An Iterative Solution for Second Order Linear Fredholm Integro-Differential Equations

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1 Malasa Joural of Matematcal Sceces 8): ) MLYSIN JOURNL OF MTHEMTICL SCIENCES Joural omeage: tt://esem.um.edu.m/oural Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos * Elaaraa rucua Moaa Sudaram Mutuvalu Jumat Sulama 4 We S o ad 5 amalrulzama Md kr Deartmet of Matematcs ad Statstcs Curt Uverst Pert W6845 ustrala Deartmet of Fudametal ad led Sceces Uverst Tekolog Petroas 75 Troo Perak Malasa Scool of Scece ad Tecolog Uverst Malasa Saba 884 ota abalu Saba Malasa 4 Deartmet of Matematcs ad Scece Nla Uverst 78 Nla Neger Sembla Malasa 5 Isttute for Matematcal Researc Uverst Putra Malasa 44 UPM Serdag Selagor Malasa E-mal: earucua@aoo.com *Corresodg autor BSTRCT Te obectve of ts aer s to aalze te alcato of te quarter-swee teratve cocet o Quadrature-Dfferece scemes amel cetral dfferece CD)-comoste traezodal CT) wt te Gauss-Sedel teratve metod to solve secod order lear Fredolm tegro-dfferetal equatos. Te formulato ad mlemetato of te Full- Half- ad Quarter-Swee Gauss-Sedel metods amel FSGS HSGS ad QSGS are reseted for erformace comarso. Furtermore comutatoal comlet ad ercetage reducto calculatos are also reseted wt several umercal smulatos. Te umercal results sow tat te roosed QSGS metod wt te corresodg dscretzato scemes s sueror comared to te FSGS ad HSGS metods. ewords: Lear Fredolm secod order tegro-dfferetal equatos quarter-swee teratos Gauss-Sedel metod secod order cetral dfferece sceme traezodal sceme.

2 Elaaraa rucua et al.. INTRODUCTION Cosder te lear secod order Fredolm tegro-dfferetal equatos a g ) P ) ) tt dt ) subect to te two-ot boudar codtos were a. tl P L are gve fuctos ad g L ad s te ukow fucto to be determed Lakesta et al. 6)). Te codtos for estece ad uqueess of soluto of suc roblems ave bee vestgated b garwal ). Solutos of lear Fredolm tegro-dfferetal equatos LFIDEs) ave bee studed b ma autors. Ma studes ave bee carred out wt Quadrature scemes b Zao ad Corless 6) rucua ad Sulama a b a b). Besdes tat metods suc as wavelet- Galerk El-Saed ad bdel-zz )) doma s Deeba et al. )) Tau Hosse ad Samorad 5)) ad Sc collocato Rsda ad Zareba 5)) are also aalsed solvg LFIDEs. However tese metods are lead to dese lear sstems ad ca be robtvel eesve to solve -t order lear sstems. Moreover tese metods are based o te stadard or full-swee teratve metods wc are more eesve terms of comutato tme. Terefore ts aer a dscretzato sceme amel quarter-swee cetral dfferece-comoste traezodal QSCD-QSCT) sceme s aled to dscretze Eq. ) to geerate a sstem of lear equatos. Te remag of ts aer s as follows. I Secto elaato of te full- alf- ad quarter-swee terato cocets ad te detals of te formulato of QSCD-QSCT dscretzato scemes are elaborated wt aromato equatos. I Secto formulatos of te FSGS HSGS ad QSGS teratve metods are sow wt te develomet of a umercal algortm. I Secto 4 several umercal tests are coducted to valdate te 58 Malasa Joural of Matematcal Sceces

3 Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos effcec of te metods. Furtermore aalss o comutatoal comlet s gve Secto 5 followed b cocluso Secto 6.. COMPLEXITY REDUCTION PPROCHES Bascall te roosed HSGS metod s sred b te cocet of alf-swee terato wc as troduced b bdulla 99) va te Elct Decouled Grou EDG) teratve metod to solve two-dmesoal Posso equatos. Te alcatos of alf-swee teratve metods ave bee mlemeted b Sulama et al. 4a) Mutuvalu ad Sulama 8) ad rucua ad Sulama a b). Otma ad bdulla ) eteded te cocet of alf-swee terato b establsg te quarterswee terato cocet va te Modfed Elct Grou MEG) metod to solve two-dmesoal Posso equatos. Furter studes to verf te effectveess of te quarter-swee terato cocet ave also bee carred out b Sulama et al. 4b) ad kr et al. ). Te quarter-swee terato erts te caracterstc of te alf-swee terato wc ts mlemetato rocess wll ol cosder earl quarter of all teror odes of te soluto doma. Fgure a) b) ad b) sow full- alf- ad quarter-swee terato cocets a) b) c) Fgure : a) b) ad c) sow dstrbuto of uform ode ots for te full- alf- ad quarterswee cases resectvel. Malasa Joural of Matematcal Sceces 59

4 Elaaraa rucua et al. Based o Fg. te full- alf- ad quarter-swee teratve metods wll comute aromate values ol at te sold odes utl te covergece crtero s reaced. Te aromate solutos at te remag ots odes odes of te ad ) ca be calculated usg te drect metod as gve Sulama et al. 9).. Formulato of Quarter-Swee Quadrature-Dfferece Scemes I ts secto cetral dfferece CD) ad comoste traezodal CT) dscretzato scemes wll be reformulated b alg te full- alf- ad quarter-swee terato cocet order to dscretze te dfferetal ad tegral terms Eq. ) to form te aromato equatos. Te full- alfad quarter-swee CD ad CT formula ca be wrtte as follows for ad '' ) - ) -) ) O ) ) b a t) dt t ) ) ) were oterwse wc t ) are te abscssas of te artto ots of te tegrato terval a b or quadrature terolato) odes; ) are umercal coeffcets tat do ot deed o te fucto t) ; s te costat ste legt betwee te ode ots as defed below b a 6 Malasa Joural of Matematcal Sceces

5 Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos Malasa Joural of Matematcal Sceces 6 were a ad b s te lower ad uer lmt of te tegral term Eq. ) ad s te umber of subterval b a ; ) O ad ) are te trucato errors of Eqs. ) ad ) wc are ot cosdered te calculatos. Meawle te value of ad 4) corresods resectvel to te full- alf- ad quarter-swee teratve metods. B substtutg Eqs. ) ad ) to Eq. ) a sstem of lear algebrac equatos are obtaed for te aromato values of ) at te odes. Terefore te full- alf- ad quarter-swee terato cocets togeter wt te CD ad CT aromato scemes eld t P g ) ) 4) for were ad 4 are resectvel for te full- alf- ad quarter-swee aroac. Te lear sstem geerated eter b te full- alf- ad quarterswee aromato equato ca be eressed b f E 5) were N N N E wc

6 Elaaraa rucua et al. 6 Malasa Joural of Matematcal Sceces P ad g g g g g g f ad ) ) ) ) ) ). obvousl E s a dese coeffcet matr. From Equato 5) t s otceable tat alcatos of te alf- ad quarter-swee terato cocets reduce te coeffcet matr E from order to ad 4 resectvel.. FORMULTION OF FMILY OF GUSS-SEIDEL ITERTIVE METHODS Te stadard GS teratve metod s also called te Full-Swee Gauss-Sedel FSGS) metod. Combatos of te GS metod wt alf- ad quarter-swee teratos are kow as Half-Swee Gauss-Sedel HSGS) ad Quarter-Swee Gauss-Sedel QSGS) metods resectvel rucua ad Sulama b)). s metoed above te geerated lear sstems of Eq. ) as smlfed Eq. 5) wll be solved b usg te FSGS HSGS ad QSGS teratve metods. Let te coeffcet matr E be decomosed to

7 Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos E D LU 6) were D L ad U are dagoal strctl lower tragular ad strctl uer tragular matrces resectvel. Terefore te geeral sceme for te FSGS HSGS ad QSGS teratve metods ca be wrtte as U k k D L f. 7) s a matter of fact te teratve metods attemt to fd a soluto to te sstem of lear equatos b reeatedl were solvg te lear sstem usg aromatos to te vector for solvg Eq. ). Iteratos for FSGS HSGS ad QSGS metods cotue utl te soluto s wt a redetermed accetable loo o te error. B determg te values of matrces D L ad U as stated Equato 6) te geeral algortm solvg Eq. ) usg te FSGS HSGS ad QSGS teratve metods ad te Gauss-Sedel metod s as follows Full- Half- ad Quarter-swee Gauss-Sedel lgortm Ste : Italze all te arameters. Set et k. Ste : for Comute k) E f Ste : Ceck te covergece Ste 4 : Sto. If te error E k ) k) k k E s satsfed terato s termated ad go to Ste 4; oterwse reeat te terato sequece.e. go to Ste ) Malasa Joural of Matematcal Sceces 6

