Implicit Second Derivative Hybrid Linear Multistep Method with Nested Predictors for Ordinary Differential Equations

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1 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) ISSN (Print) -44, ISSN (Online) -44 Global Societ o Scientiic Research and Researchers Implicit Second Derivative Hbrid Linear Multistep Method with Nested Predictors or Ordinar Dierential Equations S. E. Eoro a *, M. N. O. Ihile b, I. M. Esuabana c a Department o Mathematics, Universit o Calabar, Calabar, Cross River State, Nigeria b Department o Mathematics, Universit o Benin, Benin, Edo State, Nigeria c Department o Mathematics, Universit o Calabar, Calabar, Cross River State, Nigeria a eorosam@ahoo.com b mnoihile@ahoo.com c esuabanaita@gmail.com Abstract In this paper, we considered an implicit hbrid linear multistep method with nested hbrid predictors or solving irst order initial value problems in ordinar dierential equations. The derivation o the methods is based on interpolation and collocation approach using polnomial basis unction. The region o absolute stabilit o the method is investigated using the boundar locus approach and the methods have been ound to be A stable or step-length 6. Kewords: Linear multistep methods; hbrid; nesting; interpolation; collocation; boundar locus.. Introduction The conventional linear multistep method (LMM) is deined as α h β (.) * Corresponding author. 97

2 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 where α and β are parameter constants to be determined. The β determines i the linear multistep method is explicit or implicit. For explicit LMM (.), β and or implicit methods, β. This is a popular method or the numerical approximation o the solutions o initial value problems in ordinar dierential equations ( ), ( x ) ' x, (.) Its stabilit and order are subect to some constraints b [4]. Modiication have been made to overcome the barrier, see [,5,6,7,5,6] among others. Reerence [6] introduced a second derivative term into the Adamstpe LMM (.) to obtain the second derivative linear multistep (SDLMM) o the orm α + β + ' (.) h h O-step points have been introduced into this linear multistep method to overcome Dahlquist order and stabilit barrier. Other extension o (.) can be ound in [,,8,,,4,6]. Our interest in this paper is to construct an implicit second derivative hbrid linear multistep method o the orm ( m) ( m) ( m) n n h n v β + + β v + h λ ' m m (.4) which are o order p + with the hbrids ( l) ( l) ( l) n v ' l n h n v + β β n v h l l l v v (.5) l l o order p * + 4, where ( l) ( l) ( l) n v n h n h + α + + β + + l ' (.6) o order p ** + or l () m This method (.4) sees to approximate the solution o (.). The idea is to approximate (.) through the integration interval [ x,x N ] where ( x ) : [ x,x N ] m R in which :[,xn ] x R is smooth. m. Speciication o the hbrid methods (.4) The hbrid methods (.4) with the hbrid predictors (.5) and (.6) have constant parameters 98

3 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 ( m) ( m) { β }, β v, m ( m) λ, ( l) ( l) { β }, β v, l ( l) l, vl ( l) { l }, ( l) β and ( l) l to be determined in such a wa that the hbrid method (.4) become stable. The method (.4) is the hbrid method o Adams-tpe equipped with nested unctions evaluation o the hbrid predictors (.5) and (.6). The hbrid parameters are chosen according as m vm vl +, v l +, l () m, vl (, ), vl, (),,,,...,. Construction o the Hbrid methods (.4) We assume the solution o (.4) o the orm + (x) ax (.) + where{ a } are real constant parameters to be determined and ( x, o() ) + is the polnomial basis unction. Dierentiating (.) twice to obtain ( ) + '(x) x, a x (.) ''(x) '(x, ) ( ) ax + (.) Interpolating (.), (.) and (.) at x x n + and collocating (.) at x x n +, () and x x n + v m we obtain the sstem o equations + x x... x + xn... ( + ) xn + x ( )... + x x... ( + ) x n x... + vm ( + ) x vm... ( K + )( + ) x + a a a... a + a + a + n... v m ' (.4) Solving equation (.4) with MATHEMATICA. Sotware pacage, the coeicients 99

