Numerical Solution of a non-linear Volterra Integrodifferential Equation via Runge-Kutta-Verner Method

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1 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 ISSN -33 Nmril Solio of o-lir Volrr Igroiffril Eqio vi Rg-K-Vrr Mho Ali Filiz * * Dprm of Mhmis A Mrs Uivrsiy 9 AYDIN-TURKEY Asr- I his ppr highr-orr mril solio of o-lir Volrr igro-iffril qio is isss. Exmpl of his qsio hs solv mrilly sig h Rg-K-Vrr mho for Oriry Diffril Eqio (ODE) pr Nwo-Cos formls for igrl prs. Ix Trms- A highr-orr ry Lgrg irpolig qrr formls Rg-K mhos o-lir Volrr igro-iffril qio. I. INTRODUCTION A fiol qio i whih h ow fio pprs i h form of i is riviv s wll s r h igrl sig is ll igro-iffril qio (s [ 4 ]). I his ppr w will osir h o-lir Volrr igroiffril qio of h form (s [ 4 7 8]) ( ) ) ( ( )( ). () ' Eqio () solv mrilly sig vrios mhos (s [ 9 ]). I his ppr ) will o h x vl of h. W shll s ( ) or o o mril solio of. Howvr i his ppr w will osr highr-orr mril mho for qio (). Si h igrl o rmi xpliily i my pproxim sig fmilir mril igrio mhos. Th Nwo-Cos igrio forml whih il h -poi los Nwo-Cos forml is ll h rpzoil rl h 3-poi rl is ow s Simpso s /3 rl h 4-poi los rl is Simpso s 3/8 rl h -poi los rl is Bool s rl (Bo s rl Wl s rl highr rls il h -poi 7-poi 8-poi r wll si hr si hy s os whih wr giv i [ 37] [4 ]. II. THE NUMERICAL INTEGRATION OF A NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION I grl forml for h mril solio of igro-iffril qios rly po forml for h rlyig Oriry Diffril Eqio (ODE omi wih xiliry qrr rls pproximio of z ( ) : h ( ) ) ( s. () Of ors whrs w hv fi pproximios z ( ) i rms of qrr rls h rfl h rlyig ODE mho i is i priipl possil o mix mh. Th omiios of forml hos o h sis of orr of ovrg. Th firs ivolvs pig Rg-K mhos. W will rqir o pproxim igrl rm z ( ) : h ( ) ( ) ( s sl vls. Eqio () solv svrl wys. I his ppr w shll fos o highr-orr mril mho for qio (). Th igrl my pproxim sig fmilir mril igrio mhos. Th Nwo-Cos igrio formls whih il lf righ rgl rls h rpzoil rl Simpso s /3 rl Simpso s 3/8 rl r wll si hr si hy s os whih wr prviosly ll [ ]: (

2 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 ISSN -33 ( s h ( ) ( ) whr r h ppropri offiis for h omposi igrio shms hos. A omiio of igrio mho my s. III. NUMERICAL ROUTINE FOR NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION Now osir h o-imsiol prolm (). I orr o solv () mrilly w prpos h s of wo mhos fmilir o mos mhmiis. W osir mhos whih pproxim h solio h iiil vl prolm (IVP) im ' ( ) ) ( s ) h = 3. whr h is h os ol sp-siz i h Exmpl 3. ( r ( s. For xmpl h xplii Elr mho pproxims h solio o Exmpl 3. ( ) ( ). s h r s s Th xplii fii iffr mho giv i [] s ppli o qio () sily x o mor r prior-orror mho. Th prior sp ss ( ( h F z( )) ) o oi whih is follow y h orror sp whih ss highr orr rpzoil mho ( )) ( h F z F z ( ). This pror is ow s moifi Elr mho (so orr Rg-K-RK) is o orr mgi mor r h h xplii Elr mho. Th forh orr lssil Rg-K mho (RK4) lso p o h mril solio of qio (). Sppig from wih sp-siz h o oi h RK4 mho s ppli o his prolm i [ ]. Th sixh orr Rg-K-Vrr mhos [3] my s o rily si h irol vlio pois r iformly sp. Cosqly h igrls rig h irmi llios o sp from o my rqir h rpzoil rl or Lgrg polyomil irpolig igrio o o-iform priio ]. [ Rg-K-Vrr mho (RKV) lso p o h mril solio of (). Sppig from h RKV mho s ppli o his prolm my wri s: h z( ) (3) wih sp-siz h o oi / h / / / / z h h/ F/ / z / 4 4/ 7 7

3 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 3 ISSN -33 h 3 4/ 4/ 4/ 4/ z 4 h 3 h4/ F4/ 4/ z 4/ / 3 3 h 4 / 3 / 3 / 3 / 3 z h 4 h/3 F/3 / 3 z / / h / / / / z h h/ F/ / z / h z h h F z f / 7 8 h f f 7 / / / / z f f f h 7 h / F/ / z / 3 g h g g 8 z g g g h 8 h F z (4) () I his xmpl h rpzoil rl is s o pproxim z ( ) ( s o ] ] ] [ [ / [ 4/ [ / 3 / ] [ ] [ / ] [ ] i llig rspivly. If sir h

4 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 4 ISSN -33 rpzoil rl my s o ] (givs so orr ry S Tl ); h rpzoil rl Simpso s /3 rl [ (givig hir orr ry s [ ]) my s o ]. [ I orr o g highr-orr ry h igrl rm ms vl mor rly o ] ] [ / [ 4/ [ / 3] [ / ] [ ] [ / ] [ ] i llig s show i () (7 (8 (9 ( ( () low. Th -poi x los rl is Bool's mho my vis o ] s followig: z()= )= If = z(+)= z() + h( ) + +) ) / lsif == z(+)= z(-) + h( -) +4 ) + +) ) /3 lsif ==3 z(+)= z(-) +3h ( -) +3 -) +3 ) + +) ) / 8 lsif ==4 z(+)= z(-3) +h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 lsif == z(+)= z(-4) +h (9-4) +7-3) + -)+ -) + 7 ) +9 +) ) / 88 lsif == z(+)= z(-) + h (4-)+ -4) +7-3) +7 -)+7 -) + ) +4 +) ) / 4 lsif ==7 z(+)= z(-) + 7h (7-)+377-)+ 33-4) ) )+33 -) ) +7 +) ) / 78 lsif ==8 z(+)= z(-3) + h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 lsif mo(4)== z(+)= z(-3) + h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 lsif mo(4)== z(+)= z(-3) + h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 lsif mo(4)== z(+)= z(-3) + h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 lsif mo(4)==3 z(+)= z(-3) + h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 ls z(+)= z(-3) + h (7-3) +3 -) +-) + 3 ) +7 +) ) / 4 If w irpolig o / (spil forml rqir for h firs wo sps for xmpl w s (4) ()) Lgrg s forml for pois =- - / givs 3 h h h ) ( ) ( ) ( )( ) 3( )( )( ) ( )( ) /. 3 h h h h h h h [

5 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 ISSN -33 If w igr h xprssio w h/ w g h / ( / 3 s h 8 84 Similrly w fi = =- - 4/ 34 s h h / ( 4/ fi = =- - /3 h / 3 ( /3 4 s h fi = =- - / 8 s h h / ( / fi = =- - h 3 s h ( fi = =- - / h / ( 3 s h 9 filly fi = =- - h 3 s h ( Thrfor h Rg-K-Vrr forml om 3 (for srig vls w s qio (4) ()) h z( ) ) () (7) (8) (9) () () () / h / / / / z h 3 z h / F/ / / 4 4/ 7 7 h 3 4/ 4/ 4/ 4/ h z 34 z h / F4/ 4/ 4/

6 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 ISSN / 3 3 h 4 / 3 / 3 / 3 / 3 z h 4 z h / 3 F/ 3 / 3 / / h / / / / h z 8 z h / F/ / / h h z 3 z h F f 3 / h f f 7 / / / / z f f 3 79 f 7 / / / / h F z h g 3 7 h g g 8 z g g h F z h (3) h sixh-orr mho is s o sim h rror i h fifh-orr mho I Exmpl 3. w hv s Rg-K-Vrr mhos mril qrr rpzoil rl h 3-poi rl is ow s Simpso s /3 rl h 4-poi los rl is Simpso s 3/8 rl h -poi los rl is Bool s rl (Bo s rl Wl s rl highr rls il h -poi 7-poi 8-poi hir omiios. Exmpl 3.: Cosir firs orr o-lir Volrr igro-iffril qio of h form ( ) ' ( ) ) r s s ; ). (4)

7 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 7 ISSN -33 For lyil solio of qio (4 i (4). I qio (4 if w hoos r. w will g x solio s ( ) ). Now wriig m( ) s his is h sm s h iffril qio ) ' ( ) )( r m( ) ' ( ) r ) m( ) If m( ) s h m' ( ) ) m' '( ) '( ). Aiiolly m( ) m' () ). m' '( ) r m'( )( r m( ) m' '( ) r m'( ) m'( ) m( Eqivl o h moifi logisi qio for m () m ( ) m'( ) r m( ) m'( m ( ) m'( ) r m( ) m ( ) m' ( ) r m( ) m ( ) ( m r m( ) ( ) ' m' ( ) ( m( ) )( m( ) () r m( ) whr r roos of m ( ). If w solv qio () wih iiil oiio ( ) m w g m( ) Afr rrrgig ov solio w oi ( m( ) ( ) ( ) ) ( m( ) ) whr ( )

