Quarter-Sweep Arithmetic Mean (QSAM) Iterative. Method for Second Kind Linear Fredholm Integral. Equations

Transcription

1 Aled Mathematcal Sceces Vol. 4 o Quarter-Swee Arthmetc Mea () Iteratve Method for Secod d ear Fredholm Itegral Equatos M. S. Muthuvalu School of Scece ad Techology Uverst Malaysa Saah oced ag ota aalu Saah Malaysa J. Sulama School of Scece ad Techology Uverst Malaysa Saah oced ag ota aalu Saah Malaysa umat@ums.edu.my Astract I ths aer we cosder the umercal solutos of lear Fredholm tegral equatos of the secod d. The Quarter-Swee Arthmetc Mea () teratve method s aled to solve lear systems geerated from dscretzato of the secod d lear Fredholm tegral equatos usg quadrature method. I addto the formulato ad mlemetato of the roosed method to solve the rolem are also reseted. Numercal eamles ad comarsos wth other estg methods are gve to llustrate the effectveess of the roosed method. Mathematcs Suect Classfcato: 4A55 45A F 65Y eywords: ear Fredholm equatos Quarter-swee terato Quadrature Arthmetc Mea

2 944 M. S. Muthuvalu ad J. Sulama Itroducto Cosder the soluto of lear system A () C s a osgular matr ad where A ad are vectors of order. The classcal teratve methods roceed y solvg at each ste a smler lear system duced y a slttg of coeffcet matr A to A M N (M s osgular). The the assocated teratve scheme s of the form ( ) ( ) M N M () where M N s called the terato matr of the method. It s well ow that () coverges for all vectors ( ) to the soluto of () f ad oly f the sectral ρ M N [9]. radus ( ) < O the other had the two-stage teratve method whch s called er/outer teratve scheme was frst troduced y Nchols [6]. Two-stage teratve methods cosst of aromatg the lear system () y usg aother teratve rocedure (er teratos). More secfcally cosder the slttg M F G ad erform s ( ) er teratos each outer ste. Thus the resultg method s ( ) ( N ) ( ) ( ) ( ) s ( ) ( ) s F G ( F G) F Whe the umer of er teratos ( ). (3) s for a two-stage method s fed each of the outer stes we call the statoary two-stage method whle a ostatoary two-stage method s such that the umer of er teratos may chage wth the outer terato [5]. Actually there are may two-stage teratve methods ca e cosdered such as the Alteratg Grou Elct (AGE) [4] Iteratve Alteratg Decomosto Elct (IADE) [] Reduced Iteratve Alteratg Decomosto Elct (RIADE) [] loc Jaco [] ad Arthmetc Mea (AM) [] methods. The stadard AM method also amed as the Full-Swee Arthmetc Mea () method has ee modfed y comg the cocet of the half-swee teratve method ad the called as the Half-Swee Arthmetc Mea () method [3]. The cocet of the half-swee teratve method s troduced va the Elct Decouled Grou (EDG) teratve method for solvg two-dmesoal Posso equato []. I [4] Sulama et al. roosed aother varat of AM method Quarter-Swee Arthmetc Mea (). The teratve method s derved y comg the stadard AM wth quarter-swee teratve [7] method. The asc dea of the half- ad quarter-swee teratve methods s to reduce the \

3 Quarter-swee arthmetc mea teratve method 945 comutatoal comletes durg terato rocess. Sce the mlemetato of the half- ad quarter-swee teratos wll oly cosder early half ad quarter of all teror ode ots a soluto doma resectvely. I ths aer the erformace of the ad wll e vestgated solvg dese lear system geerated y the dscretzato of the secod d lear Fredholm tegral equatos usg quadrature method. Geerally lear secod d tegral equatos of Fredholm tye the stadard form ca e defed as follows ( y) ( y ( dt ( y) Γ Γ [ αβ ] (4) where the arameter erel ( y [ α β ] [ α β ] ad free term ( y) [ α β ] are gve ad (y) s the uow fucto to e determed. The erel fucto ( y s assumed to e asolutely tegrale ad satsfy other roertes that are suffcet to mly the Fredholm alteratve theorem. Theorem (Fredholm alteratve) [3] et χ e a aach sace ad let κ : χ χ e comact. The the equato ( κ ) has a uque soluto χ f ad oly f the homogeeous equato ( κ ) z has oly the trval soluto z. I such a case the oerator κ : χ χ oto Defto (Comact oerators) [3] et χ ad Υ e ormed vector sace ad let s comact f the set κ y y y has a ouded verse ( ) { } κ. κ : χ Υ e lear. The κ has comact closure Υ. Ths s equvalet to sayg that for every ouded sequece { y } χ the sequece { κ y } has a susequece that s coverget to some ots Υ. Comact oerators are also called comletely cotuous oerators. The outle of ths aer s orgazed followg way. I Secto the formulato of the full- half- ad quarter-swee quadrature aromato equatos wll e elaorated. The latter secto of ths aer wll dscuss the formulatos of the ad methods ad some umercal results wll e show fourth secto to assert the erformace of the teratve methods. esdes that aalyss o comutatoal comlety s metoed Secto 5 ad the cocludg remars are gve fal secto.

