Continuous Hybrid Multistep Methods with Legendre Basis Function for Direct Treatment of Second Order Stiff ODEs

Transcription

1 American Journal o Computational and Applied Matematics 06, 6(): 8-9 DOI: 0.59/j.ajcam Continuous Hybrid Multistep Metods wit Legendre Basis Function or Direct Treatment o Second Order Sti ODEs Olabode B. T. *, Momo A. L. Department o Matematical Sciences, Federal University o Tecnology, Akure, Nigeria Abstract Tis article proposed continuous ybrid multistep metods wit Legendre polynomial as basis unctions or te direct solution o system o second order ordinary dierential equations. Tis was acieved by constructing a continuous representation o ybrid multistep scemes via interpolation o te approximate solution and collocation o derivative unction wit Legendre polynomial as basis unctions. Te discrete scemes were obtained rom te continuous sceme as a by-product and applied in block orm as simultaneous numerical integrators to solve initial value problems (IVPs). Te resultant scemes are sel-starting, do not need te development o separate predictors, consistent, zero-stable and convergent. Te perormance o te metods was demonstrated on some numerical examples to sow accuracy and eiciency advantages. Te numerical results compared avourably wit existing metod. Keywords Continuous, Hybrid, Multistep metod, Legendre polynomial, Sti equations, Initial Value Problems (IVPs), Ordinary Dierential Equations (ODEs). Introduction Te matematical modeling o pysical penomena in science and engineering ield especially in mecanical systems wit several springs attaced in series or dissipation, control teory, celestial mecanics, series circuits lead to a system o dierential equations (see Landau and Lisitz (965), Libo (980)). Realistically, te analytical solutions o most dierential equations are not easily obtainable. Tis necessitated te need or approximate solution by te application o numerical tecniques. Te tecniques or te derivation o continuous linear multistep metods (LMMs) or direct solution o initial value problems in ordinary dierential equations ave been discussed in literature over te years and tese include, among oters collocation, interpolation, integration and interpolation polynomials. Basis unctions suc as, power series, Cebysev polynomials, trigonometric unctions, te r monomials x, te canonical polynomial ( Qr ( x), r 0) o te Lanczos Tau metod in a perturbed collocation approac ave been employed or tis purpose (see Abualnaja (05); Adeyea et al., (0); Awoyemi and Idowu (005); Lambert (99)). * Corresponding autor: olabodebola@yaoo.com (Olabode B. T.) Publised online at ttp://journal.sapub.org/ajcam Copyrigt 06 Scientiic & Academic Publising. All Rigts Reserved Moreover, power series as also being extensively used in literature or te same purpose. Sirisena et al., (00) proposed a new Butcer type two-step block ybrid multistep metod or accurate and eicient parallel solution o order ordinary dierential equations. Awoyemi and Idowu (005) developed a class o ybrid collocation metods or tird order ordinary dierential equations wit power series as te basis unctions and were implemented in predictor corrector mode. Eigie et al., (00) worked on generalized two-step continuous linear multistep metod o ybrid type or te direct integration o second order ordinary dierential equations. Fudzial et al., (009) constructed te explicit and implicit -point-- block (IPB) or solving special second order ordinary equations directly. Awari (0) considered te derivation and application o six-point linear multistep numerical metod or te solution o second order initial value problems wic was implemented in block mode. Yusup and Onumanyi (00) demonstrated a successul application o multiple inite dierence metods troug multistep collocation or te second order ordinary dierential equations. Furtermore, Abualnaja (05) constructed a block procedure wit linear multistep metods using Legendre polynomial or solving irst order ordinary dierential equations. Te metod depends on te perturbed collocation approximation wit Legendre as perturbation term or te solution o irst order ordinary dierential equations. In te work o Yakusak et al., (05), uniorm order Legendre approac or continuous ybrid block metods were

