The Application of the Hybrid Method to Solving the Volterra Integro-differential Equation

Transcription

1 Proceedgs of the World Cogress o Egeerg 3 Vol I WCE 3 Jul Lodo UK The Applcato of the Hbrd Method to Solvg the Volterra Itegro-dfferetal Equato G Mehdeva M Iaova ad V Ibrahov Abstract There are several wors dedcated to the vestgato of Volterra tegro-dfferetal equatos I addto there are theoretcal ad practcal represetatos of stable ethods that have a hgh order of accurac ad eteded stablt regos; these represetatos were costructed usg the u values of arthetc operatos Here hbrd ethods are proposed for the costructo of uercal ethods wth these propertes; oe of these hbrd ethods s well ow We costructed cocrete ethods wth orders of accurac of p 6 ad p 8 usg forato pertag to the soluto of the cosdered proble wth oe ad two esh pot respectvel Ide Ters Volterra tegro-dfferetal equato hbrd ethod degree ad stablt ecessar ad suffcet codto tal-value proble I I INTRODUCTION T s well ow that the earl XXth cetur to solve soe probles the feld of echacs Vto Volterra had to solve tegro-dfferetal equatos wth varable boudares I the 93 Volterra showed that atheatcal odels for soe seasoal dsease eg flueza are forulated as tegral ad dfferetal equatos (see [ pp -34]); ths wor gave petus to the developet of approate ethods for solvg tegro-dfferetal equatos Oe popular ethod for solvg what are ow ow as Volterra tegro-dfferetal equatos s the ethod of quadratures Note that the quadrature ethod was frst used b Volterra to solve tegro-dfferetal equatos wth varable boudares The authors epress ther thas to the acadeca Al Abbasov for hs suggesto that to vestgate the coputatoal aspects of our proble Ths wor was supported b the Scece Developet Foudato of Azerbaa (Grad EIF--(3)- 8/7/) G Yu Mehdeva - doctor of scece PhD professor head ad char of Coputatoal atheatcs at Bau State Uverst Bau Azerbaa (the correspodg author s phoe uber s ad hs e-al address: _bsu@alru) MN Iaova PhD teacher of departet of Coputatoal atheatcs at Bau State Uverst Bau Azerbaa (e-al: _bsu@alru) VR Ibrahov - doctor of scece PhD professor the departet of Coputatoal atheatcs at Bau State Uverst Bau Azerbaa (e-al: bvag@alco) Cosder the followg tal-value proble Volterra tegro-dfferetal equatos: F( z( )) ) X () where () s a soluto of the proble The fucto z () s defed as follows: z( ) s)) d s () X Obvousl f the fucto z () s ow the proble () ca be rewrtte as: f ( )) ) (3) Therefore usg the ow quattes ad z z z to solve proble () ca be doe b applg the -step ethod wth costat coeffcets The we have: hf( z ) (4) Here z ( ) are the approate values of the fucto () ad z () for the pots h ( ) where the paraeter h s the tegrato step whch s dvded b the seget X to N equal parts It s eas to see that f there s a wa to detere z ( ) the t should be used forula (4) Furtherore we ca calculate the values of the fucto () of the esh pots ( N ) I ths case solvg proble () s equvalet to solvg a talvalue proble wth ordar dfferetal ad tegral equatos (see eg [] - [6]) Thu b eas of ultstep ethods wth costat coeffcets we ca fd the soluto to proble () Note that solvg tegral equatos ca be accoplshed wth several dfferet approato ethods (see eg [7] - [9]) I the class of probles () the ost basc research s o the followg proble: f ( s)) d ) (5) s X To solve proble (5) oe ca use the followg ultstep ethod (see eg [] or []): ISBN: ISSN: (Prt); ISSN: (Ole) WCE 3

