Derivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations

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1 Dervato of -Pot Block Method Formula for Solvg Frst Order Stff Ordary Dfferetal Equatos Kharul Hamd Kharul Auar, Kharl Iskadar Othma, Zara Bb Ibrahm Abstract Dervato of pot block method formula wth costat step sze usg Lagrage polyomal s our am ths paper. Block method wll be troduced durg the dervato. I a pot block method formula, three soluto values wll be approxmate smultaeously at dfferet x-axs. The stablty rego of the formula derved wll be costructg ad aalyze. Keywords Block method, stff, Ordary Dfferetal Equatos. S I. INTRODUCTION cece ad techology have created tal value problems volvg systems of Ordary Dfferetal Equatos (ODEs) ad most of these problems are kow as stff ODEs. There have varous deftos of stff ODEs gve by the prevous ad preset researchers ther paper. Let cosder the lear system of frst order ODEs as follows: y' Ay φ( x), y( a) η, a x b () where T y ( y, y,..., y ), η ( η, η,..., η ) Defto: T s The lear system () s sad to be stff f t satsfed characterstcs gve by Lambert (974):-. for all, Re( λ ) < 0 s max Re( λ ). (Stffess rato) >>, where λ, A. m Re( λ ),,..., are egevalues of Amog the earlest research o block methods was troduced by Mle (95) ad had bee exteded by Rosser (967) usg Ruge-Kutta method based o the tegrato formula whch s the Newto-Cotes type. Other block methods were dscussed by other researchers such as Omar (999) wth block Adam's method of varable step sze for solvg hgher order ODEs ad followed by dervato of varable step sze varable order block Adams by Majd (004) for solvg ostff hgher order ODEs drectly. Block method wll geerates the estmates values the ew block by take all pots from the prevous block. For example, ths method wll use w, w0, w,..., w order to compute the approxmato values of w,..., w. FORMULATIO OF -POI T BLOCK METHOD I ths secto, -pot block method formula s derved by usg Lagrage polyomal. Geeral Lagrage polyomal s defed as. P ( x) L ( x) y( x ) () k k, l j j 0 where, k ( x x ) L x for each j k k k, l ( ) 0,,... 0 ( x j x ) j I the dervato, the set pots of } wll be used. x, x, x, x, x, x, x { Substtute each pot to () ad expad to become as follows:- BLOCK METHOD FOR SOLVI G ODEs Mauscrpt receved March, 00. Kharul Hamd K.A s wth the Uverst Tekolog Mara (UTM) Malaysa( e-mal:kharulhamd87@ gmal.com). Kharl I.O.,was wth Uverst Tekolog Mara (UTM) Malaysa.(e-mal: kharl7@yahoo.com). Zara B.I., s wth the Uverst Pertaa Malaysa (UPM). ISSN: ISBN:

2 Let ( X X )...( X X ) P( x) ( X X )...( X X ) s X X X ( X X )...( X X ) ( X X )...( X X ) ( X X )...( X X ) ( X X )...( X X ) ( X X )...( X X ) ( X )...( ) ( X X )...( X X ) ( X X )...( X X ) ( X X )...( X X ) ( X X )...( X X ) ( X X )...( X X) ( X X )...( X X ) X X h. The, ths expresso to equato () ( h sh)( sh)...(4 h sh) P( X sh) ( h)( h)( h)...(6 h) () X X sh ad substtute ( h sh)( sh)...(4 h sh) ( h)( h)( h)...(5 h) ( h sh)( h sh)...(4 h sh) ( h)( h)...(4 h) ( h sh)( h sh)...(4 h sh) ( h)( h)...( h) ( h sh)( h sh)...(4 h sh) ( 4 h)( h)...( h) ( h sh)( h sh)...(4 h sh) ( 5 h)( 4 h)...( h) ( h sh)( h sh)...( h sh) ( 6 h)( 5 h)...( h) (4) Smplfy equato (4) by cacellg out the h. ( s)( s)( s)( s)( s)(4 s) P( X sh) ()()()(4)(5)(6) ( s)( s)( s)( s)( s)(4 s) ( )()()()(4)(5) ( s)( s)( s)( s)( s)(4 s) ( )( )()()()(4) ( s)( s)( s)( s)( s)(4 s) ( )( )( )()()() ( s)( s)( s)( s)( s)(4 s) ( 4)( )( )( )()() ( s)( s)( s)( s)( s)(4 s) ( 5)( 4)( )( )( )() ( s)( s)( s)( s)( s)( s) ( 6)( 5)( 4)( )( )( ) Dfferetate equato (5) wth respect to s. ' P ( X sh) s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s (6) Substtute s 0 to equato (6) order to have terato at the frst pot whch represets the frst block. 7 4 hf (7) 60 (5) Substtute s to equato (6) at the secod pot whch represet secod block. ISSN: ISBN:

3 hf Substtute s to equato (6) order to have terato at the thrd pot whch represets the thrd block hf (9) 6 Solvg, (9). ad by usg equato (7), (8) ad hf hf (0) hf hf () hf hf () (8) hf hf hf ( ) STABILIT REGIO The stablty rego of equato () s determed by applyg the stadard lear test problem as follows: y ' λ y, λ > 0, λ s complex Apply expresso (4) to equatos () ad we obta y y y (5) We wrte equatos (5) the form of matrx vector ad obta. Evetually, the formula for fdg the soluto at x, x ad x smultaeously s gve by ISSN: ISBN:

4 (6) The equatos (6) are the form of Am Bm Cm wth the matrx coeffcets specfed as A m B m 0 0 C m Defto: The method () s sad to be zero- f all the roots of characterstc polyomal have modulus less tha or equal to uty,. Defto: The method () s sad to be A- f ts rego of absolute stablty cotas the whole of the plae Re( z) 0. By solvg the frst characterstc polyomal ( t) where, the followg ρ equato s obtaed ρ( t) det At Bt C ISSN: ISBN:

5 t t t ( ) t t ( ) t t ( ) t t t t t h ( ) t 0 (7) 64 I order to determe the method () s zero, let 0 to equato (7) ad obtaed t t t t t Solvg equato (8) wll produce. t e I 0 (8) 0, 0,, , e I, e 0 I λ s sutable for stff problems sce the method () s almost A- REFERE CE [] Lambert, J. D. (974). Computatoal methods Ordary Dfferetal Equatos, Joh Wley ad Sos, Lodo. [] Majd. Z. (004). Parallel Block Methods for Solvg Ordary Dferetal Equatos, PhD Thess, Uverst Putra Malaysa. [] Mle, W.E. (95). umercal Soluto of Dfferetal Equatos, Joh Wley, Nework. [4] Omar. Z. B. (999). Developg Parallel Block Methods for Solvg Hgher Order ODEs Drectly. PhD Thess. Uverst Putra Malaysa. [5] Rosser, J. B. (967). A Ruge-Kutta For All Seasos. Sam Revew 9(): The roots of polyomal have magary umbers. Therefore, we oly cosder the real umber. Thus, the method () s zero. The stablty rego of method () have bee costructed the plae usg Maple as below. u u u The stablty rego of the method () les outsdes of the bouded rego. Thus, the method () s at the etre plae except the crcle. Sce most of the rego the left plae s the rego. Therefore, the method () s almost A-. As the cocluso, the method that has bee derved ISSN: ISBN: