Derivation of a Single-Step Hybrid Block Method with Generalized Two Off-Step Points for Solving Second Order Ordinary Differential Equation Directly.

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1 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 Deivatio o a Sigle-Step Hybid Block Metod wit Geealized Two O-Step Poit o Solvig Secod Ode Odiay Dieetial Equatio Diectly. a t. Abdelaim. ad Zui. Oma, Depatmet o Matematic, College o At ad Sciece, Uiveiti Utaa Sitok, Keda, Malayia Abtact Ti pape popoe a igle-tep ybid block metod wit geealized two o-tep poit o te diect olutio o iitial value poblem o ecod ode odiay dieetial equatio. Te ue o powe eie appoximate olutio a a itepolatio polomial at te o poit i employed i developig ti metod, wile it ecod deivative i collocated at all poit i te iteval. Futemoe, ome baic popetie o te geealized metod uc a ode, zeo tability, coitecy ad covegece ae alo etablied. I additio, two example o peciic poit o te developed metod ae coideed to olve ome iitial value poblem o ecod ode odiay dieetial equatio. Te umeical eult coim tat te popoed metod poduce bette accuacy i compaed wit te exitig metod. Keywod Sigle-tep, ybid block metod, collocatio, itepolatio, ecod ode odiay dieetial equatio. I. INTODUCTION I ti aticle, te umeical olutio to te geeal ecod ode iitial value poblem o odiay dieetial equatio (ODE o te om y ( x, y, y, x ab, ( wit two iitial coditio y ( a τ, y ( a τ i coideed. Te metod o educig ( to it equivalet ytem o it ode a bee oud avig ome etback wic iclude: watage o compute time, a lot o uma eot ad computatioal bude (ee [4], [8] ad []. Teeoe, cola ave paid moe attetio o te etablimet o diect metod o olvig ige ode ODE weeby te umeical eult geeated ae bette ta te metod o eductio to ytem o it ode ODE ( ee [5], [] ad []. Some o te metod developed iclude te el tatig uge - Kutta type wic cotai may uctio to be evaluated pe tep ( [5] ad [] ad liea multitep metod wic ae ot el-tatig but equie little uctio to evaluate pe tep[]. Te implemetatio o implicit liea multitep metod i te pedicto-coecto mode i aociated wit a lot o uma eot ad compute time wic ede te metod to be ieiciet o te ue o geeal pupoe. Tee weakee i pedicto coecto metod led to developmet o block metod witout pedicto ad do ot equie may uctio to evaluate pe tep we compaed wit uge- Kutta type metod, []. It i obeved tat tee metod metioed above ae goveed by Dalquit baie coditio wic ae exteively dicued by [6] ad te itoductio o ybid metod a bee ued to cicumvet te baie (ee [7] ad [].Te developmet o ybid metod wit peciic o tep poit ave bee coideed by cola [], [], [] ad [4]. I ode to big impovemet i te exitig metod, ti pape peet a igle tep metod wit geealized two o tep poit o olvig ( diectly. Ti pape i divided ito ive ectio: ectio two cotai te deivatio o te metod, ectio tee etablie te geealized baic popetie o te metod, ectio ou iclude peciicatio o te metod ad ectio ive iclude te umeical eult geeated om te applicatio o te metod to ecod ode ODE. II. DEIVATION OF THE METHOD Suppoe te appoximate olutio i te powe eie o te om ( y x v + m i x x a i i ( wee x x, x + Fo,,,,, a ' ae coeiciet to be detemied, v i umbe o collocatio poit, m i umbe o itepolatio poit, x x i a cotat tep ize o patitio o iteval ab, wic i give by a x < x < < xn < xn b.. Te ecod deivative o ( i give by ISSN:

2 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 ( (,, y x x y y ( v + m i i i ai x x i Itepolatig ( at ad ad collocatig ( at all poit i te iteval i.e,,, give te ollowig equatio wic ca be witte i matix om a a a a a 4 a 5 y y Gauia elimiatio metod i applied o (4 to id te coeiciet ad te ubtituted ito ( to give te implicit cotiuou ybid metod o te om y ( x αi + i + β + + βi + i i, i, ( (4 (5 wee,,,, ad ae give i Appedix A Evaluatig (5 at o-itepolatig poit i.e. ad ad evaluatig it it deivative at all poit give te ollowig equatio i matix om y + ( ( y +, Y m ', y + ( ( ' y + ' y + ( ( ( ( y y y, ', y ' y ' y ( ,, , ad te elemet o ad ae give a below 4 5 ( ( ( 6 ( ( ( ( ( ( 6( ( ( ( ( 4 4 ( ( ( 4 4 ( + ( ( 6 ( ( ( + ( 6 ( ( ( ( ( ( 6 ( ( ( ( ( ( ( 6 ISSN:

