APPLICATION OF HYBRID METHODS TO SOLVING VOLTERRA INTEGRAL EQUATION WITH THE FIXED BOUNDARES G.Medeva V.Ibraov M.Iaova Doctor of scece PD rofessor ead of car of Coutatoa ateatcs of Bau State Uverst Bau Azerbaa Doctor of scece PD rofessor of te deartet of Coutatoa ateatcs of Bau State Uverst Bau Azerbaa ABSTRACT I te dde of XX cetur te scoars bega costruct te etods wc ave te best caracterstcs of te oe ad ut-ste etods wt te costat coeffcets. Ad te 955 ears tere aears te wors of Gear ad Butcer dedcated vestgato of te brd etod. Here we wat to sow tat ow ts teor deveoed ad coared te wt te ow etods. Costructed te brd etod wt te ger order of accurac ad ustrated te b te ode equato. For acato of brd etods ere roosed te se agorts wt te order of accurac te case we te aout of te used es ots s equa to or te order of te dfferece etods varabe satsfes te codto. Keword: - brd etods Voterra tegra equato accurac ad stabt of uerca etods ODE utste etods wt te secod dervatve. TITLE- Ma robes of atura sceces reduced to sovg tegra equatos wt te varabe boudares wc ear case was fudaeta vestgated b Vto Voterra. Ma scoars ave vestgated te uerca souto of te Voterra tegra equatos. But te basca use te quadrate etods or ts odfcato. But ere roosed a ew wa for costructo te uerca etods for sovg te tegra equato wt te varabe boudares. Let us to cosder te foowg tegra equato f K s s ds. Soetes tegra equato caed te tegra equato of Voterra-Urso te. Ts coveto s due to te fact tat equato te ear case was toroug vestgated b Voterra ad at a suffcet g eve was studed suc equatos occurrece ractca robes see e.g. [] []. Note tat te sguar tegra equato wt a varabe boudar a artcuar for was studed for te frst te b Abe see e.g. [.]. Gve tat eve te ear case te eact souto to equato s ot awas ossbe terefore a eerts ave used aroato etods to sove t see e.g. [3] - [7]. Here we suose tat te Voterra tegra equato as te uque souto detered o te terva X for vestgato of te uerca souto of equato of assue tat te ere K z of te tegra equato defe te doa G s X b were t as arta dervatves u to soe order cusve. But te gve suffcet soot fucto f as detered o te terva X. For deterg te aroate vaues of souto of equato we dvde te terva 456 www.are.co 49
456 www.are.co 49 X to N equa arts b te es ots... N. Deote b te aroate ad eact vaues of souto of equato at te es ots... N. As s ow oe of te ouar etods for sovg equato s te quadrature etod. Te quadrature etod wc aes to souto of equato ca wrte as foows see []: R K a f were R s te reader ter of te quadrature etod ad... a are rea ubers tese ubers caed te coeffcets of te quadrature etod. B dscardg te reader ter we obta te foowg etod: 3... f K a f 3 wc caed te quadrature etod wt te varabe boudar. Rear tat te etod 3 for a s ct but ts etod for a s ect. Ad te resut of usg etod of 3 b creasg of te vaues of te quatt te aout of coutg fuctos s K s creases aso. Cosequet at eac ste te aout of cacuato wor s creases wc s a aor dsadvatage of quadrature etods. To eate ts drawbac of etod 3 soe autors ave roosed te use of etods suc as te foowg -ste etod wt costat coeffcets see for eae [8] - []: f 4 It s ow tat f te etod 4 s stabe te te foowg s ods:. 5 Here te degree of te etod 4 quatt s order of te accurac ad s te order of te dfferece etod 4. For te costructo te stabe etods wt te degree te scoars are use odfcato of te etod 4 wt resuted aears te forward-ug etods etc. Soe of tese scoars roosed to cage te etod 4 b te foowg: " ' 6 Here te coeffcets... are te soe rea ubers ad. It s cear tat for geerazato of te Ruge-Kutta ad Adas etods oe a use te dfferet was oe of wc ca be wrtte as te foows:... ' 7 Here quattes are te brd ots. Rear tat for fucto of te etods Ruge-Kutta ad Adas. We rewrte te etod 6 as te foowg: ' ' 8 If we cotue te above roosed wa te te etod 5 ca rewrte as: " " ' ' 9 It s ot dffcut to uderstad tat te etod 9 s ore accurate ta te ow etods. Metod 4 geerazed so tat ever foowg etod s ore accurate ta te revous oe. For eae fa etods of te 8 tere are stabe etods tat are ore accurate ta te etods of te