8 Elaaraa rucua et al. 4. NUMERICL EXPERIMENT I ts secto two well-osed roblems are carred out to valdate te effectveess of te roosed metod. Tree arameters suc as umber of teratos eecuto tme ad mamum absolute error are cosdered as measuremets to evaluate te erformace of te metods. Te FSGS metod was used as te cotrol of comarso of umercal results. Trougout te umercal smulatos te covergece test was carred out wt tolerace error of wt several mes szes suc as ad 96. Te results of umercal smulatos wc were obtaed from mlemetatos of te FSGS HSGS ad QSGS teratve metods for roblems ad are sow recorded Tables ad resectvel. Te ercetage reducto of umber of teratos ad eecuto tme for te HSGS ad QSGS metods relatve to te FSGS metod s summarzed Table. Problem Delves ad Moammed 985)) Cosder te secod order lear FIDE " ) 6 t) t) dt 8) wt two ot boudar codtos Te eact soluto s ) ad ). ). Problems maal ad Sudad )) Cosder te secod order lear FIDE " ) e e ) t) t) dt wt two ot boudar codtos 9) Te eact soluto s ) ad ) e ) e. 64 Malasa Joural of Matematcal Sceces

9 Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos TBLE : Comarso of a umber of teratos eecuto tme secods) ad mamum absolute error for te teratve metods Eamle ) Number of teratos Metods Mes Szes FSGS HSGS QSGS Eecuto tme secods) Metods Mes Szes FSGS HSGS QSGS Mamum absolute error Metods Mes Szes FSGS 7.449E-5.854E-5 4.E-6 4.9E E-6 HSGS 5.58E-4.8E-4.48E E-6.68E-6 QSGS.95E- 5.58E-4.8E-4.48E E-6 TBLE : Comarso of a umber of teratos eecuto tme secods) ad mamum absolute error for te teratve metods Eamle ) Number of teratos Metods Mes Szes FSGS HSGS QSGS Eecuto tme secods) Metods Mes Szes FSGS HSGS QSGS Mamum absolute error Metods Mes Szes FSGS 9.684E-6.547E-6.4E-6.8E E-6 HSGS.688E-4 9.9E-5.45E E-6.8E-6 QSGS.84E-.E- 5.46E-4.597E-5.6E-5 Malasa Joural of Matematcal Sceces 65

10 Elaaraa rucua et al. TBLE : Reducto ercetage of te umber of teratos ad eecuto tme for te HSGS ad QSGS metods comared wt FSGS metod Metods Eamle Number of teratos Eecuto tme HSGS % % QSGS % % Metods Eamle Number of teratos Eecuto tme HSGS % % QSGS % % 5. COMPUTTIONL COMPLEXITY NLYSIS Te comutatoal comlet of te FSGS HSGS ad QSGS teratve metods s measured based o te estmato amout of te comutatoal work of artmetc oeratos erformed er terato. Based o te full- alf- ad quarter-swee Gauss-Sedel lgortm t ca be observed tat tere are addtos/subtractos DD/SUB) ad multlcatos/dvsos MUL/DIV) comutg a value for eac ode ot te soluto doma. From te order of te coeffcet matr E Equato 5) te total umber of artmetc oeratos er terato for te FSGS HSGS ad QSGS teratve metods as bee summarzed Table 4. TBLE 4: Total umber of artmetc oeratos er terato for FSGS HSGS ad QSGS metods Metods FSGS rtmetc Oerato DD/SUB MUL/DIV ) HSGS QSGS Malasa Joural of Matematcal Sceces

11 Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos 6. CONCLUSION I ts aer alcato of te quarter-swee terato cocet o umercal scemes amel CD ad CT wt GS teratve metod for solvg dese osmmetrc matr equatos arsg from te secod order tegrodfferetal equatos s eamed. Troug umercal solutos obtaed Tables ad t evdetl sows tat alcatos of te alf- ad quarterswee terato cocet reduce te umber of teratos ad comutatoal tme sgfcatl. Based o Table te ercetage reducto umber of teratos for alf- ad quarter-swee cocet are aromatel 7% ad 9% resectvel wle te comutatoal tme reduces aromatel 54% ad 75% resectvel comared to FSGS. Overall te umercal results sow tat te QSGS metod s a better metod comared to te FSGS ad HSGS metods terms of umber of te teratos ad eecuto tme. Ts s mal due to te reducto terms of comutatoal comlet; sce te QSGS metod wll ol cosder aromatel quarter of all teror ode ots soluto doma durg te terato rocess refer Table 4). CNOWLEDGMENTS Te autor taks to Professor Yogog Wu from Curt Uverst ustrala for s valuable suggestos wc greatl mroved te qualt of te aer. REFERENCES bdulla. R. 99). Te four ot Elct Decouled Grou EDG) metod: fast Posso solver Iteratoal Joural of Comuter Matematcs. 8: 6-7. garwal R. P. 98). Boudar value roblems for ger order tegrodfferetal equatos. Nolear alss Teor Metods & lcatos. 7): garwal R. P. 986). Boudar Value Problems for Hg Ordar Dfferetal Equatos. World Scetfc Sgaore. kr M.. M. Otma M. Z. Mad. Sulema M. ad Sulama J. ). Four Pot Elct Decouled Grou Iteratve Metod led to Two-Dmesoal Helmoltz Equato. Iteratoal Joural of Matematcal alss. 6: Malasa Joural of Matematcal Sceces 67

