THE CONVERGENCE AND ORDER OF THE 3-POINT BLOCK EXTENDED BACKWARD DIFFERENTIATION FORMULA

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1 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom THE ONVERGENE AND ORDER OF THE -POINT BLOK EXTENDED BAKWARD DIFFERENTIATION FORMULA H Musa M B Suleiman F Ismail N Senu and Z B Ibrahim Department o Mathematics Facult o Science Universiti Putra Malasia Serdang Selangor Malasia Institute or Mathematical Research Universiti Putra Malasia Serdang Selangor Malasia hamisuhm@ahoocom ABSTRAT In this paper we consider the ull implicit -point Bloc Etended Bacward Dierentiation Formula or solving sti initial value problems The iterative bloc method is proven to be convergent b establishing zero stabilit and consistenc conditions Numerical results are given to show the eect o zero stabilit and consistenc The accurac is seen to improve as the step length tends to zero The order o the method is also shown to be 6 Kewords: convergence order o bloc method blocs etended bacward dierentiation ormula INTRODUTION onsider the irst order sti initial value problem (IVP) ' ( ) a ( ) [ ab ] () Such dierential euations occur in man ields o engineering science and in particular the appear in electrical circuit vibrations chemical reactions inetics etc Developing methods or solving () still remains a challenge in modern numerical analsis Seuential methods among them include (urtiss et al 9; Hall et al 98; Dahluist 96; ash 98; Suleiman et al 989) Bloc methods or solving () can be ound in (Fatunla 99; Ibrahim et al 7; Musa et al ; Nasir et al ; Musa et al ) The convergence o bloc methods or solving () using bloc bacward dierentiation ormula (BBDF) has been studied in (Ibrahim et al ) The bloc etended bacward dierentiation ormula (BEBDF) that approimates the o () is proposed in (Musa et al ) and has the general orm: α i hβ i + hβ+ + i () It was developed in uest or higher order A- stable bloc methods or sti IVPs The method improves the accurac and order o the BBDF method An etra uture point n + is involved which is predicted using conventional bacward dierentiation ormula The method also approimates the at -point simultaneousl and it is A-stable For i and it is given b: h h h + h h h n n n n n n n n n n n n n n n n n n n n n n respectivel More details on the method can be ound in (Musa et al ) An acceptable linear multistep method (LMM) must be convergent onsistenc and zero stabilit are the necessar and suicient conditions or convergence o a LMM According to (Lambert 97) consistenc controls the magnitude o the local truncation error while zero stabilit controls the manner in which the error is propagated at each step o the calculation A method which is not both consistent and zero stable is reected outright and has no practical interest This paper proves the convergence o the method () b establishing zero stabilit and consistenc conditions The order o the method will also be determined ORDER OF THE METHOD The ollowing deinitions given in (Lambert 97) will be used to establish the order o the method () The general linear multistep method (LMM) is deined b: α h β () where α and β are constants α α and β cannot be zero at the same time () The order o the LMM () and its associated linear operator given b: 9

2 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom L[ ( ); h] [ α ( + h) hβ '( + h)] () is deined as a uniue integer p such that () p and p + where the are constants deined b: α + α + α + + α α + α + + α ( β + β + β + + β ) ( α + α + + α )! ( β + β + + β ) ( )! (6) We etend the above deinitions to the method () as ollows: The method () can be deined in general matri orm as: AY m h B Fm+ (7) where A deined b: A B B and B are suare matrices A A B B B and Y Y m m F m F F m m + are column vectors deined b: Y m F Y n 6 m+ + n F n m n n F n m n m Euation (7) can be re-written as: n n n n + n n n n + n + h + h + h n n + n + 6 n n + n Let A A B B and B be bloc matrices deined b ( ) A ( A A A ) ( ) B B B B ( ) and B ( B B B ) A A A A B B B B where 8 A 9 99 A A B B B B (8) A A ' A 6 99 B B B B B8 The order o the bloc method (7) and its associated linear operator given b: [ ] + (9) L ( ); h A ( + h) h B '( + h)

