Exponentially-Fitted Runge-Kutta Nystrom Method of Order Three for Solving Oscillatory Problems ABSTRACT 1. INTRODUCTION

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1 Malaysian Journal of Mathematical Sciences 8(S): 7-4 (04) Special Issue: International Conference on Mathematical Sciences and Statistics 0 (ICMSS0) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: Exponentially-Fitted Runge-Kutta Nystrom Method of Order Three for Solving Oscillatory Problems * Faranak Rabiei, Fudziah Ismail and Norazak Senu Department of Mathematics, Faculty of Sciences and Institute for Mathematical Research, Universiti Putra Malaysia, 4400 UPM Serdang, Selangor, Malaysia faranak_rabiei@upm.edu.my *Corresponding author ABSTRACT In this paper the exponentially fitted explicit Runge-Kutta Nystrom method is proposed for solving special second-order ordinary differential equations where the solution is oscillatory. The exponentially fitting is based on given Runge-Kutta Nystrom (RKN) method of order three at a cost of three function evaluations per step. Here, we also developed the trigonometrically-fitted RKN method for solving initial value problems with oscillating solutions. The numerical results compared with the existing explicit RKN method of order three which indicates that the exponentially fitted explicit Runge-Kutta Nystrom method is more efficient than the classical RKN method. Keywords: Exponentially-fitting, Runge-Kutta Nystrom method, oscillating solutions.. INTRODUCTION Consider the second order ordinary differential equation (ODEs) y( x) f ( x, y( x)), x[ x0 X ], y( x ) y, y( x ) y For approximating solution of problem in equation (E) many different methods have been studied. One of the most common method for solving numerically () is Runge-Kutta Nystrom (RKN) methods (see Dormand (996)). Exponentially fitting is a procedure for an efficient of exponential, ()

2 Faranak Rabiei, Fudziah Ismail & Norazak Senu trigonometric or hyperbolic function for solving ODEs whose solutions are oscillatory. Simos (00) and Franco (004) proposed the different approaches for exponentially fitted on explicit Runge-Kutta Nystrom methods. In this paper we propose the exponentially and trigonometrically fitting for third order, -stages Runge-Kutta Nystrom method with minimum phase-lag derived by Norazak et al. (00) also the exponentially fitting for RKN method of order three with normal set of coefficients. In section, for exponentially fitting the RKN method integrates exactly the functions exp( iwx) where w R. In Section, the construction of trigonometrically fitting RKN methods by integrating exactly the functions exp( iwx) where wrandi, is proposed. The number of test problems are experienced and the numerical results compared with existing RKN method of order derived by Dormand (996), are given in last section. The explicit RKN method is given by s s n n n i i n n i i i i y y hy h b k, y y h b k, i n n n i n i n ij j j k f ( x, y ), k f ( x c h, y hc y h a k ), i,, s. () c [ c, c,, c ], b [ b, b,, b ], b [ b, b,, b ], and ( s s) T T T where s s s matrix A [ a ij ]. The characteristic equation for RKN formulas () is given by Franco (004), tr( D( z )) det( D( z )) 0 A( z ) B( z ) where D and A, A, B, Bare polynomial in A'( z ) B'( z ) method with characteristic equation where R z ( ) tr( D) Rz ( ) ( z) z cos ( ), and Sz ( ) ( z) S( z ), z. For RKN S( z ) det( D), ( z) is phase-lag and ( z) is q dissipation. RKN method has phase-lag order q when ( z) O( z ) and r RKN method has dissipation order when ( z) O( z ). 8 Malaysian Journal of Mathematical Sciences

3 Exponentially-Fitted Runge-Kutta Nystrom Method of Order Three for Solving Oscillatory Problems Norazak et al. (00) developed the explicit RKN method of algebraic order p and phase-lag order q 6 at a cost of three function evaluations per step of integration. Here, we chose the values of free parameters from his method.. EXPONENTIALLY-FITTED THIRD ORDER RKN METHOD To construct the exponentially-fitted RKN method, we require that all stages of third order RKN method integrate exactly to the function exp( wx). We have e c v v a 0, e c v v a 0, cv cv cv cv ( ) 0, v cv cv v cv cv v cv cv ( ) 0, cv cv e c v v a a e e c v v ( a a e ) 0, e v v ( b b e b e ) 0, e v v ( b b e b e ) 0, v cv cv e v b b e b e e v( b b e b e ) 0, where v wh, w R. Using relations: v v v v e e e e cosh( v), sinh( v). the following order conditions are obtained: cosh( c v) v a, cosh( c v) v ( a a cosh( c v)), sinh( cv) cv v a sinh( cv)), cosh( v) v ( b b cosh( cv) b cosh( cv)), sinh( v) v v ( b sinh( cv) b sinh( cv)), cosh( v) v( b sinh( c v) b sinh( c v)), sinh( v) v( b b cosh( c v) b cosh( c v)). We set the values of parameters c, c and a from RKN 6 methods with minimum phase-lag derived by Norazak et al. (00), into above order conditions, so we have (EF-N method): Malaysian Journal of Mathematical Sciences 9