te 6 ad 7. Terefore researc of etods 8 ad 9 s ore rosg ta te above etods of researc. For te obectvt we ote tat brd etods are ore accurate ta te ow. However usg of te brd etod s a dffcut rocess. Terefore costructo of agorts for acato of brd etods s soetes ore dffcut rocess ta costructo of te etod tsef. I te scetfc terature soe autors suc etod caed fractoa stes.. APPLICATION OF THE METHOD OF UNDETERMINED COEFFICIENTS TO THE STUDY OF HYBRID METHODS For costructo of brd etods we ca use dfferet scees resutg of ts get etods wt te dfferet roertes. It s ow tat te roertes of uerca etods deed o te vaues of te coeffcets te foruas 5-9. Prevous te above etod aes to souto of te foowg ta vaue robe: F. I tese studes we used te etod of udetered coeffcets for deterato of coeffcets te foruas 5-9. Terefore we w tr costructo of brd etods for sovg te equato usg te etod of udetered coeffcets. Note tat oe of te ost ouar etods for fdg soutos of Voterra tegra equatos s quadrature etod. But as sow above we usg te quadrature etods te voue of coutatos creases wt te uber of quatt of te ots used costructg te. I ts regard ere we offer a to souto of equato te foowg ut-ste etod F K. Note tat fro te forua we ca obta ct ad ect etods as we as forward-ug etods. Here we wat b usg te above scees costruct brd etods for sovg of te equato. Te etod as a stragt coecto wt te etod 5. Ideed f te equato te ere of te tegra te fucto K s te et: K s F s te souto of te equato cocdes wt souto of te robe. Usg a sar coecto betwee tegra ad dffereta equatos cosder costructo of brd etods. Here to sove equato we use te fte-dfferece etod wc ca be wrtte as foows:.... O te frst et us cosder a bouders for te coeffcets of te etod. A: Te coeffcets... are soe rea ubers oreover. B: Caracterstc ooas. ave o coo uters dfferet fro te costat. C: ad. I artcuar for... fro te etod of we obta a ordar -ste etod wt te costat coeffcets. 3. ON A WAY TO CONSTRUCTION HYBRID METHOD OF TYPE Cosder te costructo of te etods of te for forua for cacuatg te vaues of te fucto.... To ts ed we costruct a. Suose tat b soe etods te souto of te equatos s foud after tag to accout tat equato oe dett s obtaed. Te fro ts dett we ca wrte: 456 www.are.co 493
Fdg te vaues of te quattes tegra: K f s s ds. 3 b eas of foruas 3 s reduced to te cacuato of te foowg K s s ds. 4 If quadrature forua 3 s aed to a evauato of tegra 4 te we obta te tegra su of te varabe boudares. But for te acato of te etod to sovg equato we gve te ow vaues ad we wat to fd te reatos betwee tese vaues wt te vaues of wc are uow. Te coutato te seget te tegra wt te varabe boudar as bee reduced to cacuatos tat te seget. Moreover ts reducto was accosed wtout creasg te coutatoa wor b creasg te vaues of te varabe. Note tat ts te of scee s aed to te deterato of te reatos betwee te varabes ad. It s ow tat for a suffcet soot fucto te foowg s ods: ere... are rea ubers. After carrg out sae oeratos we obta: z z 5 f a K. 6 ere a K Deedg o te eeds of te user te coeffcets foruas 4 ca be gve dfferet eags. For ts urose equatos 3 ca be rewrtte as foows: f a K s s ds R 7 Here... are rea ubers but R s te reader ter. B reacg te tegras foruas 7 wt a tegra su ad dscardg te reag ters we obta etods to cacuate te vaues of te quatt. However ts case te coeffcets ose te foowg addtoa codtos. It s ot dffcut to roof tat for te fdg of te coeffcets of te etod a be use te foowg sste of te oear agebrac equato:!.. 8!! B usg te souto of agebrac equatos ca costruct etods avg dfferet roertes. Fro te vewot of te acato of uerca etods te ost sgfcat roert s te stabt ad accurac. Te accurac of tese etods deeds o te uber of equatos te sste 8. However we te uber of equatos te sste 8 tere s soe dffcut fdg te souto of te sste 8. Te above etoed dffcut s ot fdg te souto of ear agebrac equatos. Note tat we costructed etods au accurac o. 456 www.are.co 494
soe artcuar cases te foowg are soe of te ood of te: ' ' 5 ' ' 5 9 8 9 ' 57 ' 4 ' /4 3 64 ' 98 ' 8 ' 36 8 ' 98 ' 64 ' / 36. 5' 8' 5' 9 5 5 As ca be see fro te above stabe brd etods are ore accurate ta corresodg etods of Ruge-Kutta ad Adas. I a secfc eae we sow tat f we use te Sso etod to sove t te resut obtaed b ste / s better ta te resut te ste. To ts ed cosder te foowg tass: ' cos eact souto s wrtte as: s. Te resut at te ste to ace te foowg: Tabe. Vaue of te varabe...3.4.5.6.7.8.9. Error for te Sso etod b te ste /.34E-8.69E-8.E-7.3E-7.6E-7.9E-7.E-7.4E-7.7E-7.9E-7 Error for te Sso etod b te ste.e-6.e-6.e-6.e-6.3e-6.3e-6.39e-6.38e-6.46e-6.44e-6 As see fro Tabe te resuts obtaed b te foowg etod of Sso ' 4' ' 6 are accurate. Now cosder te acato of te et brd etod of te 7: / / 3 / 6 / f f f To sovg foowg eaes:. cos [ ]. Eact souto: s.. []. Eact souto: e. For te cacuatg of te vaues ad a be used te etods: f f f f Reeat tese scees for :. Te resuts of cacuatos accoodated Tabe ad Tabe 3. f /... 3 / 6.4 Tabe 456 www.are.co 495
Ste sze Varabe...3.4.5.6.7.8.9. Error for te eae.e-6.e-6.e-6.e-6.3e-6.3e-6.39e-6.38e-6.46e-6 Error for te eae.6e-5.78e-5.5e-4.9e-4.8e-4.34e-4.46e-4.56e-4.7e-4 Tabe 3 Ste sze Varabe Error for te eae Error for te eae.5...3.4.5.6.7.8.9..34e-8.69e-8.e-7.3e-7.6e-7.9e-7.e-7.4e-7.7e-7.9e-7.36e-6.8e-6.3e-5.9e-5.7e-5.36e-5.47e-5.59e-5.74e-5.9e-5 4. CONSTRUCTION AND APPLICATION THE METHODS WITH THE HIGHER ORDER OF ACCURACY TO SOLVING NONLINEAR VOLTERRA INTEGRAL EQUATION As oted above oe of te a ssues oder coutatoa ateatcs s te costructo of ore accurate etods wt eteded stabt of te rego ad as a eae of suc etods s roosed a etod 6. Here cotued ts dea are costruct a brd etod wt te secod dervatve wc ca be rereseted te for 9 were te coeffcets are soe rea ubers ad besdes s od. We assue tat te coeffcets of te etod 9 satsf te foowg codtos: A. Te coeffcets... are rea ubers ad. B. Te sets of te roots of te caracterstc ooas Have ot coo factor dfferet fro te costat. C. Te foowg ods:. ad. Note tat te accurac of te etod 9 s detered b sar to te cocet of te degree of te etod 9 wc ca be foruated te foowg for: Defto. Te quatt s caed te degree of te etod 9 f ods te foowg: 456 www.are.co 496
456 www.are.co 497. [ O 5 However f te te degree of etod 9 s detered b usg sar astotc reatos to 5 were ad. Here aso we use te etod of udetered coeffcets for te deterato of te coeffcets of te etod 9. We ust tae to accout tat te basc roertes of uerca etods are detered b te vaues of ter coeffcets ad tus we cosder te deterato of te vaues of te coeffcets of etod 9. For ts urose etod 9 te aroate vaues of te fucto w be reaced b te eact vaues. Te we ave: R 6 were R s te reader ter of etod 9. If we suose tat etod 9 as te degree te fro equato 6 we ave O R. To fd te vaues of te quattes... we use te etod of udetered coeffcets.e. equato 6 we use Taor easos of te fuctos ad. Cosder te foowg easos of tese fuctos:!...! O!...! O!...! O 7 were te quattes ad... receve te foowg vaues: or.... Assue tat etod 9 as te degree ad cosder deterg te vaues of ts coeffcets. B usg easos 7 te 5 we ust assue tat te etod 9 as te degree. I ts case ts coeffcets satsf te foowg codtos: 8. 34... Te resutg correato s a oogeeous sste of oear agebrac equatos. Obvous sste 8 awas as a zero trva souto tat s ot usefu for te costructo of te etods addressed ts aer. Terefore