3 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom is a uniue integer p such that () p and p + ; where the are constant column) matrices deined b: A + A+ A + + A A+ A + + A ( β + B+ B + + B+ ) ( A + A + + A )! ( B+ B + + ( + ) B + ) ( )! For ()6 we have () A + A + A + A + A + A ( A + A + A + A + A) ( B + B + B + B + B + B + B6) ( A + A + A + A + A)! ( B + B + B + B + B + 6 B6)! ( A + A + A + A + A)! ( B + B + B + B + B + 6 B6)! ( A + A + A + A + A)! ( B + B + B + B + B + 6 B6)! ( A + A + A + A + A)! ( B + B + B + B + B + 6 B6)! ( A + A + A + A + A) 6! ( B + B + B + B + B + 6 B6)! ( A + A + A + A + A) 7! ( B + B + B + B + B + 6 B6) 6! 8 () 69 8 Thereore the ormula () is o order 6 with error constant ONVERGENE OF THE METHOD onvergence is an essential propert that ever acceptable linear multistep method must possess This section proves the convergence o the method () According to (Lambert 97) consistenc and zero stabilit are the necessar conditions or the convergence o an numerical method We shall thereore begin with the ollowing theorem and deinitions (as given in Lambert 97) which relate to the general LMM: α h β () and then establish new deinitions that relate to the ull implicit -point BEBDF method A proo o consistenc and zero stabilit o the method will then ollow Theorem The necessar and suicient conditions or the linear multistep method () to be convergent are that it is consistent and zero stable Details o the prove can be ound in (Henrici 96) A LMM is said to be consistent i its order p Thereore rom (6) it ollows that the LMM () is consistent i and onl i the ollowing conditions are satisied: α α β See (Lambert 97) () The LMM () is said to be zero stable i no root o the irst characteristic polnomial has modulus greater than one; and i ever root with modulus one is simple See (Lambert 97) Building on this we now etend the above theorem and deinitions to the BEBDF method as ollows: Theorem The necessar and suicient conditions or the BEBDF method (7) to be convergent are that it is consistent and zero stable

4 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom Proo It suices to show that (7) is consistent and zero stable These are shown in subsections and The BEBDF is said to be consistent i its order p Thereore rom () it ollows that the BEBDF method () is consistent i and onl i the ollowing conditions are satisied: A 6 A B where A and B are as previousl deined () The BEBDF method () is said to be zero stable i no root o the irst characteristic polnomial has modulus greater than one and that with modulus one is simple onsistenc o the BEBDF method In this subsection it is shown that the BEBDF satisies the consistenc conditions given in deinition From what ollowed in section it can be concluded that the order o the BEBDF method is > Let A A A be as previousl deined Then A A + A + A + A + A + A () Hence the irst condition in () is satisied A A + A + A + A + A + A (6) 6 B (7) Hence A B Thus the second condition in () is also satisied The consistenc conditions are thereore met Hence the method is consistent Zero stabilit o the BEBDF method The stabilit polnomial o the method () is given b:

5 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom 689 t 6 h t 89 t 9h t 689 h t t Rth ( ) h t 6 h t 6 h t (8) For details see (Musa et al ) The irst characteristics polnomial o the method () is given b ( t ) where Solving the polnomial obtained is: t 89 t 689 t + + (9) Solving or t gives t t 7 t888 Thus b deinition o zero stabilit the BEBDF method is zero stable Since consistenc and zero stabilit conditions are both satisied the ull implicit -point BEBDF method converges This completes the proo o conditions set in the theorem NUMERIAL RESULTS To illustrate the eect o zero stabilit and consistenc on the method the ollowing non linear problems are solved at some ied station values o The theoretical and numerical results as well as the absolute error or dierent step length h are given in Tables - Problems ( ) ' Eact () 6 ( ) + e 6 Source: (Alvarez et al ) ' () Eact ( ) + Source: (Voss et al 997) Table- Eect o zero stabilit and consistenc on the -point BEBDF method when problem is solved with h Theoretical Numerical Absolute error Table- Eect o zero stabilit and consistenc on the -point BEBDF method when problem is solved with h Theoretical Numerical Absolute error