4 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 ( ( ) ) a ' s + are obtained. Substituting these coeicients into (.) ields the discrete scheme or each.. Construction o the hbrid Predictors The corresponding hbrid predictor is obtained rom the polnomial interpolant ( ) + 4 n l+ x + v h bx (.) where { } + 4 b o + are parameter constants to be determined, { } 4 x is the polnomial basis unction. Following the approach as in section (), we obtain the sstem o equations + x x 4... x xn... ( + ) n x... ( + ) x x... ( + ) x... + x v v + v a a a n a + a + ' n + v v (.) Equation (.) is solved with MATHEMATICA. sotware pacage to obtain the coeicients o the hbrid predictor (.5) The corresponding error constants or the hbrid scheme and its hbrid predictors are obtained or each value o rom the Talor series expansion o (.4), (.5) and (.6) about x n. These are respectivel p+ p+ p+ ( ) ( ) ( ) x C h x + h (.) p+ n p ( ) ( ) ( ) * + p * + p * + * l p x C h x + h (.4) vl+ v + + n p ( ) ( ) ** + p ** + ** (x ) p x C h + h (.5) where ( ) n v v + n x +, ( + ) and ( ) x n vl + x n + v are the theoretical solutions; C p +, C p * + and C p ** + are error

5 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 constants o (.4),(.5) and (.6) respectivel. Due to the processing speed and the memor capacit o the laptop computer used in the derivation, onl ew stable members o the amil o the method could be obtained. I the method can be derived using higher processor, more stable members can be obtained rom stepnumber. Examples o A stable members o the amil o the hbrid methods (.4) with error constants are: For, m, v n h n, 5 C 88 with hbrid 7 h h, C6 84 n ' For, m, v h 8 47 h n ', C 44 with hbrids v 7,v 4 h n C, n n n n n h ' + h + + +, C For K, m, v, p 6 h 6 h n ', C 648

6 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp with hbrids v,v,v 4 8 h h n ' C 8 75 h 77 n h ' C h 5n h ' , C Stabilit o the Hbrid Schemes (.4) This section considers some important deinitions and stabilit properties o the hbrid schemes. Deinition : A numerical scheme (.4) is A stable i the region o absolute stabilit lies entirel in the open let hal o the complex plane. Deinition : The numerical scheme (.4) is ( ) { : ( ), } π A α Stable or someα,, i the wedge sα z Arg z <α z is contained in the region o absolute stabilit. The largest amax is the angle o absolute stabilit. Deinition : The numerical scheme (.4) is stil stable i (i) it is absolutel stable in the Region { } and (ii) accurate in the region R z ( z) ( z) R z: Re(z) DL { < < < } :D L Re D R; Im D, such

7 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 that the stabilit region is contained in the region R R. The numerical scheme is Zero-Stable since the roots o the irst characteristics polnomial r ( ) r r r satis r i with roots o [ r i ] being simple. To investigate the stabilit properties o the amil o the hbrid multistep methods (.4), we emplo the boundar locus approach discussed in[4]. Substituting the hbrid predictors in (.6) into (.5) then into (.4) at the hbrid points to ield a scheme, the resulting scheme or ixed is applied to the scalar test problem ' λ, ields the stabilit polnomials as ( ) λ ( ) '', Re λ < which ( rz ( ) ( ) ( ), ) r r z m m m r v H p( rz, ) z m r p β + β λ (5.) where ( ) ( ) l l ( l) ( l) ( l) ( l) Hp( rz, ) r z β r + β v... β r + zβ...( )... l + zl r + zlv T l ( ) and ( l) ( l) ( l) T β r + zβ + z l r The boundar plots are obtained rom the stabilit polnomials or various. 5. The Stabilit Plots o the hbrid method The ollowing are the boundar plots o the implicit hbrid scheme derived in: The boundar loci reveal that the scheme (.4) is zero-stable. For 6, it is A -Stable and A( α ) -Stable or > 6 to 9.