2 American Journal o Computational and Applied Matematics 06, 6(): proposed or te solution o irst order ordinary dierential equations. In tis paper, we propose te coice o Legendre polynomial witout perturbation as basis unctions or te construction o continuous scemes, wic simultaneously generate solution o (). Tey are sel-starting and do not need any predictors. Preliminaries A central notion in tis work concerns te coice o Legendre polynomial as basis unctions in te derivation o te continuous scemes, te implementation strategies employed (in block mode) and te stability analysis o te metods. For convenience o te reader, we recall te deinitions as ollows: Deinition.: Te block metod is said to be zero-stable i te roots λ j, j =,, s o te caracteristic polynomial ρ(λ) deined by ρ(λ) = s A i λ si i=0 = 0 satisies λ j and or tose roots wit λ j =, te multiplicity must not exceed te order o te dierential equation. (see Fatunla (99)). Deinition.: Te set o W equals τ C; all roots ξi ( τ) o te caracteristic equation satisy ξi( τ),multiple roots satisy ξi( τ) < is called te stability region or region o absolute stability o te metod (Hairer and Wanner (996)). Deinition:. (Widlund (967)) A metod is said to be A(α)-stable i te sector { : arg( ) α, 0} S = z z z α is contained in te stability region. Deinition. (Ele (969)): A metod is called L-stable i it is A-stable and i in addition Lim R( z) = 0. z Deinition.5 (Olagunju et al., (0)): Legendre polynomial is special case o te Legendre unction wic satisy te dierential equation ( x ) y xy + n( n + ) y = 0, n > o, x <. Te general solution can be expressed as: y = AP ( x) + BQ ( x), x <. n Pn ( x ) and Qn ( x) are respectively te Legendre unctions o te irst-and second-kind o te order n. Te nt order polynomial Pn ( x ) is generally given by te ollowing equation: n n! (n k)! Pn ( x) = ( ) x n! k!( nk)!( n k)! k= 0 n k nk were n is te order o te Legendre polynomials, n signiies te integer part o n. Legendre polynomials are ortogonal to eac oter wit respect to weigt unction w(x) = on [-,]. Te irst two polynomials are always te same in all cases but te iger orders are created wit recursive ormula: ( n + ) P ( x) = (n + ) xpn( x) npn( x), n =,,... wit initial conditions: p 0 ( x ) =, p ( x) = x. Te irst our terms o te polynomial are; p 0 ( x ) =, p ( x) = x, p ( x) x p ( x) = 5x x. = ( ), ( ) Te paper is organized as ollows. Section is o an introductory nature. Te materials and metods are described in Section. Stability analysis o te metods is discussed in Section. In Section, some numerical experiments and results sowing te relevance o te new metods are discussed. Finally, in Section 5 some conclusions are drawn.. Matematical Formulation Consider te second-order initial value problem: y = ( xyy,, ), ya ( ) = η0, y ( a) = η () were R is suiciently dierentiable and satisies a Lipscitz condition, suiciently smoot, : R m+ R m,y is an m-dimensional vector and x is a scalar variable and a set o equally spaced points on te integration interval also given by a = x 0 < x < < x n < < x k < x N = b, () wit a speciied positive integer step number k greater tan zero, can be variable or constant step-size given by = x x n, n =,, N; N = b a. Assuming an approximate solution to () by taking te partial sum o Legendre polynomial o te orm: t+ s yx ( ) = ap r r( x), xn x x () r r= 0 were x can be used only ater certain transormation. Te second derivative o () gives t+ s y ( x) = ap r r ( x) () r= 0 Substituting () into () gives t+ s ap r r ( x) = ( xyx, ( ), y( x)), xn x x r (5) r= 0 were Pr ( x ) is te Legendre polynomial o degree r, valid in x n x x r and a r s are real unknown