4 Faranak Rabiei, Fudziah Ismail & Norazak Senu a , a 0.805, b , b , b , b , b , b b. Also by choosing c, c and a 0 as free parameters, we have (EF method): a , a , b , b , b , b , b , b b.. TRIGNOMETRICALLY-FITTED THIRD ORDER RKN METHOD To construct the Trigonometrically-fitted RKN method, we require that all stages of third order RKN method integrate exactly to the function exp( iwx) where i, so we have e ic v v a 0, e ic v v a 0, icv ic v icv icv icv icv ( ) 0, e icv v a ae icv icv e icv v ( a ae ) 0, iv icv icv e iv v ( b be be ) 0, iv ( ic v ic v iv icv ic v ) 0, ie i v b be be iv icv icv ie i v( b be be ) 0, e ic v v a a e ( ) 0, e iv v b b e b e ( ) 0, iv iv iv iv e e e e where v wh, w R. Using cos( v), sin( v), the i following order conditions are obtained cos( c v) v a, cos( c v) v ( a a cos( c v)), sin( c v) c v v a sin( c v), cos( v) v ( b b cos( c v) b cos( c v)), sin( v) v v ( b sin( c v) b sin( c v)), cos( v) v( b sin( cv) b sin( cv)), sin( v) v( b b cos( c v) b cos( c v)). 0 Malaysian Journal of Mathematical Sciences

5 Exponentially-Fitted Runge-Kutta Nystrom Method of Order Three for Solving Oscillatory Problems We set the values of parameters c, c and a from RKN 6 methods with minimum phase-lag derived by Norazak et al. (00) into above order coditions, so we have (TF-N Method): a , a , b , b , b , b , b , b b. 4. NUMERICAL RESULTS To illustrate the efficiency of new methods the following problems are solved for x [0,50] and h,,, and numerical results 4 0 are compared with existing third order RKN method derived by Dormand (996). In addition, the following abbreviations are used in Figures -. EF-N: exponentially-fitted Runge-Kutta Nystrom method of order three with minimum phase-lag. EF: exponentially-fitted Runge-Kutta Nystrom method of order three when a 0. TF-N: trigonometrically-fitted Runge-Kutta Nystrom method of order three with minimum phase-lag. RKND: classical Runge-Kutta Nystrom method of order three derived by Dormand (996). Problem. (Norazak et al (00)). y y 0.00cos( x), y(0), y(0) 0. Exact solution: y( x) cos( x) xsin( x). Problem. (Franco (006)). y y 0.00cos( x), y (0), y(0) 0, y y 0.00sin( x), y (0) 0, y (0) Malaysian Journal of Mathematical Sciences

6 Faranak Rabiei, Fudziah Ismail & Norazak Senu Exact solutions: y ( x ) cos( x ) x sin( x ), y ( x ) sin( x ) x cos( x ). Figure : Logarithm of maximum global error versusnumber of function evaluations for solving problem Figure : Logarithm of maximum global error versus number of function evaluations for solving problem Figure : Maximum global error of EF- N method (dash line) and RKND (solid line) in solving problem Figure 4: Maximum global error of EF method (dash line) and RKND (solid line) in solving problem Figure 5: Maximum global error of F-N method (dashline) and RKND (solid line) in solving problem Figure 6: Maximum global error of EF- N method (dash line) and RKND (solid line) for y in solving problem Malaysian Journal of Mathematical Sciences

7 Exponentially-Fitted Runge-Kutta Nystrom Method of Order Three for Solving Oscillatory Problems Figure 7: Maximum global error of EF- N method (dash line) and RKND (solid line) for y in solving problem Figure 8: Maximum global error of EF method (dash line) and RKND (solid line) for y in solving problem Figure 9: Maximum global error of EF method (dash line) and RKND (solid line) for y in solving problem Figure 0: Maximum global error of TF- N method (dash line) and RKND (solid line) for y in solving problem Figure : Maximum global error of TF-N method (dash line) and RKND (solid line) for y in solving problem Malaysian Journal of Mathematical Sciences

8 Faranak Rabiei, Fudziah Ismail & Norazak Senu 5. CONCLUSION In this study we constructed the exponentially- fitted and trigonometrically- fitted Runge-Kutta Nystrom method of order three with - stages. The new methods give the accurate results for problems with oscillation solutions while the step size h was not chosen as small value. Numerical examples indicate that the exponentially fitted RKN methods are more efficient than given classical RKN method. REFERENCES Dormand, J. R. (996). Numerical Method for Differential Equations (A Computational Approach). CRC Press. Inc. Franco, J. M. (004). Exponentially-Fitted Explicit Runge-Kutta Nystrom Methods. Journal of Computational and Applied Mathematics. 67:-9. Franco, J. M. (006). A Class of Explicit Two Step Hybrid Methods for Second Order IVP's. Journal of Computation and Applied Mathematics. 87: Norazak, B. S., Suleiman, M, Isamil, F. and Othman, M. (00). A Zero- Dissipative Runge-Kutta Nyströ m Methods with Minimum Pahse- Lag. Mathematical Problem in Engineering. DOI:0.555/00/ 594. Simos, T. E. (00). Exponentially-Fitted Runge-Kutta Nystrom Method for the Numerical Solution of initial value problems with oscillating solutions. Applied Mathematics Letter. 5: Malaysian Journal of Mathematical Sciences

A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution of Oscillatory Problems

Applied Mathematical Sciences, Vol. 1, 16, no. 44, 157-178 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ams.16.64146 A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution