Review of Numerical Schemes for Two Point Second Order Non-Linear Boundary Value Problems

Size: px

Start display at page:

Download "Review of Numerical Schemes for Two Point Second Order Non-Linear Boundary Value Problems"

Veronica Cook
5 years ago
Views:

1 Proceedngs of e Pasan Academ of Scences 5 (: 5-58 (5 Coprg Pasan Academ of Scences ISS: (prn, (onlne Pasan Academ of Scences Researc Arcle Revew of umercal Scemes for Two Pon Second Order on-lnear Boundar Value Problems Abdul Manaf *, Musafa Habb, and M O Amad Deparmen of Maemacs, Unvers of Engneerng and Tecnolog, Laore, Pasan Deparmen of Maemacs, Unvers of Laore, Laore, Pasan Absrac: In s paper, numercal soluon of wo pons nd order nonlnear boundar-value problems was consdered Te numercal soluon was revewed w nonlnear soong meod, fne-dfference meod and four order compac meod Te resuls were compared o cec e accurac of numercal scemes w exac soluon I was found a e nonlnear soong meod s more accurae an fne-dfference meod and four order compac meod Kewords: BVPs, ODEs, onlnear Soong meod, fne-dfference meod, four order compac meod ITRODUCTIO Consder a wo pon nd order nonlnear boundar value problem (BVP of e pe f(, x,, x ab, ; ab, R ( jonl among e boundar condons a ( and b ( Were α and β are consans ( Te approac for solvng s problem as been projeced b a number of researcers suc as Robers and Spman [], Mala [], Ha [3], Auznger e al [4] and Al and Sam [5] In s researc paper, we consdered nonlnear soong meod (LSM, fne-dfference meod (FDM and four order compac meod (FOCM for e soluon of above wo pons nd order nonlnear boundar value problems (BVPs OLIEAR SHOOTIG METHOD (LSM Consder a wo pon s nd order non-lnear boundar-value problem (BVP f( x,,, a (, b ( (3 were axband α, β are consans Here, we used e soluons o a sequence of nal value problems (IVPs of [3, 5, 6] f( x,,, a (, ( a (4 Concernng a parameer, and ax b, o approxmae e soluon o our BVP (3 B coosng parameers n suc a wa a lm b (, b ( (5 Were x (, s e soluon o e IVP (4 w and (x s e soluon o e BVP (3 Ts procedure s called a Soong meod We begn w a parameer a fnd ou e nal elevaon a wc e objec s exced from e pon ( a, and besde e curve descrbed b e soluon o e IVP [6] f( x,,, a (, ( a (6 Receved, ovember 4; Acceped, Februar 5 *Correspondng auor: Abdul Manaf; Emal: manaf9@aoocom

2 5 Abdul Manaf e al If b (, s no sasfacorl close o β, we r o correc our approxmaon b coosng anoer elevaon and so on, unl b (, s sasfacorl close o srng β [3] To decde ow e parameer can be cosen, suppose a BVP (3 as a sngle soluon If (x, s e soluon o e IVP (4, en we need o deermne so a b (, (7 Snce s s a nonlnear equaon, we use ewon s meod f( xn x n x n f( x o solve e problem We need o coose nal approxmaon and en generae sequence b b (, d ( b, d (8 d s requres e nowledge of (, b Ts d presens a complex, snce an explc llusraon for (b, s no denfed; we onl now e values b (,, b (,,, b (, Suppose we modf e IVP (4, empaszng a e soluon depends on ogeer x and (, x f xx, (,, (, x, ax b, a (,, (, a (9 reanng e prme noaon o ndcae dfferenaon w respec o x d Snce we need o deermne (, b, wen d, we ae paral dervave of (9 w respec o ( x, f x, x (,, ( x, f x x n f x (, f ( x, x Snce x and are ndependen,, so f x (, ( x, f ( x, for ax b Te nal condons gve (, a, and ( a, ( If we mae smpler e noaon b usng zx (, o denoe (, x and suppose a e order of dfferenaon of x & can be reversed, Eq ( w nal condons becomes IVP [6] f f z( x, zx (, z( x,, ax b, za (,, z(, a ( Terefore, one requres a wo IVPs be solved for eac eraon, (4 and ( Ten from Eq (8, b (, ( Zb (, In pracce, none of ese IVPs s solved exacl; nsead, e soluons are approxmaed b one of e IVP solvers [3] Hence, n soong meod for nonlnear BVPs, we use classcal Runge-Kua four-order meod o approxmae bo soluons requred b ewon s meod 3 FIITE-DIFFERECE M ETHOD (FDM Meods nvolvng fne dfferences for solvng boundar value problems (BVPs replace eac of e dervaves n e dfferenal equaon b an approprae dfference-quoen approxmaon Te

