On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations

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1 Pure and Applied Matematics Journal 7; 6(5: 74 ttp://wwwsciencepublisinggroupcom/j/pamj doi: 648/jpamj765 ISSN: 6979 (Print; ISSN: 698 (Online On One Justiication on te Use o Hybrids or te Solution o First Order Initial Value Problems o Ordinary Dierential Equations Kamo Nataniel Mawas, Gyemang Dauda Gyang, Soomiyol Mrumun Comort Department o Matematics/Statistics, Bingam University, Karu, Nigeria Department o Matematics and Computer Science, Benue State University, Makurdi, Nigeria Department o Matematics/Statistics, Plateau State Polytecnic, BarkinLadi, Nigeria address: mawas477@gmailcom (K N Mawas, daudagyemang@yaoocom (G D Gyang, mrumunsoomiyol@yaoocom (S M Comort To cite tis article: Kamo Nataniel Mawas, Gyemang Dauda Gyang, Soomiyol Mrumun Comort On One Justiication on te Use o Hybrids or te Solution o First Order Initial Value Problems o Ordinary Dierential Equations Pure and Applied Matematics Journal Vol 6, No 5, 7, pp 74 doi: 648/jpamj765 Received: September 7, 7; Accepted: September 8, 7; Publised: October, 7 Abstract: Tis paper is aimed at discussing and comparing te perormance o standard metod wit its ybrid metod o te same step number or te solution o irst order initial value problems o ordinary dierential equations Te continuous ormulation or bot metods was obtained via interpolation and collocation wit te application o te sited Legendre polynomials as approximate solution wic was evaluated at some selected grid points to generate te discrete block metods Te order, consistency, zero stability, convergent and stability regions or bot metods were investigated Te metods were ten applied in block orm as simultaneous numerical integrators over nonoverlapping intervals Te results revealed tat te ybrid metod converges aster tan te standard metod and as minimum absolute error values Keywords: Hybrid Metod, Collocation, Interpolation, Sited Legendre Polynomials Approximation, Continuous Block Metod, Order, Consistency, Zero Stability, Convergent Introduction Most pysical penomena in science and engineering used matematical models to elp in te understanding o te pysical world problems Tese models oten yield equations tat contain some derivatives o an unknown unction o one or several variables Suc equations are called dierential equations Dierential equations play an important role in te modeling o pysical problems arising rom almost every discipline o study suc as economics, medicine, psycology, operation researc, space tecnology and even in areas suc as biology and astronomy Interestingly, dierential equations arising rom te modeling o suc pysical penomena oten are very diicult or impossible to solve analytically Hence, te need or te development o numerical metods to obtain approximate solutions becomes inevitable Many scolars ave worked extensively on te solution o dierential equations Tese autors proposed dierent metods ranging rom predictor corrector metod to block metod using dierent polynomials as basis unctions evaluated at some desired points In tis work, two types o block metods are proposed, te irst is te ivestep standard block metod and te second is te ivestep ybrid metod wit our o grid points, using te sited Legendre polynomials evaluated at some grid and ogrid points to give te needed discrete scemes Consider a numerical metod or solving general irst order initial value problems o ordinary dierential equations o te orm: y ( x, y, y( y ( were is a continuous unction and satisies Lipscitz condition o te existence and uniqueness o solution Many autors ave proposed solution or irst order initial value problems o ordinary dierential equations using dierent approaces Tis work will adopt te block metod tecnique or solving ( as suggested by researcers [4], considered

2 8 Kamo Nataniel Mawas et al: On One Justiication on te Use o Hybrids or te Solution o First Order Initial Value Problems o Ordinary Dierential Equations te Continuous implicit ybrid one step metods or te solution o initial value problems o general second order ordinary dierential equations [5], introduced te application o two step continuous ybrid Butcer s metod in block orm or te solution o irst order initial value problems; tis approac eliminates requirements or a starting value [], introduced a new ybrid metod or systems o sti equations [], developed a new Butcer type twostep block ybrid multistep metod or accurate and eicient parallel solution o ordinary dierential equations [], used ybrid ormula o order our to generate starting values or Numerov metod [], developed linear multistep ybrid metods wit continuous coeicients or solving sti ordinary dierential equations [], introduced a ybrid linear collocation multistep sceme or solving irst order initial value problems o ordinary dierential equations [9], developed a tree step implicit ybrid linear multistep metod or te solution o tird order ordinary dierential equations Derivation o te Metod In tis section, two metods are developed namely ivestep block metod and ivestep wit tree ogrid points, by interpolating and collocating at some selected points Consider te sited Legendre approximation o te orm y( x C P( t ( i were, and are interpolation and collocation points Te irst derivative o ( gives substituting ( in (, to get Five Step Metod i i i y ( x C P ( t ( i i i ( x, y C P ( t (4 Interpolating ( at and collocating ( at,,,, and gives te system o nonlinear equations o te orm were i i AX B (5 p(4 p(4 p(4 p(4 p( p( p( p ( p( p( p( p ( A p( p( p( p( p( p( p( p ( p(4 p(4 p(4 p(4 p(5 p(5 p(5 p(5 c y c X B 4, 5 c n Solving or te using inverse o a matrix metod and substituting in ( gives te continuous ormulation were 4 5 (6 y( x α ( x y β ( x j j j j j j, ⁴ 4 ⁵ 7 ⁶ ³ 6 96 ⁴ 7 6 ⁵ 44 ⁶ ⁴ 7 ⁶ ⁵ ³

