An Accurate Self-Starting Initial Value Solvers for. Second Order Ordinary Differential Equations

Transcription

1 International Journal of Contemporar Matematical Sciences Vol. 9, 04, no. 5, HIKARI Ltd, ttp://dx.doi.org/0.988/icms An Accurate Self-Starting Initial Value Solvers for Second Order Ordinar Differential Equations * E.O. Adeefa, + R.B. Adenii, ^Y. Haruna, $ J.A. Oladunoe * Department of Matematics and Statistics, Federal Universit Wuari, Taraba State, Nigeria + Department of Matematics, Universit of Ilorin, Ilorin, Kwara State, Nigeria ^Department of Matematics, Sa adatu Rimi College of Education Kumbotso, Kano, Kano State, Nigeria $ Department of Computer Science, Federal Universit Wuari Taraba State, Nigeria Coprigt 04 E. O. Adeefa et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in an medium, provided te original wor is properl cited. Abstract Tis paper focuses on te development of continuous and discrete algoritms for te numerical solution of ordinar differential equations. A continuous brid two-step metod is developed for second order initial value problems wit Cebsev polnomial as basis function troug collocation and interpolation tecniques. B selection of equall-spaced points for bot interpolation and collocation, new efficient, self-starting and zero-stable finite difference metod coupled as bloc metod is produced wic is generall more accurate wen compared wit existing metods. Te metod is analzed, its order and error constants are determined wit investigations made on consistenc and stabilit. Matematics Subect Clarification: 65L05 Keword: Hbrid, Collocation, Interpolation, Bloc Metod Introduction Te continuous integration algoritms for te numerical solution of te initial value problems (IVPs) ave been discussed in te literature. Te wors of Lie and Norsett (989) and Onumani et al (994) focused on te construction of continuous multistep metods b emploing te multistep collocation approac. Te use of finite difference to construct continuous implicit scemes troug wic

2 78 E. O. Adeefa et al. te bloc formulae are derived as been discussed extensivel b some scolars suc as Awoemi et al (0), Olabode (009), Adesana et al (009) to mention a few. Te traditional multistep metods including te brid ones can be made continuous troug te idea of multistep collocation, Norsett (989) and Onumani et al (994). To obtain multiple discrete brid metod, te continuous implicit brid metod is evaluated at some selected points involving grid and off-grid points along wit its first derivative. For te derivation of te bloc metods, te multiple discrete brid formulae obtained are solved simultaneousl and te resulting equations constituted a bloc from wic a number of explicit metods will be obtained. Materials and Metods We consider ere te derivation of te proposed continuous brid two-step bloc metods. Tis we do b approximating te analtical solution of f ( x,, ), ( a) z, ( a) () 0 0 were f is a continuous function, wit a Cebsev polnomial in te form rs ( x) a T ( x) () 0 on te partition a = x0 < x <... < xn < xn+ <... < xn = b of te integration interval [a, b], wit a constant step size, given b = xn+ - xn; n = 0,,..., N -. Te second derivative of () is given b rs ( ) ( ) 0 x a T x () were x[a, b], te a s are real unnown parameters to be determined and r + s is te sum of te number of collocation and interpolation points. We need to interpolate at at least two points to be able to approximate () and, to mae tis appen, we proceed b selecting two equall spaced offstep points. So, () is interpolated at x = & x = and its second derivative is collocated at xn+i, i = 0, v and, so as to obtain a sstem of seven equations wic are solved b Gaussian elimination metod.

3 An accurate self-starting initial value solvers a a a fn a f a4 f a 5 f a 6 f n (4) Solving for a s and substituting te resulting equations into (), we ave ( x) ( t) ( t) ( t) ( t) ( t) ( t) ( t) (5) were 0 t ( t) t () t t t t t t 0() t t t 5t 5t t t () t t t t t t () t t t t t () t t t t t t () t (6) and t = x x. Evaluating (5) at xn, xn+ and xn+, we ave

4 70 E. O. Adeefa et al. n (7 fn 4 f f f f) 90 ( fn 4 f 94 f 4 f f) (7) 90 ( fn f f 4 f 7 f) 90 Differentiating (5) at xn, x, x, x and x, n we ave ( x) ( t) ( t) ( t) ( t) ( t) ( t) ( t) (8) 0 were () t () t 4 5 0() t () t () t () t () t t t t t t t 5t 7 5t t t 9 t t t t t t t t t t t (9) Evaluating (8) at xn, x, x, x and xn, we ave

5 An accurate self-starting initial value solvers 7 ( 9 f 98 f 9 f 06 f 5 f ) n n 440 fn f f f 60 n 440 fn 60 ( 68 4 ) ( 7 f 74 f 74 f 7 f ) ( 4 f 68 f f) ( 5 f 06 f 9 f 98 f 9 f ) n 440 (0) Combining and solving (7) and (0) simultaneousl, explicit scemes () are obtained. n n (67 fn 540 f 8 f 6 f f) 5760 n n (47 fn 468 f 54 f 60 f 9 f) 640 n n (5 fn 44 f 0 f 6 f f) 60 n n (4 fn 48 f f 6 f ) 45 n (5 fn 646 f 64 f 06 f 9 f) 440 n (9 fn 4 f 4 f 4 f f) 80 n (7 fn 0 f 7 f 4 f f) 60 n (7 fn f f f 7 f) 45 () Analsis of te Metod Here, te order, error constant and consistenc of te metod are discussed.