3 0 Olabode B. T. et al.: Continuous Hybrid Multistep Metods wit Legendre Basis Function or Direct Treatment o Second Order Sti ODEs parameters to be determined and ( t + s ) is te sum number o collocation and interpolation points. Te well-known Legendre polynomials are deined on te interval [-,]... Derivation o te Continuous Hybrid Multistep Metods Our objective ere is to construct a continuous ormulation o te general linear multistep metod yx ( ) o degree r = t+ s t > 0, s > 0. Two cases were considered one-step and two-step metods. CASE : One-Step Continuous Hybrid Multistep Metod (OSCHMM). Collocating (5) at points, and interpolating () at points x= x s, s = 0,, respectively lead to a system o equations expressed in matrix orm: MD = U (6) were, M = D= a0 a a a a a5 a6 and U = yn y / n / / /. Solving (6) using Gaussian Elimination metod in Maple sot environment produces te ollowing values o a r s: [,,,,,,,] T [,,,,,, ] T 07 8 a0 = yn + y + n n a = yn + y + n n a = n n a = n + + n a = n + + n a5 = n + n a6 = n n Substituting (7)-() into equation () and ater some manipulation gives te continuous sceme (7) (8) (9) (0) () () ()

4 American Journal o Computational and Applied Matematics 06, 6(): 8-9 xx ( ) n xx yx y n = n + y xx xx xx xx xx xx xx 5 n xx 80 n n n n n n n 5 6 xxn xxn xxn xx xx xx xx xx n n n n n 5 6 xx 9 n xx 80 n xx 0 80 n xx 576 n xx 8 n + Evaluating () at x= x n + gives te discrete sceme y = y y n n 90 Te discrete sceme (5) is consistent, zero-stable and o order p = 5 wit error constant C p + = Here, it is our intention to get additional discrete scemes, so, we evaluated () at te points x= x i, i =, to obtain: y = y yn + 7 n n + 80 y = y yn + 67 n n Te irst derivative o () is ound and evaluated at points x= x i, i = 0,,,, yields te ollowing derivative scemes: y n = y yn 67 n n y = y yn + 5 n n y = y yn + 97 n n y = y yn + 8n n y y y = n + 8 n n () (5) (6) (7) (8) (9) (0) () ()

5 Olabode B. T. et al.: Continuous Hybrid Multistep Metods wit Legendre Basis Function or Direct Treatment o Second Order Sti ODEs.. Implementation o te One-Step Continuous Hybrid Multistep Metod (OSCHMM). In tis section, te implementation strategy o tis work is discussed. Following Fatunla (99, 99), te general discrete block ormula is given as: 0 µ µ AY = ey + dfy ( ) + BFY ( ) () m n m m 0 were e, d are vectors, B are RxR matrix and A identity matrix, µ is te order o dierential equation. Expressing equations (0) - () and () in orm o () and solving wit matrix inversion metod gives: Writing () explicitly y y y n n y y y n n = y y y n n y y n y n n n n n y = yn yn + 67 n n + 00 y = yn yn + 5 n n + 0 y = yn yn + 9 n n y y y = n n + 7 n Te block metod is o uniorm order ( 5, 5, 5, 5) T p = wit error constant C p+ = 07 9,,, Substituting (5) into (9)-() yields y = yn + 5n n y = yn + 9 n n + 60 y = yn + 9 n n + 0 () (5) (6) (7) (8) T (9) (0) ()

6 American Journal o Computational and Applied Matematics 06, 6(): 8-9 y y = n + 7 n Equations (5)-() are ten applied in block orm as simultaneous numerical integrators to solve (). CASE : Two-Step Continuous Hybrid Multistep Metod (TSCHMM). Similarly, collocating (5) at points, and interpolating () at points orm (6) were, M = () x= x s, s = 0, lead to a system o equations o D= a0 a a a a a5 a6 and U = yn y / n / /. Te continuous sceme is as ollows : [,,,,,,,] T xx ( ) n xx yx y n = n + y [,,,,,, ] T xx xx xx xx xx xx n n n n n n xx 5 n xx 960 n xx n xx n xx 6 n xx xx xx xx xx n n n n n xx 9 n xx 0 n xx n xx 6 n xx 6 n xx xx xx xx xx n n n n n Te discrete sceme is obtained as: y = y y n n 90 Te discrete sceme () is consistent, zero-stable and o order p = 5, wit te error constant Evaluating () at te points x= x i, i =,, we obtained te ollowing discrete scemes: C p + =. 560 n () ()