456 www.are.co 498 to vestgate te ozero otrva soutos of oear sste 8 wc we observe tat te uber of equatos s equa to but te uber of uows s equa to 7 7. It ca be assued tat sstes 8 w ave a otrva souto for te vaues 7 5. Tus we see tat b usg souto of te sste 8 we ca costruct etods of te 9 wt te degree 7 5. 9 It s ot cosequet tat tere est te etods of te 9 wt te degree 5 7. However etods costructed b us obe te rue 5. Cosder te foowg oe-ste etod: /36 64 98 8 /36 8 98 64 / / / 3 ere 4 / 4 /. Ts etod s stabe ad as a degree 8. Note tat ts etod s ot obtaed fro te forua 9 because at te costructo te etod of te te 9 s used o es ots. But te etod 3 artcate tree es ots. Terefore te etod 3 artcuar s ot cotaed te cass of te etod 9.For sct we cosder a seca case ad set. Te b usg of te etod of te 9 we obta:. ` 3 I acatos of etod 3 to sove certa robes t s ecessar to detere te vaues of te quattes ad. Soe autors refer to costruct etods of te 9 for wc te foowg codto ods:. B usg te codtos oe ca be obtaed fro sste 8 we obta te foowg: 3 4 3 3 3 5 4 4 4 3 3 6 5 5 5 4 4 7 6 6 6 5 5 3 8 7 7 7 6 6 9 8 8 8 7 7. 9 9 9 8 8
B sovg te resutg sste oe ca costruct dfferet etods wt te degree. For eae et us cosder to te foowg etods: 98655857 8764.36.997683 45667 83988.73444779336 33 84773474 67545 589 38 477777 6655 43937763 76437.6543976 9438 34 57877833449 644948974 84386469 5897363 99745.498844 78379 497394 4843 9883 54537.3939 7986 35 476778378497 797984557 45437 3595969 3998569.58597969.79537 9 5476 778637 66386 45835 55939.7593 9396 36.655364 5689.493756659 94499.339967353 74657 9945438 59378 957963 5746666 344 456 www.are.co 499
56984773.445543.736 79964 69775 577375.437366 978566 37.7546 756786.34887938 9786.379689589557 794647883 34586935 44968 7746973 Cosder te acato of te etod 36 ad 37 to sovg of te foowg equato: Eact souto for wc t te foowg: s s. e ds 3 6 were te user decdes to ove. Te resuts ca be foud Tabe. 38 Nuber of eae Metod 36 Metod 37.E-9.4.E-5.53E-9.7..4E-5.3E-4.E-8.8E-8 I te sovg tegra equato of 38 ave used te agort fro te [3]. 5. CONCLUSIONS Cosderg tat te souto of ordar dffereta equatos s fudaeta vestgated. Usg te eaes above we ave sow soe of te advatages of brd etods. As s ow te order of accurac of te etod of Sso cocdes wt te above gve order of accurac of te brd etod but te coarso of te resuts set fort tese tabes route coes tat te stabt rego for te brd etod s ore eteded. Severa autors b usg te advatages of brd etods aed te to te sovg of soe robes ecacs ad caed te te fractoa stes sze etods. It s ot dffcut to uderstad tat te use of brd etods to te sovg of te equato gves better resuts ta te corresodg ow etods see e.g. [8]-[]. Cosequet te use of brd etods te scetfc terature for 6 ears sows tat tese etods are ore rosg. Fro te foregog t foows tat te teor of brd etods tere are a usoved robes. ACKNOWLEDGEMENT Te autors ws to eress ter tas to acadeca A Abbasov for s suggesto to vestgate te coutatoa asects of our robe ad for s frequet vauabe suggesto. 456 www.are.co 5