3 Revew of umercal Scemes 53 dfference quoen s cosen o manan a specfed order of runcaon error [7] Te nonlnear second order BVP of e form (3 requres a dfference quoen approxmaons be used o approxmae bo and Frs, we ae >, wc s an neger, and brea up ab, no + equal subnervals, wose end pons are mes pons b a x a, for,,,, were A neror mes pons, x for,,,,, followng dfferenalequaon s o be approxmaed ( x f x, x (, ( x (3 Expandng, n e rd Talor-polnomal, abou x evaluaed a x and x, we ge x x x x x! ( ( ( ( ( ( for some nx, x, and x x x x x ( ( ( ( ( ( for some x, x, assumng 4 C x, x B addng e above wo equaons, we ave n 4 4 ( x ( x ( x ( x ( ( 4 Usng nermedae value eorem, s can be smplfed even furer as x x x x 4 ( ( ( ( ( (4 for some n x x, Ts s called e cenered-dfference formula for ( x A cenered dfference formula for ( x s obaned n a relaed manner resulng n ( x x ( x ( ( (5 6 for some n( x, x Te use of e cenered dfference formulas n equaon (3 gves ( x ( x ( x ( x ( x f x x 6 (4, (, ( ( for some and n e nerval ( x, x As n lnear case, e dfference meod resuls wen e error erms are deleed and boundar condons emploed [7]: w, w, and w w w,, w w f x w for eac,,, Te nonlnear ssem obaned from s meod s w w w fx, w,, w 3w w w w 3fx, w,, w w w w w fx, w,, w w w fx, w, To approxmae e soluon o s ssem, we use ewon s meod for nonlnear ssem A sequence of eraes ( ( ( w, w,, w s generaed a converges o e soluon of ssem, provded a e nal approxmaon ( ( ( w, w,, w s suffcenl close o e soluon, w, w,, w, and a e Jacoban marx for e ssem s nonsngular For e ssem, e Jacoban marx s rdagonal ewon s meod for nonlnear ssems requres a a eac eraon, e lnear ssem [8]

4 54 Abdul Manaf e al,,,,,, J w w w v v v w w w f x, w,, w w,,,, w w w w w f x, w,, 3 w w w 3fx w w w w f x, w, be solved for v, v,, v, snce w w v, for eac,,, (6 ( ( 4 FOURTH ORDER COMPACT METHOD (FOCM A four order general dfferencng sceme, proposed b Kress [9], s developed and s appled on e ree es problems o confrm accurac and applcabl of s ecnque Burger s equaon, e Howar s rearded boundar flow, and e ncompressble drven cav, are e solved es problems [6, 8, 9] A number of researcers ave conrbued n s meod, some of em are Amad [8], Orsazag and Israel [], Rcard [], Levenal and Cmen [] and Pegrew and Rasmussen [3] Te dealed wor on compac meod was done b Amad [8] We follow e procedure of Pegrew and Rasmussen [3] o derve e compac meod for nonlnear ODEs [4, 5] Consder e nonlnear wo pon second order BVP of e form (3 Le us denoe frs and second dervaves w respec o x of b F and S respecvel, e F, and S (7 F S Consder F Inegrang bo sdes from x o x, we ge x x x F( d (8 F( d x Approxmang negral b Smpson s rule [8] 5 4 F4 F F F ( F 4 F F F ( 3 9 mulplng bo sdes b 3, we ge F 4 F F F( 3 and o e four order, we ave 3 F 4F F (9 Tus we ave a relaonsp beween and F and s e frs dfference-equaon ow for e second equaon, we sar b evaluang equaon (eq ( a e mdpon Tus equaon ( becomes [7] f x, ( x, ( x Snce S, so we ave S fx, x (, F ( ow, we requre an expresson fors If we expand n e 5 Talor s polnomal on x evaluaed a x and a e x, we ge (4 (5 (6 (! 3! 4! 5! 6! for some nx, x, and (4 (5 (6 (! 3! 4! 5! 6! for some n x, x, assumng 6 C x, x If we add ese equaons, (5 erms nvolvng, and are elmnaed and we ge (6 (6 ( ( (6 (6 S ( ( 7 Here nermedae-value eorem s used o furer smplf s as (6 S ( ( 36

5 Revew of umercal Scemes 55 for some nx, x ow expandng F n Talor seres 3 4 (5 (4 (5 F F F F F F F ( (! 3! 4! 5! 3 4 (5 (4 (5 F F F F F F F ( (3! 3! 4! 5! On subracng eq(3 from eq(, we ge 3 (5 (5 (5 F F F F F ( F ( 3 B usng nermedae value eorem, we ge 3 (5 (5 F F F F F ( 3 6 For a few nx, x (4 (5 (6 Snce, F S, F, F, So, 3 (5 (4 (6 F F S ( (4 3 6 Elmnang (4 from eq ( and (4, we ge S F F 36 (4 (6 ( B a smlar procedure, we ge (4 (6 S F8 F F ( 9 (4 (6 S F8 F 6 F ( 9 and o e four order, we ave S F F (5 S F 8F F (6 S F F F (7 From eq (5, pung e value ofs n eq (, we ge FF fx, x (, F (8 We ave replaced now dfferenal equaon (3 b wo dfference-equaons (9 and (8 ow, consder s boundar condon e a x=a, and denoe e pons x a, a, a b,, e frs dfference-equaon, we oban from e boundar condon, s [7] (9 ow o oban e second equaon, we begn w dfferenal equaon a pons and For S f x,, F (3 S f x,, F (3, eq(6 mples S F F F (3 Eq (5 mples S F F (33 Eq (9 mples 3 F 4FF (34 ow, we ave fve equaons from (3 o (34 We elmnaes, S,, F from ese equaons From eq (3 and (3, we ave f x,, F F F F (35 From eq (3 and (33, we ave 4 f x,, F F F (36 From eq (34, we ge 4 F F F ( Pung s value of n equaons (35 and (36, we ge f x,, F F F F ( f x,, F F F F ( Subracng eq (38 from eq (39, we ge 6 6 f x,, F f x,, F F F (4 In a smlar approac, we derve e followng wo dfference-equaons for and F a x=m e a e rg boundar pon [7] m (4 6 f xm, m, Fm f xm, m, Fm m m Fm Fm (4 Tus for eac pon we ave wo dfference equaons If we wre em all ogeer, we ave e followng Four Order Compac Sceme