7 Olabode B. T. et al.: Continuous Hybrid Multistep Metods wit Legendre Basis Function or Direct Treatment o Second Order Sti ODEs y = y y n n 80 y = y y n n 5760 Te irst derivative o () is ound and evaluated at points x= x i, i = 0,,,, yield te ollowing derivative scemes: y y y n = n 67 n y y y = n + 5 n y y y = n + 97 n y y y = n + 9 n y y y = n + 8 n Te implementation o Two-Step Continuous Hybrid Multistep Metod (TSCHMM) is as ollows: Combining (), (5), (6), (7) and solving by using matrix inversion metod gives: y y y n n y n yn y + n = y y + n n y n y y n y n n n n + n Te above block metod is o uniorm order ( 5, 5, 5, 5) T n p = wit te error constant C p+ = 07 9,,, Equations (8) -() wit () are applied in block orm as simultaneous numerical integrators to solve (). (5) (6) (7) (8) (9) (0) () () T

8 American Journal o Computational and Applied Matematics 06, 6(): Stability Analysis In te spirit o Sommeijer et al., (99), te linear stability o block metod can be investigated by applying te metod to te test equation y = λ y. Tis leads to a recursion o te orm: Y = M( zy ), n M ( z) : = [ I zd] [ A + zb], z : = λ M is called te ampliication matrix and its eigenvalues te ampliication actors. By requiring te elements o te diagonal matrix D to be positive, te matrix I zd is nonsingular or all z on te negative real axis. Tereore, in te sequel, we assume tat te (diagonal) elements o D are positive. We sall use te result on te power o a matrix N (Varga 96), n q n N = on ( [ ρ( N)] ) as n, were. and ρ ( N) are te spectra norm and te radius o N and were all diagonal sub-matrices o te Jordan normal orm o N wic ave spectral radius ρ ( N) are at most qxq. I te spectra radius ρ ( N) <, ten N is called power bounded. Te region were te ampliication matrix M(z) is power bounded is called te stability region o te block metod. I te stability region contains te origin, ten te metod is called te zero- stable. Below are te grapical representations o stability o OSCHMM and TSCHMM respectively.. Numerical Experiments and Results In tis section, we applied te new metods to some problems: te irst is Undamped Duing s equation o Fang and Wu (008), two body problem o Fatunla (990), sti problem, linear second order initial value problem, Stieel and Bettis problem and Implicit -point -block (IPB) o Fadzial (009). Problem : Te Undamped Duing s equation: 0 0 ( ε ) y + y = y + cost+ sin0t 99εsin0t 0 yt ( ) =, y( t ) = 0 ε, ε = 0. Te exact solution yt ( ) = cost+ ε sin0t. It describes a periodic motion o low requency wit a small perturbation o ig requency. Te numerical results are sown on tables and below. Problem : Consider te given two-body problem y y =, y (0) =, y(0) = 0, r y y =, y(0) = 0, y(0) =, r ( ) [ 0,5 ] r = y + y x π y ( x) = cos x; y ( x) = sinx Teoretical solution: Problem : Consider te sti problem y + 00y + 000y = 0, y(0)=, y (0)=, = 0. Teoretical solution: y(x) = exp(-x). Problem : y y + 8 y = x, y(0)=, y (0) =, x [0,], Teoretical solution: y( x) = e x (cos( x ) sin( x)) + x + x + x Problem 5: Consider te system o equations o Stieel and Bettis problem: y + y = 0.00cos( x), y (0) =, y(0) = 0 y + y = 0.00sin( x), y(0) =, y(0) = , x 0, 0, [ π ] Te exact solutions are given as: y ( x) = cos( x) xsin x, y( x) = sin( x) xcos x. Table. One-step metod or problem Undamped Duings Equation X y-exact solution y-approximate Error e e e e e e e e e e e-05