REFERENCES [] E.M. Posu Vto Voterra. Legrad Naua 977 4. [] V. Voterra Teor of fuctoa ad of tegra ad tegro-dfferesa equatos Dover ubcatos. Ig New Yor Naua Moscow 98.34 Russa. [3] A.V. Mazrov A.D. Poa Hadboo of Itegra Equatos: Metods of soutos. Moscow: Pubsg House of te "Factora Press" 384. [4] A.F. Vera V.S. Szov. Itegra equatos: etods agorts rogras. Kev Nauovo Dua 986 384. [5] Y.D Maedov. Metods of cacuato 978. [6] A.Quartero R. Sacco F. Saer. Nuerca Mateatcs Secod Edto Srger 656. [7] R.L.Burde J.D.Fares Nuerca aass. Cegege Lerg 85 7-t edto. [8] G.Yu. Medeva M.N. Iaova V.R. Ibraov O oe acato of forward ug etods. Aed Nuerca Mateatcs Voue 7 October 3. 34 45. [9] P.Lz Lear Mutste etods for Voterra Itegro-Dffereta equatos Joura of te Assocato for Coutg Macer Vo.6 No. Ar 969. 95-3. [] G.Yu Medeva M.N. Iaova V.R. Ibraov Acato of te brd etod wt costat coeffcets to sovg te tegro-dffereta equatos of frst order. 9t Iteratoa coferece o ateatca robes egeerg aerosace ad sceces AIP Vea Austra -4 Ju. 56-5. [] G.Yu Medeva M.N. Iaova V.R. Ibraov O a Researc of Hbrd Metods. Nuerca Aass ad Its Acatos Srger 3. 395-4. [] V.R.Ibraov V.D.Aeva Te costructo of te fte-dfferece etod ad acato Proceedgs of te Iteratoa Coferece o Nuerca Aass ad Aed Mateatcs 4 ICNAAM-4 AIP Cof. Proc. 648 8549-8549-5. [3] A.A.Marogou Boc - b-boc etod for te uerca souto of Voterra dea tegro-dffereta equatos Coutg 3 983 3.49-6. [4] A.Fedste J.R. Soa Nuerca etods for oear Voterra tegro-dffereta equatos. SIAM J. Nuer. Aa. 974 V. 86-846. [5] V.R. Ibraov M.N. Iaova O a Researc of Setrc Equatos of Voterra Te. Iteratoa oura of ateatca odes ad etods Aed sceces Voue 8 4.434-44. [6] M.N. Iaova Oe te utste etod of uerca souto for te Voterra tegra equato. Trasactos ssue ateatcs ad ecacs seres of sca -tecca ad ateatca scece ХХВI 6. [7] A.Marogou Hbrd etods te uerca souto of Voterra tegro-dffereta equatos. Joura of Nuerca Aass 98.-35. [8] V.R.Ibraov V.D.Aeva Te costructo of te fte-dfferece etod ad acato Proceedgs of te Iteratoa Coferece o Nuerca Aass ad Aed Mateatcs 4 ICNAAM-4 AIP Cof. Proc. 648 8549-8549-5. [9] E. Harer S.P.Norsett G.Waer Sovg ordar dffereta equatos. Russa М. Mr 99 5. [] Moder uerca etods for sovg ordar dffereta equatos. Edtors J.Ho ad J.Watt Pubsg House "Mr" Moscow 979. [] G.Daqust Covergece ad stabt te uerca tegrato of ordar dffereta equatos. Mat. Scad. 956 4.33-53. [] G. Daqust Stabt ad Error Bouds te Nuerca Itegrato of Ordar Dffereta Equatos. Tras. Of te Roa Ist. Of Tec. Stoco Swede 959 3.3-87. [3] G.Yu Medeva V.R. Ibraov O te researc of utste etods wt costat coeffcets LAP LAMBERT Acadec Pubsg 3 34. Russa. [4] G. Yu. Medeva M. N. Iaova V. R. Ibraov A acato of te brd etod of utste te. Advaces Aed ad Pure ateatcs Proceedgs of Iter.Cof. o Mat.Co ad Aqtatst. Scece MCSS 4. 7-76. [5] G.K. Guta. A ooa reresetato of brd etods for sovg ordar dffereta equatos.mateatcs of co. voue 33 uber 48 979.5-56. [6] L.M. Svortsov Ect two-ste Ruge-Kutta etods. Mat. odeg 9 9. 54-65. [7] G.Yu. Medeva M.N. Iaova V.R. Ibraov Soe acato of te brd etods to sovg Voterra tegra equatos Advaces Aed ad Pure ateatcs Proceedgs of Iter.Cof. o Mat.Co ad Aqtatst.Scece MCSS 4. 35-356. [8] E.A.Areo R.A. Adeu P.O. Babatoa Accurate coocato utste etod for tegrato of frst order ordar dffereta equatos. J.of Moder Mat.ad Statstcs : -6 8.-6. 456 www.are.co 5
[9] V. Ibraov O te aa degree of te -ste Obrecoff s etod. Buet of Iraa Mateatca Socet Vo.8. -8. [3] V.R. Ibraov O a reato betwee order ad degree for stabe forward ug forua. Z.Vcs.Mat. 7 99.45-56. [3] G.Yu. Medeva V.R. Ibraov M.N. Iaova Te Acato Dfferece Metods to Sovg Voterra Itegra Equato Pesee Joura Pars Vo. 75 Issue. 3. 393-4. 456 www.are.co 5