss nterpolaton on(,). hese equatons generate n ( ) lnear or nonlnear algebrac equatons. he nonlnear systems resulted can be solved usng ewton teratve method. he remander of ths paper s organzed as follows: he Jacob polynomals and ther ntegral and product operatonal matrces to ntegral equatons are obtaned n Secton. Secton 3 s devoted to applyng the Jacob operatonal matrces for solvng system of ntegral equatons. In Secton 4, the proposed method s appled to several eamples. A concluson s presented n Secton 5.. JACOBI POLYOMIALS AD HEIR OPERAIOAL MARICES.. Propertes of shfted Jacob polynomals he Jacob polynomals, assocated wth the real parameters (, ) are a sequence of polynomals P ( t),(,,,...), each of degree, are orthogonal wth Jacob weghted functon, w( ) ( ) ( ) over I [,], and P ( t) P ( t) w( t) dt h, n m n nm where s Kroneker functon and h nm n ( n ) ( n ) (n ) n! ( n ) hese polynomals can be generated wth the followng recurrence formula; ( ) t( )( ) P ( t) P ( t) ( )( ) ( )( )( ) P ( ),, 3,..., t ( )( ) where, P ( t ), and P ( t) t..

3 7 A. Borhanfar and K. Sadr In order to use these polynomals on the nterval [,], shfted Jacob polynomals are defned by ntroducng the change of varable t. Let the shfted Jacob polynomalsp ( ) be denoted by from followng formula; R ( ), then R ( ) can be generated ( ) ( )( )( ) R ( ) R ( t) ( )( ) () ( )( )( ) R ( ),, 3,..., t ( )( ) where, R ( ), and R ( ) ( ). Remark. Of ths polynomals, the most commonly used are the shfted Gegenbauer polynomals, C ( ), the shfted Chebyshev polynomals of the frst knd, ( ), the S, S, shfted Legendre polynomals, P ( ), the shfted Chebyshev polynomals of the second S, knd, U ( ). hese orthogonal polynomals are related to the shfted Jacob S, polynomals by the followng relatons.! ( )! ( ) C ( ) R ( ), ( ) R ( ), S, S, ( ) ( ) ( )! ( ) (, ) P ( ) R ( ), U ( ) R ( ). S, S, 3 ( ) he analytc form of the shfted Jacob polynomals, R R (), s gven by k k ( ) ( ) ( k ) ( ), ( k ) ( )( k)! k! k Some propertes of the shfted Jacob polynomals are as follows, () R () ( ), () R () ( ), d ( n ) (, ) (3) ( ) ( ). R n R n d ( n ) he orthogonalty condton of shfted Jacob polynomals s R ( ) R ( ) ( ) d, (3) k k k ()

4 Shfted Jacob Collocaton Method Based on Operatonal Matr 79 where ( ), shfted weghted functon, s as follows, hk ( ) ( ), and. k Lemma. he shfted Jacob polynomal where Proof. p are ( n ) p ( n) p can be obtaned as, ( n) n Rn n ( n) n R ( ) p, n n ( ). n ( ) can be obtaned n the form of: ow, usng propertes () and (3) n above, the lemma can be proved. Lemma. For m, one has ( ) l l m R ( ) ( ) d p B( m l, ), where B( s, t ) s the Beta functon and s defned as Proof. Usng Lemma and ( n) d p R ( ). n! d s t ( s) ( t) ( ). B s t v v dv ( ) ( ) one has ( s t) m ( ) m l l l ( ) m l l l ( ) l l R ( ) ( ) d p ( ) d p ( ) d p B( m l, )... he appromaton of functons Let (,), and for r ( s the set of all non-negatve ntegers), the r weghted Sobolov space H ( ) s defned n the usual way and s denoted nner product, sem-norm and norm by ( uv, ), v and v, respectvely. In r, r, r, partcular, L ( ) H ( ), ( u, v ) ( u, v ) and v v,,,

5 A. Borhanfar and K. Sadr H ( ) f f s ntegrable and v, r r k ( k, k) r, k v v v r r, (, ) A functon u( ) H r ( ) can be epanded n (,, ). r, P span R ( ), R ( ),..., R ( ) as the below formula,, where the coeffcents c are gven by u( ) c R ( ), (4) c R ( ) u( ) ( ) d,,,,.... (5) By notng n practce, only the frst ( ) terms shfted polynomals are consdered, then one has where Snce from u( ) u ( ) c R ( ) ( ) C, (6) C [ c, c,..., c ], ( ) [ R ( ), R ( ),..., R ( )]. (7) (,, ) P s a fnte dmensonal vector space, u ( ) has a unque best appromaton (,, ) P, say (,, ) u () P, that s (,, ) y P, u( ) u ( ) u( ) y. In [37] s shown that for any u( ) H r ( ), r and r, a postve constant C ndependent of any functon,, and est that r, r, u ( ) u ( ) c ( ( )) u. () Let u ( ) s tmes contnuously dfferentable. he followng heorem can present an upper bound for estmatng the error. heorem. Let u( ) : [,] s tmes contnuously dfferentable for, and (,, ) P span R ( ), R ( ),..., R ( ). If u ( ) ( ) A s the best appromaton to u ( ) from follows: (,, ) P then the error bound s presented as

6 Shfted Jacob Collocaton Method Based on Operatonal Matr MS u( ) u ( ) B(, ), ( )! ( ) where M ma u ( ) and S ma,. [,] Proof. Let consder the aylor epanson ( ) ( ) y( ) u( ) u ( )( )... u ( )!. herefore u u u ( )! ( ) ( ) ( ) ( ), (,), Snce ( ) A s the best appromaton to u ( ) from P (,, ), and has (,, ) y () P one M ( ) ( ) ( ) ( ) ( ) ( ) ( ). (( )!) u u u y d () Snce s always postve n (,), by choosng S ma, one has ( ) ( ) M S M S ( ) ( ) ( ) (, ). (( )!) (( )!) u u d B hs error bound shows appromaton of polynomals converges to u ( ) as..3. he Jacob ntegral operatonal matr In ths subsecton, Jacob operatonal matr of the ntegraton s derved. Let ( t) dt P ( ), (9) where matr P s called the Jacob operatonal matr of the ntegraton. heorem 3. Let P s ( ) ( ) operatonal matr of ntegral. hen the elements of ths matr are obtaned as P p p B( m n, ),,,,.... ( ) ( ) m n m n m Proof. Usng Eq. (9) and orthogonalty property of Jacob polynomals one has, where ( ( t) dt, ( )) and follows, ( ( ), ( )), P t dt are two( ) ( ) matrces defned as

7 A. Borhanfar and K. Sadr Set t dt R t dt R, ( ( ), ( )) ( ( ), ( )), dag. ( R ( t) dt, R ( )) R ( t) dt R ( ) ( ) d. R () t dt and R ( ) by usng Lemma can be obtaned as m R ( t) dt p, m ( ) n n () m m n R ( ) p,,,,...,. herefore, by usng Lemma can be obtaned as follows, ( ) ( ) m n m n m n m ( ) ( ) p p B m n m n m n m So, the elements of matr P s obtaned as ( ) ( ) m n m n m p p ( ) d (, ). P p p B( m n, ),,,,.... ow the followng theorem can present an upper bound for estmatng the error of ntegral operator. he error vector E s defned as, where E ( t) dt P ( ) [ E, E,..., E ], k k k E R ( t) dt P R ( ), k,,...,. r heorem 4. If E R ( t) dt P R ( ) H ( ), then an error bound of ntegral operator of vector k k k can be epressed by k k r ( k) ( k) k, E c ( ( )) B( r 3, r ).

8 Shfted Jacob Collocaton Method Based on Operatonal Matr 3 Proof. By usng nequalty (), Lemma, and settng u( ) R ( t) dt one has where r, r () k ( r, r) k r ( k ) D p ( r, r) k! ( k ) r p ( r ) ( r, r) k k ( k ) r ( k ) r ( r, r) k k ( k ) ( k ) r r ( ) d k k ( k) ( k) B r r u D R t dt! p ( r ) ( k) ( k) ( )( ) ( ) d ( 3, ), and the theorem can be proved..4 he product operatonal matr he followng property of the product of two Jacob functon vector wll also be appled to solve the Volterra ntegral equatons. ( ) ( ) Y Y ( ) () where Y s a ( ) ( ) product operatonal matr and t`s elements are determned n terms of the vector Y `s elements. Usng Eq.() and by the orthogonalty property of Jacob polynomals the elements Y can be calculated as follows, Y Y ( ( )) ( ( )) ( ( )) ( ) d k k k Y h Y R ( ) R ( ) R ( ) ( ) d k k k k where h R ( ) R ( ) R ( ) ( ) d. k k k, k

9 4 A. Borhanfar and K. Sadr 3. APPLICAIOS OF HE OPERAIOAL MARICES OF IEGRAIO AD PRODUC In ths secton, the presented operatonal matrces are appled to solve the system of lnear and nonlnear Fredholm and Volterra ntegral equatons. 3.. he system of Fredholm ntegral equatons A system of Fredholm ntegral equatons can be presented as follows; m u ( ) f ( ) k (, t) G ( u ( t), u ( t),..., u ( t)) dt,,,,..., n, () n where k (, t) L [,] [,], f are known functons, and G are lnear or nonlnear functons n terms of unknown functons u ( ), u ( ),..., and u ( ). o solve system n (), the functons u ( ), G () t and k (, t) can be appromated as follows, u ( ) c R ( ) ( ) C, G ( t) ( t) Y, k (, t) ( ) K ( t), where K and vectors respectvely and Y are ( ) ( ),,..., n,,,..., m, () known matrces and ( ) unknown C [ c, c,..., c ], ( ) [ R ( ), R ( ),..., R ( )]. th substtutng appromatons (3) n system () one has m ( ) C f ( ) ( ) K ( t) ( t) Y dt m f ( ) ( ) K ( t) ( t) dt Y m f ( ) ( ) K DY,,,..., n, where D s the followng ( ) ( ) known matr, D ( t) ( t) dt. he system (3) have n ( ) unknown coeffcents (3) c. For collocatng, ( ) roots of Jacob polynomals R ( ) are appled and the equatons are collocated at ( ) them. Unknown coeffcents are determned wth solvng the resulted system of lnear or nonlnear algebrac equatons. 3.. System of Volterra ntegral equatons A system of Volterra ntegral equatons of the frst knd can be presented as follows,

10 Shfted Jacob Collocaton Method Based on Operatonal Matr 5 m f ( ) k (, t) G ( u ( t), u ( t),..., u ( t)) dt,,,,..., n. (4) n By usng the appromate relatons () one has m ( ) ( ) ( ) ( ) m ( ) ( ) ( ) m ( ) K Y ( t) dt m f K t t Y dt K t t Y dt ( ) K Y P ( ),,,..., n, where Y and P are product and ntegral operatonal matrces, respectvely. Also, a system of Volterra ntegral equatons of the second knd can be presented as m u ( ) f ( ) k (, t) G ( u ( t), u ( t),..., u ( t)) dt,,,,..., n. (6) n In the same way, the system of followng equatons s resulted. m (5) ( ) C f ( ) ( ) K Y P ( ),,,..., n, (7) By usng the frst ( ) roots of Jacob polynomals (7), unknown coeffcents c are determned. R ( ) and collocated system ( ) 4. ILLUSRAIVE EXAMPLES In ths secton, the presented method s appled to solve some eamples. Comparson between the results of present method wth the correspondng analytc solutons s gven. For ths purpose, the mamum of absolute error s computed. Eample. he followng system of lnear Volterra ntegral equatons of the second knd s consdered, t ( 3) u ( ) f ( ) ( t) u ( t) dt ( t ) u ( t) dt t u ( t) dt, ( 3 ) u ( ) f ( ) t( ) u ( t) dt t( ) u ( t) dt ( t ) u ( t) dt, () (3 6) u ( ) f ( ) ( t ) u ( t) dt t( t ) u ( t) dt (t ) u ( t) dt,

11 6 A. Borhanfar and K. Sadr he eact solutons are u ( ) 5, u ( ) 5 and u ( ). By 3 the applyng the technque descrbed n pervous secton wth 4, solutons and kernels are appromated as: u ( ) ( ) C, u ( ) ( ) C, u ( ) ( ) C, t ( ) K ( t ), 3 3 t K t t K t t K t 3 4 ( ) ( ), ( ) ( ), ( ) ( ) ( ), t( ) ( ) K ( t), t ( ) K ( t), t ( ) K ( t ), t( t ) ( ) K ( t), t t ( ) K ( t ). 3 9 he system () by usng above equatons s rewrtten as, ( 3) ( ) C ( ) K C K C K C P ( ), 3 3 ( 3 ) ( ) C ( ) K C K C K C P ( ), (3 6) ( ) C ( ) K C K C K C P ( ), where C, C and C are the operatonal matrces of product correspondng to unknown 3 vectors C, C and C. ow usng the roots of R ( ) and collocatng the system 3 5 (), reduces the problem to solve a system of algebrac equatons. Unknown coeffcents are obtaned for some values of parameters and. Mamum absolute error for 4 and dfferent values of and has been lsted n able. () able. Mamum absolute error for 4 and dfferent values of and for Eample. Error( ) Error( ) Error( u ) 4.33 / / 7.7 / /.7 / / 6. / / 4.63 / 4 / /. / 3/ 3. u u able shows that a good appromaton can be acheved for the eact solutons by usng a few terms of shfted Jacob polynomals for varous values of parameters and.

12 Shfted Jacob Collocaton Method Based on Operatonal Matr 7 Eample. Consder the followng system of lnear Fredholm ntegral equaton of second knd, u ( ) 3 u ( ) f ( ) ( t) u ( t) dt t u ( t) dt, 3 u ( ) 4 u ( ) f ( ) (t ) u ( t) dt ( t) u ( t) dt, he eact solutons are u () e and u (). th 5, solutons and kernels are appromated as: u ( ) ( ) C, u ( ) ( ) C, t ( ) K ( t), t ( ) K ( t ), () 3 4 t ( ) K ( t), t ( ) K ( t), D ( t) ( t) dt. he system () by usng above equatons s rewrtten as, ( ) C 3 ( ) C ( ) K DC ( ) K DC f ( ), ( ) C 4 ( ) C ( ) K DC ( ) K DC f ( ). () ow usng the roots of R ( ) and collocatng the system (), reduces the problem 6 to solve a system of lnear algebrac equatons. Solvng the system (), the unknown coeffcents wll be obtaned. able dsplays the mamum absolute errors for varous and wth 5. able. Mamum absolute error for 5 and dfferent values of and for Eample. Error( ) Error( u ) 5. / / / /.95 5 / / / 4 / / / / 4. u

13 A. Borhanfar and K. Sadr Eample 3. Consder the followng system of lnear Fredholm ntegral equaton of second knd, u ( ) f ( ) t cos( ) u ( t) dt c sn( t) u ( t) dt, t ( ) ( ) ( ) ( ) ( ). u f e u t dt t u t dt he eact solutons are u ( ) and u ( ) cos( ). th 5, solutons and () kernels are appromated as: u ( ) ( ) C, u ( ) ( ) C, t cos( ) ( ) K ( t), sn( t) ( ) K ( t), ep( t ) ( ) K ( t), ( t) ( ) K ( t), D ( t) ( t) dt. 3 4 he system () by usng above equatons s rewrtten as, ( ) C ( ) K DC ( ) K DC f ( ), 3 4 ( ) C ( ) K DC ( ) K DC f ( ). (3) ow usng the roots of R ( ) and collocatng the system (3), reduces the problem 6 to solve a system of lnear algebrac equatons. Solvng the system (3), the unknown coeffcents wll be obtaned. able 3 dsplays the mamum absolute errors for varous and wth 5. able 3. Mamum absolute error for 5 and dfferent values of and for Eample 3. Error( ) Error( u ).63 / /.966 / / / /.57 / 4 / 4.5 / / / u

14 Shfted Jacob Collocaton Method Based on Operatonal Matr 9 Eample 4. In ths eample, the followng nonlnear Volterra ntegral equaton of frst knd s consdered, ( u ( t) ( t) u ( t) u ( t)) dt f ( ), ( u ( t) ( t) u ( t) u ( t)) dt f ( ). he eact solutons are () u e and () u e. th 5, solutons and kernels are appromated as: u ( ) ( ) C, u ( ) ( ) C, ( ) K ( t), t ( ) K ( t), u ( ) u ( ) C C ( ), he system (4) by usng above equatons s rewrtten as, ( ) K C K E P ( ) f ( ), ( ) K C K E P ( ) f ( ), where C, C and E are the operatonal matrces of product correspondng wth the vectors C, C and E C C, respectvely. ow usng the roots of R ( ) and collocatng the each equaton of system (5), reduces the problem to solve a system of nonlnear algebrac equatons. Solvng the system (5) by ewton teratve method, the unknown coeffcents wll be obtaned. able 4 dsplays the mamum absolute errors for varous and wth. able 4. Mamum absolute error for and dfferent values of and for Eample 4. Error( ) Error( u ) / / 5.4 / / 7.76 / /.67 / 4 / / / / u (4) (5) Eample 5. Ffth eample covers the system of nonlnear Volterra ntegral equaton of second knd,

15 9 A. Borhanfar and K. Sadr f ( ) u ( t) u ( t) dt u ( ), f ( ) u ( t) dt u ( t) dt u ( ). he eact solutons are u ( ) cos( ) and u ( ) sn( ). Solutons and kernels are appromated as: u ( ) ( ) C, u ( ) ( ) C, ( ) K ( t ), (6) u ( ) u ( ) C C ( ), u ( ) C C ( ), u ( ) C C ( ), he system (6) by usng above equatons s rewrtten as, ( ) C ( ) K E P ( ) f ( ), 3 ( ) C ( ) K E E P ( ) f ( ), where E, E and E are the operatonal matrces of product correspondng wth the 3 vectors, CC (7) CCand E C C, respectvely. he mamum absolute errors for and 7,,5 are lsted n able 5. Also, usng the roots of R ( ) 6 and collocatng the each equaton of system (7), reduces the problem to solve a system of nonlnear algebrac equatons. Solvng the system (7) by ewton teratve method, the unknown coeffcents wll be obtaned. able 6 dsplays the mamum absolute errors for varous and wth 5. able 5. Mamum absolute error for 7,,5 and for Eample 5. Error( ) u u Error( ) able 6. Mamum absolute error for 5 and dfferent values of and for Eample 5. Error( ) Error( u ) / / / / / / / 4 / u

16 Shfted Jacob Collocaton Method Based on Operatonal Matr 9 5. COCLUIO In ths paper, the shfted Jacob collocaton method was employed to solve a class of systems of Fredholm and Volterra ntegral equatons of frst and second knds. Frst, a general formulaton for the Jacob operatonal matr of ntegral has been derved. hs matr s used to appromate numercal soluton of system of lnear and nonlnear Volterra ntegral equatons. Proposed approach was based on the shfted Jacob collocaton method. he solutons obtaned usng the proposed method shows that ths method s a powerful mathematcal tool for solvng the ntegral equatons. Provng the convergence of the method, consstency and stablty are ensured automatcally. Moreover, only a small number of shfted Jacob polynomals are needed to obtan a satsfactory result. 6. ACKOLEDGEMES e would lke to epress our grattude to the anonymous revewer for the careful readng of the manuscrpt and for hs nvaluable comments that helped us to mprove t consderably. 7. REFERECES. A. Borhanfar and M. M. Kabr, ew perodc and solton solutons by applcaton of applcaton of ep-functon method for nonlnear evoluton equaton, Computatonal and Appled Mathematcs, 9, 5-67, 9.. A. Borhanfar, M. M. Kabr and L. Maryam Vahdat, ew perodc and solton wave solutons for the generalzed zakharov system and (+)-dmensonal nzhnk-novkovveselov system, Chaos, Soltons and Fractals, 4, , A. Borhanfar and M. M. Kabr, Solton and perodc solutons for (3+)-dmensonal nonlnear evoluton equatons by ep-functon method, Applcaton and Appled Mathemahcs, 9, 5-67, A. Borhanfar and A. zamr, Applcaton of G / G epanson method for the Zhber -Shabat equaton and other related equatons, Mathematcal and Computer Modellng, 54, 9-6,. 5. A. Borhanfar, A. zamr and M. M. Kabr, Eact travellng wave solutons for the generalzed shallow water wave (GS) equaton, Mddle-East Journal of Scentfc Research, (3), 3-35,. 6. A. Borhanfar, M. M. Kabr and R. Abazar, Applcaton of G / G epanson method to regularzed (RL) equaton, Computer and Mathematcs wth Applcatons, 6, 44-47,. 7. A. Borhanfar and R. Abazar, General soluton of generalzed (+)-dmensonal Kadomtsev-Petvashvl (KP) equaton by usng the G / G epanson method, Journal of Computer Mathematcs,, -5,.. A. Borhanfar, H. Jafar and S. A. Karm, ew soltary wave solutons for generalzed and regularzed long-wave equaton, Internatonal Journal of Computer Mathematcs, 7 (3), 59-54,.

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