3 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, ( ( ( ( ( 4 4 ( ( ( ( ( ( ( + ( + ( ( ( ( ( ( Multiplyig Equatio (6 by metod m give te ybid block Y + + (8, ( ( ( ( 6 + ( ( 6 + ad ( ( 5 ( 5 6( ( 6( ( 6( ( 4 4 ( 5 ( ( 5 6( ( 6( ( 6( ( ( 5 5 ( 5 ( ( ( 6( ( 6( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Equatio (8 ca alo witte a 4( 5 ( 5 ( ( 4( 5 4 ( 5 ( ( ( 5 ' y + y 6 ( ( ' y + y + 6 ( ( ( ( ( ( ' y + y ( ( + ( ( + ( 6 + ( ' ( ISSN:

4 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 ( ( ( ( ( 6 ' ' ( ' ( 6 + ( ( ( ( ( ( ( ( ( III. POPETIES OF THE METHOD A. Ode o te Metod Deiitio : Te liea dieece opeato L aociated wit (7 i deied a ( ; L y x Y m (9 wee y(x i a abitay tet uctio cotiuouly dieetiable o ab,. Expadig ad compoet epectively i Taylo eie ad collectig tem i powe o give ( ; ( ( p p C y ( x C p y ( x L y x C y x + C y x wee ae vecto ( Deiitio : Te ybid block metod (7 ad te aociated liea dieece opeato (9 ae aid to ave ode i C C C C p C p + ad C p +. C p + i called vecto o eo cotat. Expad (9 i Taylo eie about give ( 5 5 ( ( 5 + ( + + y! ( 6 6( (! ( 5 ( ( y y y y 6( (! 6( (! ( ( ( 5 ( + y y y! 6 6( (! 4 ( ( ( y + y y y 6( (! 6( (! + ( ( 5 ( +! 6 6 ( (! + ( 5 ( ( ( (! 6( (! ( ( 6 + ( ! ( (! + ( ( ( + ( + + y ( ( y! ( (! ( ( ! 44! ( ( + ( ( + ( (! ( (! + ( 6 + ( y! ( (! + + ( ( + ( ( + + y ( (! ( (! Compaig te coeiciet o led to C C C C C 4 C 5,,,,, T Teeoe, te ode o te metod i o all , (, \ wit geeal eo cotat vecto ISSN:

5 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 ( ( 5 44 ( C 6 ( ( ( B. Zeo tability o te metod Deiitio 4: Te ybid block metod (7 i aid to be zeo table i te it caacteitic polomial π ( z avig oot uc tat z ad i z te te multiplicity o mut ot geate ta two. π ( z z z z 4 ( z woe olutio i z,,,,,. Hece, ou metod i zeo table o all, (, C. Coitecy Deiitio : Te oe tep ybid block metod (7 i aid to be coitet i ode o te metod geate ta o equal oe i.e. P Sice i ou ew metod C 6 C 4+,, ti implie P 4. Hece, ou metod i coitet by deiitio. D. Covegece Teoem. [9]: Coitecy ad zeo tability ae uiciet coditio o a liea multitep metod to be coveget Teeoe, ice te ew ybid block metod i coitece ad zeo tabile, it ca be cocluded tat te metod i coveget. F. egio o Abolute Stability I ti aticle, te locu metod wa adopted to detemie te egio o abolute tability. Te metod (7 i aid to be abolutely table i o a give all oot o te caacteitic π z, ρ z σ z, atietie z <. Te tet polomial ( ( ( equatio y λ y i ubtituted i (7 wee λ ad d λ. Subtitutig z coθ i iθ ad coideig eal dy pat yield ( θ ( 44co θ 44 ( coθ IV. SPECIFICATION t ( Ti ectio coide two peciic umeical metod o two ybid poit. Metod A Subtitutig, ito equatio (8, te ollowig 5 block o oe tep wit two ybid poit ad it deivative ae obtaied ' 57 y ' 49 y ' 7 y + y ' 7 y ' ' y ' ' ISSN:

INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 eplacig, ito Equatio( give te ode o 5 te metod A to be 4,4,4,4,4,4 T wit eo cotat 8 5.47e 8. e 4.47 e C 6 7 9.9556e 7.e.8556e Ate ubtitutig te value o ad i (, we get te tability iteval o (-4654, a ow i Figue.

6 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 eplacig, ito Equatio( give te ode o 5 te metod A to be 4,4,4,4,4,4 T wit eo cotat e 8. e 4.47 e C e 7.e.8556e Ate ubtitutig te value o ad i (, we get te tability iteval o (-4654, a ow i Figue. Uig te ame pocedue a peviouly decibed, te ode o te block metod B i 4,4,4,4,4,4 T wit eo cotat 5.5e e e C e e e, 4 eplacig i ( give te tability iteval o (-5,. ee to Figue. Fig. Stability egio o igle tep ybid block metod A Metod B ' 4 7 y y + y ' 7 y y + y ' 9 y + y ' 7 y ' 7 9 y ' Fig. tability egio o igle tep ybid block metod B V. NUMEICAL EXPEIMENT Te peomace o te two peciic ybid block metod(metod A ad metod B i tated o te ollowig two ecod iitial value poblem Poblem : Exact olutio: y x ( y, y (, y (. + x y + l wit. x Table Exact olutio ad computed olutio o te ew metod A o olvig Poblem x Exact olutio Computed olutio i A ISSN:

7 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 Table Exact olutio ad computed olutio o te ew metod B o olvig Poblem x Exact olutio Computed olutio Table Compaio o te ew metod wit [] o olvig Poblem x EO IN A Eo i [] Eo i B Table 5 Exact olutio ad computed olutio o te ew metod B o olvig Poblem x Exact olutio Computed olutio i B Table 6 Compaio o te ew metod wit [] o olvig Poblem Poblem : Exact olutio: 6 4 y + y + y, y (, y (. x x y 5, wit. 4 x Table 4 Exact olutio ad computed olutio o te ew metod A o olvig Poblem x x Exact olutio Computed olutio A x EO IN A Eo i[] Eo i B VI. COONCLUSION Ti pape a ucceully developed a ew igle-tep ybid block metod wit geealized two o-tep poit o olvig ecod ODE. Te zeo tability, coitecy, covegece, ode, egio o abolute tability ad eo cotat o te developed metod ae alo examied. Te popoed metod ot oly poee good popetie o a umeical metod, it a alo bee pove to be upeio ta te exitig metod i tem o accuacy we olvig te ame iitial value poblem o ecod ode ODE diectly. Hece, ti metod ould be coideed a a viable alteative o olvig iitial value poblem o ecod ode ODE. Futemoe, te developed metod ca be exteded to olve a ytem o iitial value poblem o ige ODE diectly. EFEENCES [] T. A. Aake, D. O. Awoyemi ad A. O. Adeaya, A oe tep metod o te olutio o geeal ecod ode odiay dieetial Equatio, Iteatioal Joual o Sciece ad Tecology, Vol., o. 4, pp. 59-6,( b. ISSN:

8 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 [] T. A. Aake, D.O. Awoyemi ad A. O. Adeaya, Oe-Step Implicit Hybid Block Metod o Te Diect Solutio o Geeal Secod Ode Odiay Dieetial Equatio, IAENG Iteatioal Joual o Applied Matematic, 4(4, 4 8, (a. [] D.O. Awoyemi, A P-table liea multitep metod o olvig geeal tid ode o odiay dieetial equatio, It. J. Comput. Mat,vol 8,pp ,. [4]. Bu ad Y.D. Vail yev, A umeical metod o olvig dieetial equatio wit deivative o ay ode, Computatioal matematic ad matematical pyic,vol (, pp. 7, 99. [5] J. Butce, Numeical metod o odiay dieetial equatio. J. Wiley Ltd., Cicete. [6] G. Dalquit, covegece ad tability i te umeical itegatio o odiay dieetial equatio Matematic cadiavia,vol 4, - 5,959. [7] C. W. Gea, Hybid metod o iitial value poblem i odiay dieetial equatio, Joual o te Society o Idutial ad Applied Matematic, Seie B: Numeical Aalyi, vol (, pp.69 86,965. [8] E. Haie ad G. Wae, A teoy o ytom metod. Numeice Matematik, 5(4, pp.8 4,975 [9] P. Heici, Dicete vaiable metod i odiay dieetial equatio, 96. [] A. Jame, A. Adeaya,ad S. Joua, Cotiuou block metod o olutio o ecod ode iitial value poblem o odiay dieetial equatio, Iteatioal Joual o Pue ad Applied Matematic, vol 8(, pp.45 46,. [] A. Jame, A. Adeaya, ad J. Suday, Uiom ode cotiuou block ybid metod o te olutio o it ode odiay dieetial equatio, IOS Joual o Matematic, vol (6 pp.8-4,. [] S.N. Jato, Solvig ecod ode iitial value poblem by a ybid multitep metod witout pedicto, Applied Matematic ad Computatio, vol 7(8,pp ,. [] J. D. Lambet, Computatioal Metod i ODE. New Yok: Jo Wiley, 97. [4] F. Ngwae ad S. Jato, Block ybid ecod deivative metod o ti ytem, Iteatioal Joual o Pue ad Applied Matematic,vol 8(4, pp ,. [5] Z. Oma ad M. Sulaima, Paallel -poit implicit block metod o olvig ige ode odiay dieetial equatio diectly. Joual o ICT,vol (, pp.5 66, 4. ISSN:

9 INTENATIONAL JOUNAL OF MATHEMATICS AND COMPUTES IN SIMULATION Volume, 6 Appedix A ISSN:

Advances in Mathematics and Statistical Sciences. On Positive Definite Solution of the Nonlinear Matrix Equation

Advances in Mathematics and Statistical Sciences. On Positive Definite Solution of the Nonlinear Matrix Equation Advace i Mathematic ad Statitical Sciece O Poitive Defiite Solutio of the Noliea * Matix Equatio A A I SANA'A A. ZAREA Mathematical Sciece Depatmet Pice Nouah Bit Abdul Rahma Uiveity B.O.Box 9Riyad 6 SAUDI