3 Pure and Applied Matematics Journal 7; 6(5: ³ 6 48 ⁴ 5 ⁵ 7 ⁶ 4 ⁵ ³ 4 ⁶ ⁴ 5 8 ² ( (,( ( ( / (7 Evaluating (6 wit coeicients (7 at,,, and wit ( te ollowing discrete scemes is respectively obtained as; , 44 (8 Five Step Metod wit Tree OGrid Points Interpolating ( at and collocating ( at,,,,, 5, 6, 5 and gives a system o nonlinear equations o te orm were p (4 p(4 p (4 p (4 A p (4 p (4 p (4 p( p( p( p( p ( p ( p ( p ( p ( p ( p ( p ( p ( p ( p ( p ( AX B (9 p (4 p p p p p p p p p p p p p(5 p(5 p(5 p(5 c y 4 c n 4 X, B 9 4 c 5 Solving or te using inverse o a matrix metod and substituting in ( gives te continuous ormulation

4 4 Kamo Nataniel Mawas et al: On One Justiication on te Use o Hybrids or te Solution o First Order Initial Value Problems o Ordinary Dierential Equations were ² k k ( y( x α ( x y β ( x β ( x j j j j ωi ωi j j ω 7 5, 8,8,,8, and 95 ⁸ ³ 55 ⁹ i ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ 8 7 ⁴ ⁵ ⁶ ⁷ ² ³ ⁴ ⁵ 6 ⁶ ⁷ ⁸ 68 4 ⁹ ³ ⁴ ⁵ ⁷ ⁶ ⁸ 6 4 ⁹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ 4 4 ⁹ ³ ⁴ ⁶ ⁷ ⁹ ⁸ ⁵ ² 9 85 ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ³ 498 ⁴ 8 ⁵ 464 ⁶ 4 ⁷ 896 ⁸ 66 4⁹, 4,,, ² ( (³ ( (, ( / 4( < ( 4 ( >, ( Evaluating ( wit coeicients ( at,,,, 5, 6, 5 and wit ( te ollowing discrete scemes are respectively obtained as;

5 Pure and Applied Matematics Journal 7; 6(5: Analysis o te Metod In tis section te error constant, order, consistency, zero stability, convergent and region o absolute stability o te scemes generated are discussed Order and Error Constant Expanding (8 and ( in Taylor s series and collecting like terms in powers o, te order and error constant are respectively obtained as ollows; Table Order and Error Constants o te Discrete Scemes o te Block Metod (8 Sceme Order Error constant ³ 6 75 ³ ³ ³ ² Table Order and Error Constants o te Discrete Scemes o te Hybrid Block Metod ( Sceme Order Error constant ⁴ ⁵ ⁶ ⁸ ⁸ ⁷ Hence te block metods are o order BC 6, 9 and error constant o D, D respectively Consistency Following [8] and [6], te block metods are said to be consistent i tey satisy te ollowing conditions: (i te order E, 4 5 4,, 5 ( J (ii IK C I (iii B( B L ( (iv B LL (!N( Were B(O and N(O are te irst and second caracteristics polynomials o te block metod According to [8] and [6], condition (i above is a suicient condition or te block metods to be consistent Hence te block metods are consistent since E 6,9 > Zero Stability Te block metods are said to be zero stable i te roots Q R ; O,,U o te irst caracteristic polynomial E(Q, deined by E(Q VW QY Z satisies Q R and every root wit Q R as multiplicity not exceeding two in te limit as Calculations rom all available inormation revealed tat te block metods (8 and ( ave te ollowing roots respectively Q (Q and Q 4 (Q Hence te block metods are zero stable, since all roots wit modulus one do not ave multiplicity exceeding te order o te dierential equation in te limit as 4 Convergence According to [7], we can saely assert te convergence o te block metods (8 and ( 5 Region o Absolute Stability Reormulating te block metods (8 and ( as a General Linear Metods (GLM containing a partition o matrices A, Band C and ten substituting into te stability polynomial O(] Q ^ Using a MATLAB code based on te idea o Newton s iteration, te regions o absolute stability o te block metods are respectively sown below