6 56 Abdul Manaf e al 6 6 F F f x F f x F 3 3 F 4F F 4 F F f x,, F,,,, 5 RESULTS AD DISCUSSIO Tes Problem-: Consder e wo pons nd order nonlnear BVP of e form 3 ( x,x 5, (, (5 m m m Fm Fm f x m, m, Fm f x m, m, Fm and s exac soluon s ( x x 3 Table Comparson of resuls x Exac values FDM resuls FOCM resuls LSM resuls Te above Table sows a resuls of LSM are more accurae o e exac soluon as compared w e resuls of FDM and FOCM Fg Comparson of resuls Table Comparson of absolue errors x FDM resuls FOCM resuls LSM resuls Te above able sows e comparson of absolue errors for FDM, FOCM and LSM I s found a an absolue error for LSM s less as compared o e oer wo meods

7 Revew of umercal Scemes 57 Tes Problem-: Consder e wo pons nd order nonlnear BVP of e form x, (, ( and s acual soluon s ( x 3 x 4 Table 3 Comparson of resuls x 3 (, x Exac values FDM resuls FOCM resuls LSM resuls Te above able sows a resuls of LSM are more accurae o e exac soluon as compared w e resuls of FDM and FOCM Fg Comparson of resuls Table 4 Comparson of absolue errors x FDM resuls FOCM resuls LSM resuls Te above able sows e comparson of absolue errors for FDM, FOCM and LSM I s found a an absolue error for LSM s less as compared o e oer wo meods

8 58 Abdul Manaf e al 6 COCLUSIO In s paper, we consdered wo dfferen nonlnear nd order wo pons BVPs for ordnar dfferenal equaons (ODEs and solved ese problems usng LSM, FDM and FOCM All e above meods are approprae for solvng wo pons nd order nonlnear BVPs, bu was found a LSM s more accurae an FDM and FOCM 7 REFERECES Robers, SM, & JS Spman Two Pon BVPs; Soong Meods Amercan Elsever, ew Yor, USA (97 Mala, V Solvng Boundar Value Problems for Ordnar Dfferenal Equaons Usng Drec Inegraon and Soong Tecnques PD ess, Unvers Pura Malasa, Malasa (999 3 Ha, S A nonlnear soong meod for wo-pon BVPs Compuer and Maemacs Applcaons 4: 4-4 ( 4 Auznger W, G Knesl, O Koc, & E Wenmuller A collocaon code for boundar value problems n ordnar dfferenal equaons umercal Algorms 33: 7 39 (3 5 Al, BS & MI Sam Effcen soong meod for solvng wo pon BVPs Caos, Solons and Fracals 35: (8 6 Fares, JD & RL Burden umercal Analss Inernaonal Tomson Publsng Inc, ew Yor, USA (998 7 Musafa, H Four Order Compac Meod for onlnear Second Order BVPs MPl ess, Unvers of Engneerng and Tecnolog, Laore, Pasan (7 8 Amad, MO An Exploraon of Compac Fne Dfference-meods for umercal Soluon of PDE PD ess, Unvers of Wesern, Onaro, Canada (997 9 Kress, HO Meods for e Approxmae Soluon of Tme Dependen Problems GARP Repor o 3 (975 Orszag, SA & M Israel umercal smulaon of vscous ncompressble flows Annual Revews Inc 6: 8-38 (974 Rcard, SH & DH Rud Te role of dagonal domnance and cell Renolds number n mplc dfference meods for flud mecancs problems Journal of Compuaonal Pscs 6: 34-3 (974 Levenal, SH & M Cmen A noe on e operaor compac mplc meod for e wave equaon Maemacs of Compuaon 3(: (978 3 Pegrew, MF & H Rasmussen A compac meod for second order boundar value on neon unform gads Compuer and Maemacs Applcaons 3: -6 (996 4 Scllng, RJ & SL Harres Appled umercal Meods for Engneers Tomson Busness Informaon Inda, Bangalore, Inda (6 5 Sasr, SS Inroducor Meods of umercal Analss Prence Hall of Inda, ew Del, Inda (3

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: www.modernscenfcpress.com/journals/jmms.aspx ISSN: 66-86X Florda, USA Approxmae