8 Iriol Jorl of Siifi Rsrh Pliios Volm 3 Iss 9 Spmr 3 8 ISSN -33. W ow h or x solio ws (). Wh m' ( ) ) ) ( ) lyil solio of qio ( ) r m (4). Hr r roos of h qio m ( ) ( ) (whr ) so h is pproh y m lrg vls of. Th xpo is fi y ( ). Tl : Nmril solios (4) for RKV mho ( r. ) Nmril solio h= Nmril solio h= Al Solio Rsls mx Error wih h= Rg-K-Vrr mho (RKV) mril qrr rls rsls. REFERENCES Error wih h= [] M. Armowiz I. A. Sg (Es Igrio..4 i Hoo of Mhmil Fios wih Formls Grphs Mhmil Tls 9 h priig Nw Yor: Dovr 97 pp [] A. Asov Uiqss of h solio of sysms of ovolio-yp Volrr igrl qios of h firs i I: Ivrs prolms for iffril qios of h mhmil physis (Rssi) Novsiirs: A. N SSSR Siirs. Ol. Vyhil. Tsr 978 Vol pp. 34. [3] R. L. Br J. D. Firs Nmril Alysis. Nw Yor: Broos/Col Plishig Compy USA 997 h.. [4] C. T. H. Br Th Nmril Trm of Igrl Eqios. Clro Prss; Oxfor Uivrsiy Prss 977. [] C. T. H. Br G. A. Bohorov A. Filiz N. J. For C. A. H. Pl F. A. Rih A. Tg R. M. Thoms H. Ti D. R. Will Nmril Mollig y Rr Fiol Diffril Eqios Nmril Alysis Rpor Mhsr Cr for Compiol Mhmis No:33 ISS [] C. T. H. Br G. A. Bohorov A. Filiz N. J. For C. A. H. Pl F. A. Rih A. Tg R. M. Thoms H. Ti D. R. Will Nmril Mollig y Dly Volrr Fiol Diffril Eqios Nmril Alysis Rpor I: Compr Mhmis is Apliios-Avs & Dvlopms (994- Elis A. Lipiis (Eior LEA Plishrs Ahs Gr pp [7] R. Bllm A Srvy of h Thory of h Boss Siliy Asympoi Bhvior of Solios of Lir No-lir iffril iffr qios Wshigo D. C [8] K. L. Coo Fiol Diffril Eqios Clos o Diffril Eqio Amr. Mh. So. 9 Vol 7 pp [9] A. Filiz O h solio of Volrr Lo-Volrr Typ Eqios LMS sppor O Dy Mig i Dly Diffril qio (Livrpool UK h Mrh. [] A. Filiz Nmril Solio of Som Volrr Igrl Eqios PhD Thsis Th Uivrsiy of Mhsr. [] A. Filiz Forh-Orr Ros Nmril Mho for Igro-iffril Eqios Asi Jorl of Fzzy Appli Mhmis 3 Vol I pp [] P. Liz Alyil Nmril Mhos for Volrr Eqios SIAM Phillphi 98. [3] C. W. Urhr Nmril Compio : Mhos Sofwr lysis. Brli: Sprigr-Vrlg 997. [4] V. Volrr Lços Sr l Thori Mhmiq l L Por L Vi. Ghir-villrs Pris 93. [] V. Volrr Thory of Fiol of Igro-Diffril Eqios. Dovr Nw Yor 99. [] V. Volrr Sll Eqzioi Igro-iffrzili Dll Tori Dll lsi Ai Dll Rl Ami i Lii 8 (99 Rpri i Vio Volrr Opr Mhmih; Mmori No Vol 3 Ami i Lii Rom 97. [7] Wolfrm MhWorl Nwo-Cos Formls vill hp://mhworl.wolfrm.om/nwo-cosformls.hml AUTHORS Firs Ahor Ahor Nm: Dr. Ali Filiz BS (Eg Uivrsiy TR MS (Th Mhsr Uivrsiy UK PhD (Th Mhsr Uivrsiy UK Assis Profssor Dprm of Mhmis A Mrs Uivrsiy 9 AYDIN- TURKEY E-mil: filiz@..r Corrspo Ahor Dr. Ali Filiz E-mil: filiz@..r. Co Nmr: (+9) x. 4