6 4 Kamo Nataniel Mawas et al: On One Justiication on te Use o Hybrids or te Solution o First Order Initial Value Problems o Ordinary Dierential Equations 4 Numerical Illustrations Te ollowing numerical experiments are perormed wit te aid o MAPLE 8 sotware package in order to urter airm te earlier establised convergence o te metods Example Te ordinary dierential equation ( Figure Te regions o absolute stability or metods (8 and ( y 5 y, x, y, ( Te exact solution is ( W _ ([] Example Te ordinary dierential equation Table Absolute Error Values or Example o Metods (8 and ( ( y x y, x, y, (4 Te exact solution is ( W`_ ([] Example Te ordinary dierential equation ( y y x, x, y 5, (5 Te exact solution is ( ( 5W _, ([] a Exact Solution Result o Metod (8 Absolute error o (8 Result o Metod ( Absolute error o ( ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Table 4 Absolute Error Values or Example o Metods (8 and ( a Exact Solution Result o Metod (8 Absolute error o (8 Result o Metod ( Absolute error o ( `, ` ` ` ` ` ` ` `, ` `, ` `, ` `, ` `, ` `, `

7 Pure and Applied Matematics Journal 7; 6(5: 74 4 Table 5 Absolute Error Values or Example o Metods (8 and ( a Exact Solution Result o Metod (8 Absolute error o (8 Result o Metod ( Absolute error o ( ` ` ` ` ` ` ` ` ` ` ` ` ` `, ` `, ` `, ` `, 5 Discussion o Result Two dierent metods or solving irst order initial value problems o ordinary dierential equations ave been proposed in tis work, te conventional or standard metod and te ybrid metod wit te ybrid aving more advantages over te conventional metod te ybrid iger order and accuracy Te metods does not require developing separate predictors to implement tis makes it simple and attractive or solving initial value problems o ordinary dierential equations A careul observation o Tables, and sowed tat bot standard and ybrid metods perormed well as teir absolute error values are convergent, ence airming te earlier establised convergence o te metods Te ybrid metod is useul as it reduces te step number o a metod and still remains zero stable; in addition, te absolute error values presented in Tables, 4 and 5 indicated tat te ybrid metod perormed better tan te standard metod wen applied to sti and nonsti dierential equations respectively Te results also revealed tat te ybrid metod converges aster tan te standard metod, since it as minimum absolute error values 6 Conclusion Justiying rom te numerical calculations, it as been observed tat te ybrid metod perormed well tan te conventional metod, it as also been establised rom te calculations tat te ybrid metod as ig order and relatively small error constants tan te conventional metod Finally, it as been establised tat ybrid metods gives better results tan te standard metods wen applied to eiter sti or nonsti initial value problems Reerences [] Adee, S O, Onumanyi, P, Sirisena, U W andyaaya, Y A (5 Note on Starting te Numerov Metod More Accurately by a Hybrid Formula o Order Four or Initial Value Problems Journal o Computational and Applied Matematics, 75: 697 [] Ademiluyi, R A (987 New Hybrid Metods or Systems o Sti Equations, Benin City, Nigeria: PD Tesis, University o Benin [] Akinenwa, O A, Jator, S N and Yao, N M ( Linear Multistep Hybrid Metods wit Continuous Coeicients or Solving Sti Ordinary Dierential Equations, Journal o Modern Matematics and Statistics, 5(: 4757 [4] Anake, T A ( Continuous Implicit Hybrid OneStep Metods or te Solution o Initial Value Problems o Second Order Ordinary Dierential Equations Ogun State, Nigeria PD Tesis, Covenant University, Ota [5] Awari, Y S, Abada, A A, Emma, P M and Kamo, N M ( Application o Two Step Continuous Hybrid Butcer s Metod in Block Form or te Solution o First Order Initial Value Problems Natural and Applied Sciences, 4(4: 98 [6] Fatunla, S O (988 Numerical Metods or Initial Value Problems or Ordinary Dierential Equations USA, Academy press, Boston 95 [7] Henrici, P (96 Discrete Variable Metods or ODE`s, New York, USA, Jon Wiley and Sons, (96 [8] Lambert, J D (99 Numerical Metods or Ordinary Dierential Equations, New York: Jon Wiley and Sons pp 9 [9] Moammed, U and Adeniyi, R B (4 A Tree Step Implicit Hybrid Linear Multistep Metod or te Solution o Tird Order Ordinary Dierential Equations General Matematics Notes, 5(: 674 [] Odejide, S A and Adenira, A O ( A Hybrid Linear Collocation Multistep Sceme For Solving First Order Initial Value Problems Journal o te Nigerian Matematical Society : 94 [] Serisina, U W, Kumleng, G M and Yaaya, Y A (4 A New Butcer Type TwoStep Block Hybrid Multistep Metod or Accurate and Eicient Parallel Solution o Ordinary Dierential Equations, Abacus Matematics Series : 7 [] Yakusak, N S, Emmanuel, S and Ogunniran, M O (5 Uniorm Order Legendre Approac or Continuous Hybrid Block Metods or te Solution o First Order Ordinary Dierential Equations IOSR Journal o Matematics, (: 94 [] Zill, D G and Warren, W S ( Dierential Equations wit Boundary Value Problems, Cengage Learning: Eigt edition books/cole

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