2 Proceedgs of the World Cogress o Egeerg 3 Vol I WCE 3 Jul Lodo UK h f h z (6) ˆ z h ) (7) Ths ethod s obtaed b usg a ultstep ethod to solve both tegral equato () ad tal-value proble (5) Therefore solvg proble () ca be accoplshed wth oe of the approate ethods of ordar dfferetal equatos b eplog soe cobatos of the ethods proposed for solvg tegral equatos wth varable boudares The order of accurac of the stable ethod whch s costructed b the schee (6) - (7) does ot eceed ; ths result was establshed b Dahlquste (see []) Therefore scetsts have proposed varous was to costruct stable ethods wth a order of accurac greater tha To ths ed wor [3] a hbrd ethod that was frst vestgated b Gear ad Butcher was appled to solve the proble () (see [4] [5]) However [] the estece of stable forward upg ethods wth a hgher order of accurac tha was prove ad a ethod to solve Volterra tegral equatos was proposed We rear that [6] stable hbrd ethods wth a hgher order of accurac tha were costructed but [7] a hbrd ethod was appled to eted Maroglou s deas for solvg equato () Thu we fd that the uercal ethods of ordar dfferetal equatos ca be appled to solve both tegral equatos of tpe () ad tal-value probles wth the for of () Note that f oe wshes to solve Volterra tegral equatos usg quadrature or other ethods that are dfferet fro ethod (7) the oe caot eclusvel use the ethods of ordar dfferetal equatos to solve proble () However f the erel of the tegral s degeerate e f z a ( ) b ( z (8) the proble () ca be reduced to a sste of ordar dfferetal equatos Obvousl ths case proble () ca be solved usg the ethods of ordar dfferetal equatos I ths wor we costructed stable hbrd ethods wth a hgh order of accurac that used forato about the soluto of proble () ol all The proposed wor s a cotuato of the vestgatos coducted [6] Cosder the applcato of the followg ethod for solvg proble (): h (9) ; fro whch we a obta a well-ow hbrd ethods I [8] ethod (9) s appled to solve proble (3) ad t was proved that there est stable ethods of tpe (9) wth degree p 3 II THE APPLICATION OF THE GENERALISED HYBRID METHOD TO SOLVE PROBLEM () Aog the uercal ethods of both theoretcal ad practcal terest are covergg ethods It s ow that stablt s a ecessar ad suffcet codto for the ISBN: ISSN: (Prt); ISSN: (Ole) covergece of ultstep ethods Thu we vestgate the stable hbrd ethods that are appled to solve proble () Usuall the stud of ultstep ethods poses certa restrctos o the coeffcets (see eg []) These costrats o the coeffcets of ethod (9) ca be wrtte the followg for: A The values of the varables ( ) are real ubers ad B The characterstc poloals ( ) ; ( ) ; ( ) of ethod (9) have o coo ultpler that s ot a costat C ( ) () ad p Here p s the order of accurac of ethod (9) whch s defed the followg for: Defto For a suffcetl sooth fucto () ethod (9) has the degree p f the followg holds: h) h h) ( ) h) p O( h ) h () Codto A s obvous Therefore we cosder codto B ad assue the coverse The the poloals () () ad () have a coo ultpler whch we deote b () After tag to accout the shft operator E ( E ) h)) the fte-dfferece equato (9) ca be rewrtte as follows: ( E) ( ) ( ) h E h E () Let us use the gve assuptos to rewrte equato () the followg for: ( E)( ( E) ( ) ( ) h E h E ) Here ( ) ( ) / ( ); ( ) ( ) / ( ); ( ) ( ) / ( ) Hece we fd that ( E) ( ) ( ) h E h E () because () cost Obvousl to have a uque soluto of fte-dfferece equato () there should be o ore tha tal data However fro the theor of fte-dfferece equatos t s ow that for a ftedfferece equato of order to have a uque soluto tal data are requred However the dfferece equatos () ad (9) are equvalet Hece dfferece equato (9) has a uque soluto despte havg o ore tha tal data whch cotradcts the above-etoed theor Cosequetl the assupto that there s a coo factor of the poloals () () ad () s correct Now cosder the valdt of codto C Assue that ethod (9) coverges The as (9) approaches the lt ad as h we have: ) h (3) Because ( ) fro equato (3) we have: WCE 3

3 Proceedgs of the World Cogress o Egeerg 3 Vol I WCE 3 Jul Lodo UK ( ) (4) Equato (4) s a ecessar codto for the covergece of the ethod defed b forula (9) ad b usg t we ca wrte ( ) ( ) ( ) Furtherore b usg () we obta: ( E)( ) ( ) ( ) h E h E (5) Here b chagg the value of varable fro to ad sug the resultg equato we obta: ( E)( ) ( ) ( ) h E h E The as h we have: () () ( ) d ()( ) ) (6) However fro proble () we ca wrte: ) ) f ( )) d (7) B coparg (6) ad (7) t s clear that () () () () It s eas to prove that due to the codtos ( ) ; () () () p Now we ust prove that ( ) () Assue otherwse The fro the codtos ( ) ad ( ) we obta that s a double root of the poloal () Cosder the hoogeeous fte-dfferece equato whose geeral soluto ca be wrtte the followg for: c c c33 c where ( ) are the roots of the poloal () Hece as h we ow that because Thu f ( ) () the the ethod does ot coverge It follows that ( ) () If we use the codtos ( ) ad ( ) () () asptotc relato () the we obta that p Now cosder the applcato of ethod (9) to solve proble () To ths ed we vestgate the uercal soluto of proble () b usg the followg ethods: h F h F (8) h z h ( ) ) ) (9) To stud ethod (9) we suggest that the erel of the tegral K ( s a cotuous fucto that s defed the rego G s X b ad that has cotuous dervatves up to ad cludg soe order p If ethod (9) we tae to accout the propertes of the fucto K ( the we have followg (see eg [7]): ; ( ) Method (9) as a uercal ethod for solvg Volterra tegral equatos s studed [7] We rear that ethod (8) s a geeralsato of hbrd ethods I the past few ear scetsts have thoroughl studed the applcato of hbrd ethods to solvg tal-value probles wth ordar dfferetal equatos ad Volterra tegrodfferetal equatos (see eg [3] - []) Let us cosder fdg the coeffcets ethods (9) ad (9) It ca be show that b usg the Talor epasos ( h) h) ) h ) )! () p ( h) ( p) p ( ) O( h ) p! ( lh) lh) ) lh ( ) ( )! () p ( lh) ( p) p ( ) O( h ) ( p )! asptotc equato () we ca obta the ecessar ad suffcet codtos for equato () where h s the fed pot ad l ( ) These codtos ca be wrtte the for of sstes of equatos that cosst of the followg olear equatos: ( ) p p p ( ) ( ) p! ( p )! ( p )! () ( l p) It s eas to detere that sste () for the values ( ) s lear ad cocdes wth ow sstes that are used to detere the coeffcets of the ultstep ethod wth costat coeffcets Furtherore for the codtos sste () s olear; b solvg t we detere the coeffcets of ethod (9) I ths sste the uber of uows s equal to 4 4 ad the uber of equatos s equal to p Because sste () s hoogeeou t alwas has a trval soluto but to esure that sste () wll have a soluto that s dfferet fro zero the codto 4 4 p ust hold Thu oe ca be wrte the followg: p 4 Note that f we tae ( ) the the relatoshp betwee the degree ad the order of ethod (9) wll be as follows: ISBN: ISSN: (Prt); ISSN: (Ole) WCE 3