4 946 M. S. Muthuvalu ad J. Sulama Quadrature Aromato Equatos As afore-metoed a dscretzato scheme ased o method of quadrature was used to costruct aromato equatos for rolem (). Geerally quadrature method ca e defed as follows β α () t dt ( t ) ε ( ) where t ( ) tegrato terval [ α β ] ( ) ot deed o the fucto ( ad ( ) s the ascssas of the artto ots of the (5) s umercal coeffcets that do ε s the trucato error of Eq. (3). Meawhle Fg. shows the fte grd etwors order to form the full- halfad quarter-swee quadrature aromato equatos. a) ) c) Fgure : a) ) ad c) show dstruto of uform ode ots for the full- halfad quarter-swee cases resectvely. ased o Fg. the full- half- ad quarter-swee teratve methods wll comute aromate values oto ode ots of tye oly utl the covergece crtero s reached. The other aromate solutos at remag ots (ots of the dfferet tye) ca e comuted usg the drect method [ 7 3 4]. y alyg Eq. (5) to Eq. (4) ad eglectg the error ε ( ) lear equatos ca e formed for aromato values of ( () where h h h a system of as show Eq.

5 Quarter-swee arthmetc mea teratve method 947 A M O M M M [ ] T ad [ ] T. I order to facltate the formulato of the full- half- ad quarter-swee quadrature aromato equatos for rolem () further dscusso wll e restrcted oto reeated traezodal (RT) scheme whch s ased o lear terolato formula wth equally saced data. ased o RT scheme umercal coeffcet wll satsfy the followg relato otherwse h h (6) where the costat ste sze h s defed as follows h β α (7) ad s the umer of sutervals the terval [ ] β α. Meawhle the value of whch corresods to ad 4 reresets the full- half- ad quarter-swee cases resectvely. 3 Arthmetc Mea teratve methods As stated revous secto AM methods are oe of the two-stage teratve methods ad the teratve rocess volves of solvg two deedet systems such as ad. To develo the formulato of AM methods let the coeffcet matr A e eressed as the matr sum U D A (8) where D ad U are the strctly lower tragular dagoal ad strctly uer tragular matrces resectvely. Thus y addg ostve accelerato arameter the geeral scheme for ad methods s defed y

6 948 M. S. Muthuvalu ad J. Sulama ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( D U D U D D (9) where () s a tal vector aromato to the soluto ad < <. The AM methods requre a slght addtoal comutatoal effort of the sum of two matrces at each terato ut ts rate of covergece s relatvely sestve to the eact choce of the arameter []. Practcally the value of wll e determed y mlemetg some comuter rograms ad the choose oe value of where ts umer of teratos s the smallest. y determg values of matrces D ad U as stated Eq. (8) the geeral algorthm for ad teratve methods to solve rolem (4) would e geerally descred Algorthm. Algorthm : ad schemes ) evel () For ad Calculate ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) others ) evel () For ad Calculate ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) others ) For

7 Quarter-swee arthmetc mea teratve method 949 Calculate ( ) ( ) The ad algorthms are elctly erformed y usg all equatos at level () ad () alteratvely utl the secfed covergece crtero s satsfed. Geerally the asc dea for the covergece aalyss of the AM methods has ee rove y []. 4 Numercal Results I order to comare the erformaces of the teratve methods descred the revous secto several umercal eermets were carred out o the followg two Fredholm tegral equatos. Eamle [5] Cosder the Fredholm tegral equato of the secod d ( y) y (4yt y ) ( dt y () ad the eact soluto s gve y ( y) 4y 9y. Eamle [8] Cosder the Fredholm tegral equato of the secod d ( y 6 3 ( y) y 5y y t ) ( dt y () wth the eact soluto ( y) y 5y y y There are three arameters cosdered umercal comarso such as umer of teratos eecuto tme ad mamum asolute error. As comarsos the Full-Swee Gauss-Sedel () method acts as the cotrol of comarso of umercal results. Throughout the smulatos the covergece test cosdered the tolerace error ε. The smulatos were carred out o several mesh szes ad 893. Results of umercal smulatos whch were otaed from mlemetatos of the ad teratve methods for Eamles ad have ee recorded Tales ad resectvely.

8 95 M. S. Muthuvalu ad J. Sulama Tale : Comarso of a umer of teratos eecuto tme ad mamum asolute error for the teratve methods at otmum value of (Eamle ) Numer of teratos Mesh sze Eecuto tme (secods) Mesh sze Mamum asolute error Mesh sze E E E-6.73 E E E E-4.73 E E E E-4.73 E E E E E-3.83 E E E E-5 5 Comutatoal Comlety Aalyss I order to measure the comutatoal comlety of the ad methods a estmato amout of the comutatoal wor requred for teratve methods have ee coducted. The comutatoal wor s estmated y cosderg the arthmetc oeratos erformed er terato. ased o Algorthm t ca e oserved that there are 7 addtos/sutractos (ADD/SU) 4 ad 9 multlcatos/dvsos (MU/DIV) comutg a value for each ode ot the soluto doma for secod d lear Fredholm tegral equatos. From the order of the coeffcet matr A the total umer of arthmetc oeratos er terato for the ad teratve methods for solvg Eq. (4) has ee summarzed Tale 3.

9 Quarter-swee arthmetc mea teratve method 95 Tale : Comarso of a umer of teratos eecuto tme ad mamum asolute error for the teratve methods at otmum value of (Eamle ) Numer of teratos Mesh sze Eecuto tme (secods) Mesh sze Mamum asolute error Mesh sze E E E-6.94 E E E E-4.84 E E-5.48 E E-4.84 E E E-4.48 E E E E E E-5 Tale 3: Total umer of arthmetc oeratos er terato for ad methods Arthmetc Oerato ADD/SU MU/DIV Coclusos I ths aer we reset a alcato of the teratve method for solvg dese lear systems arsg from the dscretzato of the secod d

10 95 M. S. Muthuvalu ad J. Sulama lear Fredholm tegral equatos y usg RT scheme. Through umercal results otaed Tales ad t clearly shows that y alyg the AM methods ca reduce umer of teratos comared to the method. I terms of eecuto tme for oth eamles t ca e cocluded that ad methods are much faster tha method (refer Tales ad ). Reducto ercetage of the eecuto tme for ad methods comared wth method have ee summarzed Tale 4. The accuracy of the teratve methods s also good agreemet. Tale 4: Reducto ercetage of the eecuto tme for ad methods comared to the method Eamle % % % % % % Overall the umercal results show that the method s a etter method comared to the ad methods the sese of umer of teratos ad eecuto tme. Ths s maly ecause of comutatoal comlety of the method whch s aromately 5% ad 75% less tha ad methods resectvely (refer Tale 3). Refereces [] A. R. Adullah The four ot Elct Decouled Grou (EDG) method: A fast Posso solver Iteratoal Joural of Comuter Mathematcs 38 (99) 6-7. [] T. Allahvraloo E. Ahmady N. Ahmady ad. S. Aletay loc Jaco two-stage method wth Gauss-Sdel er teratos for fuzzy system of lear equatos Aled Mathematcs ad Comutato 75 (6) 7-8. [3]. E. Atso The Numercal Soluto of Itegral Equatos of the Secod d Camrdge Uversty Press Uted gdom 997. [4] D. J. Evas The Alteratg Grou Elct (AGE) matr teratve method Aled Mathematcal Modellg (4) (987) [5] Z. -Y. u H. -. Wu ad. The two-stage teratve methods for symmetrc ostve defte matrces Aled Mathematcs ad Comutato 4 () -. [6] N.. Nchols O the covergece of two-stage teratve rocess for solvg lear equatos SIAM Joural o Numercal Aalyss (973)

11 Quarter-swee arthmetc mea teratve method 953 [7] M. Othma ad A. R. Adullah A effcet Four Pot Modfed Elct Grou Posso solver Iteratoal Joural of Comuter Mathematcs 76 () 3-7. [8] A. D. Polya ad A. V. Mazhrov Hadoo of Itegral Equatos CRC Press C Florda 998. [9] A. Quartero R. Sacco ad F. Saler Numercal Mathematcs Srger-Verlag New Yor. [] V. Ruggero ad E. Gallga A teratve method for large sarse systems o a vector comuter Comuters & Mathematcs wth Alcatos () (99) 5-8. [] M. S. Sahm A. Ahmad ad A. A. aar The Iteratve Alteratg Decomosto Elct (IADE) method to solve the heat coducto equato Iteratoal Joural of Comuter Mathematcs 47 (993) 9-9. [] M. S. Sahm ad M. hatm The Reduced Iteratve Alteratg Decomosto Elct (RIADE) method for dffuso equato Pertaa Joural of Scece ad Techology 9() () 3-. [3] J. Sulama M. Othma ad M.. Hasa A ew Half-Swee Arthmetc Mea () algorthm for two-ot oudary value rolems. Proceedgs of the Iteratoal Coferece o Statstcs ad Mathematcs ad Its Alcatos Develomet of Scece ad Techology [4] J. Sulama M. Othma ad M.. Hasa A ew Quarter Swee Arthmetc Mea () method to solve dffuso equatos Chamchur Joural of Mathematcs () (9) [5] W. Wag A ew mechacal algorthm for solvg the secod d of Fredholm tegral equato Aled Mathematcs ad Comutato 7 (6) Receved: Jauary