8 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 Figure 4 Figure 5 Figure 6 4

9 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp Numerical implementations This section considers numerical implementation o the new hbrid methods (.4) on some sti initial value problems in ordinar dierential equations. Since the method is an implicit method, the implicitness is resolved b appling the Newton scheme [ ] ( ) ( ) [ r+ ] [ r] [ r] r J F, r,,,,... (6.) [ r] or a modiication o (6.) where J( ) n + is the Jacobian matrix o the new hbrid method. The (6.) requires starting value and is generated rom the explicit scheme r h n + ( n + n), p (6.) Using ixed step-size h. The ollowing problems are considered or implementation. Problem [] The Chemical reaction problems in [7] 4 '.4, + ( ) 4 7 '.4., ( ) 7 '., ( ) h 6, x [,] Problem [] The non linear moderatel sti problems in [9] '. 99.9, ( ) ', ( ).x x h. with exact solution ( x) e + e and (x) x e 5

10 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 Problem [] The Van der pol equation in [] ', ( ) (( ) ) ' / ε, ( ) h., ε ode5s ode5s ode5s -axis x-axis Figure : Graphical solution o problem.5 E E Figure : Graphical solution o problem 6

11 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp data (ode5) (ode5) 5 unction axis x-axis Figure : Graphical solution o problem 7. Conclusion This paper has presented a class o hbrid linear multistep methods (.4) with nested hbrid predictors (.5) or sti initial value problems in ordinar dierential equations. The hbrid scheme has high order stabilit and is seen to overcome Dahlquist order barrier on linear multistep methods (.). The scheme has been implemented on three sti problems and the results in igures and show that the scheme (.4) compares avourabl with ODE5s o MATLAB in []. In igure, the graph is in alignment with the exact solution o the ODE. Reerences [] Brugnano, L & Trigiante, D.; Solving Dierential Problems b Multistep Initial and Boundar Value Methods, Amsterdan: Gordon and Breach Science Publishers, 998. [] Butcher, J.C; A modiied multistep method o numerical integration o ordinar Dierential equations, J Ass. comput. Math; 965, vol;, pp.4-5. [] Butcher, J.C; A Transormed implicit Runge-Kutta Method, J. Ass. comput. Math., 979, Vol.6, pp [4] Dahlquist, G. A; special stabilit problems or linear multistep methods, BIT. 96, vol., pp. 7 [5] Donelson, J. & Hansen, E.; Cclic Composite Multistep Predictor-Correctors Methods, SIAM, J. Num. Anal.,97, Vol.8,pp [6] Enright,W.H., Second Derivative Multistep Methods or sti ODE s, SAIM. J. Num.Anal., 974, vol., ISS. pp. -. [7] Enright, W.H., continuous numerical methods or ODE s with deect control, J. computational. Appl. 7

12 American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) (8) Volume 4, No, pp 97-8 math., Vol.5, (), pp [8] Esuabana I. M & Eoro S. E.; Hbrid Linear Multistep Methods with Nested Hbrid Predictors or Solving Linear and Nonlinear Initial Value Problems in Ordinar Dierential Equations, IISTE ournal o Mathematical Theor and Modeling, 7, vol. 7,iss., pp [9] Fatunla, S. O.; Numerical Methods or Initial Value Problems in Ordinar Dierential Equations, New Yor: Academic Press, 988. [] Gear, C. w.; Hbrid methods or IVP s in ODEs, SIAM Journal on Numerical Analsis, vol.,(965), pp [] Gragg, W. B & Shetter, H. J.; Generalised Multistep Predictor-Correctors methods, J. Assoc. Comput. Mach.,964, Vol., pp [] Hairer E. & Wanner G.; solving ordinar dierential equation : Sti and Dierential Algebraic problems, nd rv.ed. springer-verlag, New Yor,996. [] Higham, D. J.; Higham, N.J. MATLAB Guide, Societ o industrial and applied Mathematics (SIAM), Philadelphia, PA,. [4] Ihile, M. N. O & Ouonghae, R. I.; Stil Stable Continuous Extension o Second Derivative Linear Multistep Method with an o-step point or IVPs in ODEs, J. Nig. Assoc., Math. Phs., 7, vol., pp [5] Lambert, J. D.; Computational methods or Ordinar Dierential Sstems, Chichester; Wile, 97, pp.9. [6] Ouonghae, R. I., & Ihile M. N. O., A class o Hbrid Linear Multistep Methods With A( α ) -Stable Properties or Sti IVPs in ODEs, J. Num. Math. 4, Vol. 8, iss. 4, pp [7] Robertson, H. H, The solution o a set o reaction rate equation in: Numerical Analsis: An introduction (J. Walsh, Ed.), academic Press, New Yor, 966, pp