12 Elaaraa rucua et al. maal. M. ad Sudad. I. ). romated Soluto of Hger Order Lear Fredolm Itegro Dfferetal Equatos b Comutg of Sgular Value Decomosto SVD). Egeerg ad Tecolog Joural. 8 4). rucua E ad Sulama J. ). Numercal Soluto of Secod-Order Lear Fredolm Itegro-Dfferetal Equato Usg Geeralzed Mmal Resdual GMRES) Metod. merca Joural of led Sceces Scece Publcato. 7: rucua E. ad Sulama J. a). Half-swee Cougate Gradet Metod for Solvg Frst Order Lear Fredolm Itegro-dfferetal Equatos. ustrala Joural of Basc ad led Sceces. 5: 8-4. rucua E. ad Sulama J. b). Quarter Swee Gauss-Sedel Metod for solvg Frst Order Lear Fredolm Itegro-Dfferetal Equatos. Matematka. 7 ): rucua E. ad Sulama J. a). lcato of te Cetral- Dfferece Sceme wt Half-Swee Gauss-Sedel Metod for Solvg Frst Order Lear Fredolm Itegro-dfferetal Equatos Iteratoal Joural of Egeerg ad led Sceces. 6: 96-. rucua E. ad Sulama J. b). Comarso of Closed Reeated Newto-Cotes Quadrature Scemes wt Half-Swee Iterato Cocet Solvg Lear Fredolm Itegro-Dfferetal Equatos Iteratoal Joural of Scece ad Egeerg Ivestgatos. 8): 96-. rucua E. ad Sulama J. a). Half-Swee Quadrature-Dfferece Scemes wt Iteratve Metod Solvg Lear Fredolm Itegro- Dfferetal Equatos Progress led Matematcs. 5): -. rucua E. Mutuvalu M. S. ad Sulama J. b). lcato of Quarter-Swee Iterato for Frst Order Lear Fredolm Itegro- Dfferetal Equatos IP Coferece Proceedgs 5: Malasa Joural of Matematcal Sceces

13 Iteratve Soluto for Secod Order Lear Fredolm Itegro-Dfferetal Equatos Deeba E. ur S.. ad Sse X. ). algortm for solvg a olear tegro-dfferetal equato. led Matematcs ad Comutato. 5:. Delves L. M. ad Moamed J. L. 985). Comutatoal Metods for Itegral Equatos. Cambrdge Uverst Press Lodo. El-Saed S. M. ad bdel-zz M. R. ). comarso of doma s decomosto metod ad wavelet-galerk metod for solvg tegro-dfferetal equatos. led Matematcs ad Comutato. 6:5 59. Hosse S. M. ad Samorad S. ). Numercal soluto of a class of tegro-dfferetal equatos b te Tau metod wt a error estmato. led Matematcs ad Comutato. 6 ) Hosse S. M. ad Samorad S. 5). Numercal ecewse aromate soluto of Fredolm tegro-dfferetal equatos b te Tau metod. led Matematcal Modellg Lakesta M. Razzag M. ad Dega M. 6). Semortogoal sle wawalets aromato for Fredolm tegro-dfferetal equatos. Matematcal Problems Egeerg:. Mutuvalu M. S. ad Sulama J. 8). Half-Swee Geometrc Mea metod for soluto of lear Fredolm equatos Matematka. 4 ): Otma M. ad bdulla. R. ). effcet Four Pots Modfed Elct Grou Posso solver. Iteratoal Joural of Comuter Matematcs. 76): -7. Rasda J. ad Zareba M. 5). Numercal soluto of lear tegral equatos b usg Sc collocato metod. led Matematcs ad Comutato. 68: Sulama J. Hasa M.. ad Otma M. 4a). Te Half-Swee Iteratve lteratg Decomosto Elct HSIDE) metod for dffuso equato Lectures Notes Comuter Scece: Malasa Joural of Matematcal Sceces 69

14 Elaaraa rucua et al. Sulama J. Otma M. ad Hasa M.. 4b). Quarter-Swee Iteratve lteratg Decomosto Elct algortm aled to dffuso equatos. Iteratoal Joural of Comuter Matematcs. 8): Sulama J. Otma M. ad Hasa M.. 9). New Quarter-Swee rtmetc Mea QSM) Metod to Solve Dffuso Equatos Camcur Joural of Matematcs. ):9-. Zao J. ad Corless R. M. 6). Comact Fte Dfferece Metod for Itegro-dfferetal Equatos. led Matematcs ad Comutato. 7: Malasa Joural of Matematcal Sceces