4 Proceedgs of the World Cogress o Egeerg 3 Vol I WCE 3 Jul Lodo UK p 3 It s ow that f we cosder the case ( ) the the degree of the stable ethod receved fro forula (9) satsfes the codto p (see []) To detere the coeffcets ethod (9) cosder a specal case ad let F( The fro () we have F( ( ) (3) If we appl ethod (9) to solve proble (3) the we obta: ˆ h ˆ F h ˆ F (4) where ˆ ; ˆ ( ) (5) Frst fro sste () we detere the values of ˆ ˆ ˆ ( ) ad the b solvg sste (5) we fd the coeffcets of ethod (9) Note that sste (5) the uber of equatos s equal to ad the uber of uows s greater tha Cosequetl the soluto of sste (5) s ot uque Therefore although the ethod of tpe (9) a be uque the correspodg ethod of tpe (9) s ot uque Ths fact allows us to select soe of the coeffcets to costruct the ethod wth a eteded rego of stablt Cosder specal cases ad let The b solvg sste () ad usg the soluto sste (5) we obta a few ethods of degree p 6 Oe of the s the followg: h( F F ) 5h( F F ) (6) where the correspodg ethod of tpe (9) oe varable ca be wrtte as: z z h( ) )/ 4 5h( ) (7) ) ))/ 4 ( 5 ) Note that the ethod wth degree p 8 for s as follows: ( h ) 8 (8) h( ) 8 ) where the value 4 For the sae of splct let us cosder the applcato of the followg ethod: h ( / / ( ) / 3 / 6) (9) Ths ethod wll be used to solve proble (); to use t oe ust defe the values of / / whch ca be doe as follows: 3 ˆ / / ((4 6 ) (3) 3 3 (8 4 ) / (4 6 ) )/ 4 B the forula ˆ h (3) ˆ / we ca fd the varable The we ca detere 3 / b the followg forula: h( 7 / ) / 6 3 / / (3) B usg the et sequece of ethod oe ca solve tal-value proble () Step Calculate ˆ b forula (3) Step Calculate b forula (3) / Step 3 Calculate b forula (9) Step 4 Calculate 3 / b forula (3) To llustrate these result cosder the followg table Oe a also cosder ad copare the results obtaed b ethods of tpe (9) wth other ow ethods usg the followg odel probles: ep( ) (the eact soluto s ) Nuber of eaple I h 5 II h / tep( ( t)) dt ) ) B the wor of Gragg & Stetter (see[3]) 3E-9 5E-8 5E-6 B the wor of Kohfeld & Thoso (see[3]) 35E- 8E-8 4E-7 B the ethod fro [3] Ma error 8E-7 For hbrd ethod (9) 33E- E-8 E-6 63E-9 7E-7 5E (4 ep( ) 3 ep( ( )) 3 s s ds ) (the eact soluto s ) l ) Note that the receved results are cosstet wth the theoretcal results preseted here Rear It s ow that scetsts have vestgated the uercal solutos of ordar dfferetal equatos because the wshed to solve tegral ad tegrodfferetal equatos b applg the ethods of dfferetal equatos For the sae of deostrato suppose that the erel s degeerate ad has the followg for: a( ) b( (33) I ths case oe ca rewrte the proble as follows: f a( ) v( ) ) (34) ( ISBN: ISSN: (Prt); ISSN: (Ole) WCE 3

5 Proceedgs of the World Cogress o Egeerg 3 Vol I WCE 3 Jul Lodo UK ( ) b( v( ) v (35) Thu oe ca replace solvg proble () wth solvg probles (34) ad (35) whch are tal-value probles that ca be solved wth ordar dfferetal equatos It s ow that the proble ecapsulated b (34) ad (35) cossts of two ODEs of the frst order Ufortuatel ths splfcato s ot alwas correct Ideed b the dervatve of (33) we obta df ( )) a ( ) a( ) b( d The above equato s a tegro-dfferetal equato I the cosdered eaple the erels have the for (33) Cocluso I ths paper soe forato about solvg tegro-dfferetal equatos s gve We vestgated solvg a tal-value