6 7 E. O. Adeefa et al. Te explicit scemes (7) derived are discrete scemes belonging to te class of LMM of te form f 0 0 () Associated wit () is te linear differential operator L defined b [ ( ); ] [ ( ) ( )] () 0 L x x x Expanding () b Talor series, we ave q ( q) L[(x);] C0( x) C ( x)... Cq ( x)... (4) were C C... C (... ) ( 0... )!... p p q q Cp (... ) (... ), q p! ( q )! Definition Te LMM () is said to be of order p if C0 = C = C =... = Cp = Cp+ = 0 and Cp+ 0 is te error constant, see Lambert (97). According to tis definition, te discrete scemes (7) ave order p = (5, 5, 5) T wit error constants T (,.00 0 and ) Definition Te LMM () is said to be consistent if it is of order p and its first and second caracteristic polnomials defined as were z satisfies 0 0 ( z) z and ( z) z

7 An accurate self-starting initial value solvers 7 ( i) 0,( ii) () () 0,( iii) ()! (), see Lambert (97). 0 Te discrete scemes derived are all of order greater tan one and satisf te conditions (i) - (iii). Definition Te LMM () is said to be zero-stable if no root of te first caracteristic polnomial as modulus greater tan one, and if ever root of modulus one as multiplicit not greater tan two. All te roots of te derived scemes ave been verified to be less tan or equal to and z =, simple. Numerical Examples Here, we consider te application of te derived scemes to tree test problems for te efficienc and accurac of te metod implemented as bloc metod. Pr oblem 5 xe 4x Exact Solution : ( x). x e Source : Adesana et al (009). x, (0), (0), Problem x( ) 0, (0), (0), x Exact Solution : ( x) ln x Source : Awoemi et al (0). Pr oblem , (), (), x x 5 Exact Solution : ( x). 4 x x Source : Yaaa and Badmus (009).

8 74 E. O. Adeefa et al. Table of Results Table : Te exact solutions, te computed results and te absolute errors from problems X Proposed Metod Exact Solutions Absolute Errors e e e e e e e e- 07 Table : Te exact solutions, te computed results and te absolute errors from problems X Proposed Metod Exact Solutions Absolute Errors e e e e e e- 06

9 An accurate self-starting initial value solvers 75 Table : Te exact solutions, te computed results and te absolute errors from problems X Proposed Metod Exact Solutions Absolute Errors e e e e e e e e- 06 References [] A.O. Adesana, T. A. Anae, S. A. Bisop, and J. A. Osilagun, Two Steps Bloc Metod for te solution of general second order Initial Value Problems of Ordinar Differential Equations, Journal of Natural Sciences, Engineering and Tecnolog, 8, no, (009), 5 -. [] D. O. Awoemi, T. A. Anae, and A. O. Adesana, A One Step Metod for te Solution of General Second Order Ordinar Differential Equations, International Journal of Science and Tecnolog,, no 4, (0), [] D. O. Awoemi, E. A Adebile, A. O. Adesana, and T. A. Anae, Modified bloc metod for te direct solution of second order ordinar differential equations, International Journal of Computational and Applied Matematics,, no : [4] J. D. Lambert, Computational Metods in Ordinar Differential Equations, Jon Wile, New Yor, 97. ttp://dx.doi.org/0.00/zamm

10 76 E. O. Adeefa et al. [5] I. Lie and Norsett, Superconvergence for Multistep Collocation, Mat. Comp., 5, (989), ttp://dx.doi.org/0.090/s [6] B. T. Olabode, An accurate sceme b bloc metod tor tird order ordinar differential equations, Te Pacific Journal of Science and Tecnolog, 0, no, (009), 6 4. [7] P. Onumani, D. O. Awoemi, S.N. Jator and U. V. Sirisena, New linear multistep metods wit continuous coefficients for first order initial value problems, Journal of Nig. Mat. Soc.,, (994), 7-5. [8] Y. A. Yaaa and A. M. Badmus, A Class of Collocation Metods for General Second Order Ordinar Differential Equations, African Journal of Matematics and Computer Science researc,, no 4, (009), Received: Ma 5, 04; Publised: December, 04