9 6 Olabode B. T. et al.: Continuous Hybrid Multistep Metods wit Legendre Basis Function or Direct Treatment o Second Order Sti ODEs Figure. Stability Domain o Block o OSCHMM wic is A(α)-stable by Deinition. Figure. Stability Domain o Block TSCHMM wic is L(α)-stable by deinition. and. Table. Te y-exact, y-approximate and error o TSCHMM or problem Two-step metod, = 0.0 problem Undamped Duings Equation X y-exact solution y-approximate Error e e e e e e e e e e-0

10 American Journal o Computational and Applied Matematics 06, 6(): Table. Te y-exact, y-approximate and error o OSCHMM or problem Table : Te y-exact, y-approximate and error o TSCHMM or problem X y -exact y -exact y - approximate y - approximate Error in y Error in y E-.09E E- 6.79E E- 9.05E E-.807E E-.77E- 5.8E-.6E-.667E-.8576E E-.08E Table. Te y-exact, y-approximate and error o TSCHMM or problem 9.05E-.0865E-.06E-.E- X y -exact y -exact y - approximate y - approximate Error in y Error in y E-.97E E-.980E E E E E E-.7968E-.789E- 9.57E-.065E-0.85E E-.9E Table 5. Te y-exact, y-approximate and error o OSCHMM or problem 5.87E E-.657E-0.E-0 X y-exact y-approximate Error in OSCHMM (o problem ) E E E E E E-.577E E Table 6. Accuracy Comparison o TSCHMM or Problem.55966E-.5067E- X y-exact y-approx (o problem) Error in TSCHMM p = 5 Error in Jator & Li (009), p = E E E E E E E E E E-06.0E E E E E E E E E E-0

11 8 Olabode B. T. et al.: Continuous Hybrid Multistep Metods wit Legendre Basis Function or Direct Treatment o Second Order Sti ODEs Table 7. Accuracy Comparison o OSCHMM and TSCHMM wit Implicit -point- block (PIB) H METHOD MAX ERR IPB.98(-8) OSCHMM 9.555(-0) TSCHMM (-9) IPB.99(-9) OSCHMM.95(-0) TSCHMM 9.55(-0) IPB 8.670(-) OSCHMM 9.59(-) TSCHMM (-) IPB.9577(-0) OSCHMM.9(-) TSCHMM 9.59(-) Table 8. y-exact, y-approximate and error in TSCHMM or problem 5 X y -exact y -exact y -approx (prob5) y -approx (TSCHMM) Error y Error y E-8 7.0E E-8.0E E-8.E E-8 6.0E E-8.09E E-8.5E E-8.85E E-7.8E E-8.79E E-7.6E-9 On tables and, y-exact, y-computed and error o OSCHMM and TSCHMM or problem are sown wile te y-exact, y-computed and error o TSCHMM or problem is sown on table 5. On table 6, it is observed tat te maximum absolute error o te TSCHMM is E-08 is (smaller) more accurate tan E-06 o Jator & Li (009) or problem. Te accuracy comparison o te new metods and IPB are sown on table 7. Te new One-step metod (OSCHMM) and Two-step metod (TSCHMM) are substantially more accurate tan te numerical solution o initial-value problems (IVPs) using IPB, as te maximum absolute error is smaller wit variable is 9.555(-0) wile tat te maximum absolute error o TSCHMM is (-9) wic is smaller tan.98(-8) o IPB or = Conclusions We ave presented continuous ybrid multistep metods wit Legendre polynomial as basis unction or te direct solution o system o second order ODEs. Te derived metods were implemented in block mode wic ave te advantages o being sel-starting, uniormly o te same order o accuracy and do not need predictors, aving good accuracy as sown on table 7. It sould be noted tat accuracy and eiciency rate o a metod is dependent on te implementation strategies. I economical computation is required, ten te new metods are te better coice. Te new metods are tereore recommended or general purposed use. Finally, te region o absolute stability o te block metods o One-step and Two-step metods were presented in igures and. Maple and Matlab sotware package were employed to generate te scemes and results. REFERENCES [] Abualnaja, K. M (05) A Block Procedure wit Linear Multistep Metods using Legendre Polynomial or Solving ODEs. Applied Matematics SciRes, 6, pp77-7 ttp://dx.doi.org/0.6/am [] Adeyea, E. O. Folaranmi O. R. and A. F Adebisi, (0), A Sel-Starting First Order Initial Value Solver, In. J. Pure Appl. Sc. Tecnol., 5(), 8-.