proble the class of Volterra tegro-dfferetal equatos usg hbrd ethod ad we bega wth the wor of Maroglou (see [3]) Note that the costructed hbrd ethods are setrcal However asetrc hbrd ethods are usuall ore accurate tha setrc oes We costructed a asetrc stable hbrd ethod wth degree p 9 for the case However the applcato of ths ethod to practcal probles s ore dffcult tha usg setrc ethods I a sgle artcle t s ot possble to vestgate all aspects of ths proble We beleve that the proposed ethod wll have a applcatos the future Note that to appl (6) - (8) to solve certa proble oe ca use bloc ethods or ethods of predctor-corrector [3] A Maroglou Hbrd ethods the uercal soluto of Volterra tegro-dfferetal equatos Joural of Nuercal Aalss 98 pp-35 [4] CS Gear Hbrd ethods for tal value probles ordar dfferetal equatos SIAM J Nuer Aal v 965 pp [5] Butcher JC A odfed ultstep ethod for the uercal tegrato of ordar dfferetal equatos J Assoc Coput Math v 965 pp4-35 [6] GMehdeva MIaova VIbrahov Applcato of the hbrd ethods to solvg Volterra tegro-dfferetal equatos World Acade of Scece egeerg ad Techolog Par 97- [7] G Mehdeva VIbrahov MIaova O the costructo test equatos ad ts Applg to solvg Volterra tegral equato Metheatcal ethods for forato scece ad ecooc Motreu Swtzerlad pp 9-4 [8] GYuMehdeva MN Iaova VR Ibrahov O a wa for costructg uercal ethods o the ot of ultstep ad hbrd ethods World Acade of Scece egeerg ad Techolog Par 4-43 [9] GK Gupta A poloal represetato of hbrd ethods for solvg ordar dfferetal equato Matheatcs of cop volue 33 uber pp5-56 [] Areo EA RA Adelu Babatola PO Accurate collocato ultstep ethod for tegrato of frst order ordar dfferetal equatos // Jof Moder Mathad Statstc (): -6 8 P -6 [] OAAfewa NMYao SNJator Iplct Two step cotuous hbrd bloc ethods wth four off steps pots for solvg stff ordar dfferetal equato WASET 5 p45-48 REFERENCES [] VVolterra Theor of fuctoal ad of tegral ad tegrodfferesal equato Dover publcatos Ig New Yor p [] PLz Lear Multstep ethods for Volterra Itegro-Dfferetal equato Joural of the Assocato for Coputg Macher Vol6 No Aprl 969 pp95-3 [3] Feldste A Sopa JR Nuercal ethods for olear Volterra tegro dfferetal equatos // SIAM J Nuer Aal 974 V P [4] HBruer Ilct Ruge-Kutta Methods of Optal oreder for Volterra tegro-dfferetal equato Metheatcs of coputato Volue 4 Nuber 65 Jauar 984 pp 95-9 [5] Maroglou AA Bloc - b-bloc ethod for the uercal soluto of Volterra dela tegro-dfferetal equato Coputg p49-6 [6] Bulatov МB Chstaov EB Chsleoe reshee teqrodfferesal sste s vrodeo atrse pered prozvodo oqoshaqov etoda Df Equato pp8-55 [7] AFVerla VS Szov Itegral equatos: ethod algorth progras Kev Nauova Dua 986 (Russa) [8] OSBudova MV Bulatov The uercal soluto of equatos tegroalgebrachesh ultstep ethod Joural of Coput Math ad atfz т5 5 p (Russa) [9] ЯДМамедов ВА Мусаев Исследование решений системы нелинейных операторных уравнений Вольтера-Фредгольма ДАН СССР т [] G Mehdeva VIbrahov MIaova Research of a ultstep ethod appled to uercal soluto of Volterra tegro-dfferetal equato World Acade of Scece egeerg ad Techolog Asrterda pp [] ГЮМехтиева Ибрагимов ВР МНИманова Applcato of A Secod Dervatve Mult-Step Method to Nuercal Soluto of Volterra Itegral Equato of Secod Kd Pastatoperres VolVIII No [] GDahlqust Covergece ad stablt the uercal tegrato of ordar dfferetal equatos Math Scad p33-53 ISBN: ISSN: (Prt); ISSN: (Ole) WCE 3