6 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom Table- Eect o zero stabilit and consistenc on the -point BEBDF method when problem is solved with h Theoretical Numerical Absolute error Table- Eect o zero stabilit and consistenc on the -point BEBDF method when problem is solved with h Theoretical Numerical Absolute error From the above tables the zero stabilit o the method is indicated b the decrease in error as the step length h tends to zero The accurac also improves as the step length is reduced Thus the error is not propagated in an eplosive manner Similarl the at an ied point improves as the step length is reduced This can be seen when we compare Tables and or problem and Tables and or problem The absolute error also indicates that the numerical becomes close to the eact Thus the computed tends to the theoretical as the step length tends to zero This shows the consistenc o the method ONLUSIONS The paper studied the ull implicit -point bloc etended bacward dierentiation ormula and proved that the method is consistent and zero stable This indicates that the method is convergent The numerical results

7 VOL 7 NO DEEMBER ISSN Asian Research Publishing Networ (ARPN) All rights reserved wwwarpnournalscom presented illustrated the eect o zero stabilit and consistenc o the method when a sti IVP is solved There is no evidence o eplosive error propagation in the method The method was also proven to be o order 6 These added advantages mae the BEBDF method to be numericall acceptable method or solving sti initial value problems AKNOWLEDGEMENT We are thanul to the Institute or Mathematical Research (INSPEM) and the Department o Mathematics Universiti Putra Malasia or the support and assistance in the course o this research We also want to than the anonmous reviewers or their insightul comments which improved the ualit o the paper REFERENES urtiss and JO Hirschelder 9 Integration o sti euations Proceedings o the National Academ o Sciences o the United States o America 8: - NAAM Nasir ZB Ibrahim and MB Suleiman Fith order two-point bloc bacward dierentiation ormula or solving ordinar dierential euations Appl Math Sci : -8 P Henrici 96 Discrete variable methods in ordinar dierential euations John Wile and Sons SO Fatunla 99 Bloc methods or second order ODEs International Journal o omputer Mathematics : -6 ZB Ibrahim KI Othman and M Suleiman 7 Implicit r-point blocs bacward dierentiation ormula or solving irst-order sti ODEs Applied Mathematics and omputation 86: 8-6 ZB Ibrahim M Suleiman NAAM Nasir and KI Othman onvergence o the -Point Bloc Bacward Dierentiation Formulas Applied Mathematical Sciences : 7-8 D Voss and S Abbas 997 Bloc predictor-corrector schemes or the parallel o ODEs omputers and Mathematics with Applications : 6-7 G Hall and M Suleiman 98 A single code or the o sti and nonsti ODE's SIAM Journal on Scientiic and Statistical omputing 6: GG Dahluist 96 A special stabilit problem or linear multistep methods BIT Numerical Mathematics : 7- H Musa M B Suleiman and N Senu Full implicit -point bloc etended bacward dierentiation ormula or sti initial value problems Applied Mathematical Sciences 6: -8 H Musa M B Suleiman and F Ismail A-Stable - point bloc etended bacward dierentiation ormula or solving sti ordinar dierential euations AIP on Proc : -8 J Alvarez and J Roo An improved class o generalized Runge-Kutta methods or sti problems Part I: The scalar case Applied Mathematics and omputation : 7-6 J D Lambert 97 omputational Methods in Ordinar Dierential Euations hi hester New Yor USA JR ash 98 On the integration o sti sstems o ODEs using etended bacward dierentiation ormulae Numerische Mathemati : -6 M Suleiman and W Gear 989 Treating a single sti second-order ODE directl Journal o omputational and Applied Mathematics 7: -8