12 American Journal o Computational and Applied Matematics 06, 6(): [] Awari (0) Derivation and Application o Six-Point Linear Multistep Numerical Metod or te Solution o Second Order Initial Value Problems. IOSR Journal o Matematics, e-issn: , p-issn 9-765X, vol. 7, Issue, PP -9. [] Awoyemi D. O and Idowu, O. M (005) A Class o Hybrid Collocation Metods or Tird Order Ordinary Dierential Equations. International Journal o Computer Matematics vol.8, No 00.pp-7. [5] Eigie, J.O (00) On Generalized -Step Continuous Linear Multistep Metod o Hybrid Type For Te integration o Second Order Ordinary Dierential Equations, Arc. Appl. Sci. Res., 00,(6):6-7. [6] Ele, B. L. (969): On Pade Approximations to te Exponential Junction and A-stable Metods or te Numerical Solution o Initial Value Problems. Researc Report CSRR 00, Dept. AACS, Univ. o Waterloo, Ontario, Canada. IV.5. [7] Fang, Y. L; Songa, Y and Wu, X. Y (008) A Robust Trigonometrically Fitted Embedded Pair For Perturbed Oscillators. Journal o Computational and Applied Matematics 5 (009) [8] Fatunla, S. O (990) Block Metod For Second Order ODEs. Inter. J. Compter Mats, (990), pp55-6. [9] Fudzial, I ;Yap Lee Ken and Moamad Otman (009) Explicit and Implicit -Point Block Metods or Solving Special Second Order Ordinary Dierential Equations Directly. Int. Journal o Mat. Analysis, Vol.. No5, 9-5, pp9-5. [0] Hairer, E and Wanner, G (996) Solving Ordinary Dierential Equations II, Sti and Dierential Algebraic Problems second revised edition ISBN Springer-Verlag Berlin Heidelberg New York. pp. [] Jator S. N and Li (009) A Sel-Starting Linear Multistep Metod or te Direct Solution o General Second Order Initial Value Problem. International Journal o Computer Mat. 86(5), [] Lambert, J. D. (99) Numerical Solution or Ordinary Dierential Systems, New York Jon Wiley, 99. [] Landau, L. D and Lisitz, F. M. (965) Quantum Mecanics, Pergamon, New York. [] Libo, R. L (980) Introductory Quantum Mecanics, Addison-Wiley, Reading M. A 980. [5] Olagunju, A.S; Josep, F. L and Raji, M. T (0) Comparative Study o Te Eect o Dierent Collocation Point on Ledendre Collocation Metods o Solving Second Order Boundry Value Problems, IOSR Journal o Matematics vol.7() pp 5-. [6] Onumanyi, P; Oladele, J. O; Adeniyi, R. B and Awoyemi, D. O. (99) Derivation o Finite Dierence Metods by Collocation, Abacus, (), 7-8. [7] Sirisena, U. W; Kumleng, G. M and Yaaya, Y.A (00) A New Butcer Type Two-Step Block Hybrid Multistep Metod For Accurate and Eicient Parallel Solution o ODEs. ABACUS Journal o Matematical Association o Nigeria. Volume, Number A Matematics series NR-ISSN pp-7. [8] Sommeijer, B. P; Couzy, W. and Van De Houwen (99) A-Stable Parallel Block Metods or Ordinary and Integro-Dierential Equations, Applied Numerical Matematics 9(99) 67-8 Nort-Holland. [9] Yusup Y and Onumanyi, P (00) New Multiple FDMS Troug Multistep Collocation or y = (x,y). Abacus vol. 9 No Matematics series. Te Journal o Matematical Association o Nigeria pp [0] Varga, R. S (96) Martix Iterative Analysis. Prentice-Hall, Englewood Clis, NJ, pp 65. [] Widlund, O. B (967): A note on unconditionally stable linear multistep metods. BIT, vol. 7, pp