Adebayo O. Adeniran1, Saheed O. Akindeinde2 and Babatunde S. Ogundare2 * Contents. 1. Introduction

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1 Malaya Journal of Matematik Vol. No An accurate five-step trigonometrically-fitted numerical scheme for approximating solutions of second order ordinary differential equations with oscillatory solutions Adebayo O. Adeniran Saheed O. Akindeinde and Babatunde S. Ogundare * Abstract In this paper class of second order ordinary differential equation with oscillatory solutions is considered. By employing the trigonometric basis function a continuous five-step scheme known as five-step trigonometrically fitted scheme is derived to approximate solutions to the class of considered equation. Consistency and zero stability of the developed method were proved. Stability and convergence properties of this new scheme were also established. The scheme so obtained is used to solve standard initial value problems with oscillatory solutions. From the numerical results obtained it was revealed that the proposed method performs better than some of the existing methods in the literature. Keywords Continuous schemes Multistep collocation Trigonometrically fitted method Initial value problem. AMS Subject Classification Primary L Secondary L Department of General Studies Federal Polytechnic Ile-Oluji Nigeria. Group in Computational Mathematics (RGCM) Department of Mathematics Obafemi Awolowo University Ile-Ife Nigeria. *Corresponding author: bogunda@oauife.edu.ng dareadeniran7@yahoo.com soakindeinde@oauife.edu.ng c Article History: Received July 8 Accepted 3 September 8 8 MJM. Research Contents Introduction Development of the method Error and Stability Analysis of the Scheme Local truncation error Zero stability Convergence of the method Implementation of the Scheme Numerical Examples Conclusion References Introduction Differential equations arising from the modeling of physical phenomena often do not have exact solutions. Hence the development of numerical methods to obtain approximate solutions becomes necessary to the extent that several numerical methods such as finite difference methods finite element methods and finite volume methods among others have been developed based on the nature and type of the differential equation to be solved. Here we are concerned with solutions of second order initial value problem of the form y = f (x y y ) y(a) = η y (a) = η (.) where f : R Rm Rm Rm and y y y R are given real constants. Many scholars such as Henrici[] Jeltsch[3] Twizel and Khaliq [] Awoyemi[] Simos[] Yusuph and Onumanyi[8] Adeniran Odejide and Ogundare[] have devoted lots of attention to the development of various methods for solving directly (.) without reducing to system of first order equations. Hairer and Wanner[] developed Nystromtype methods for (.) in which conditions for the determination of the parameters of the method were listed. Gear[8]

2 differential equations with oscillatory solutions 737/73 Hairer[] Chawla and Sharma[] independently developed explicit and implicit Runge-Kutta Nystrom type methods. Dormand and Prince[7] also developed two classes of embedded Runge-Kutta- Nystrom methods for the direct solution of (.). Several numerical methods based on the use of polynomial functions (Power series Legendre Chebyshev e.t.c) have been used as basis function to develop numerical methods for direct solution of (.) using interpolation and collocation procedure. However it is well known that polynomial of high degree tend to oscillate strongly and in many cases they are liable to produce very poor approximations. Psihoyious and Simos[] developed a trigonometrically fitted predictorcorrector method for numerical solution of IVPs with oscillating solutions. Numerical experiment showed that the method is efficient. Vigo-Aguiar and Ramos[7] in their paper titled On the choice of the frequency in trigonometricallyfitted method use the trigonometrically-fitted method to obtain an approximate solution to some nonlinear oscillators and also presented a strategy for the choice of frequency in trigonometrically-fitted methods. The main focus of this article is to employ trigonometric function as basis function to develop a new five-step numerical methods using interpolation and collocation procedure for the direct solution of (.). Equations (.3) and (.) leads to a system of eight equations which is solved by any linear system solvers such as Crammer s rule to obtain a j j = The a j s obtained are then substituted into (.) to obtain the continuous form of the method U(x) = α yn α yn h β j (u) fn j where w is the frequency α j and β j are continuous coefficients. The continuous method (.) is used to generate the main method of the form (.). That is we evaluate at x = xn and letting u = wh we obtain our main method yn = α yn α yn h β j (u) fn j with coefficients α = 8 9q 8q (q q ) α = q q (q q ) β = 9u q 8q 3q u (q q ) β = u q 8u q 8q q (q q ) The main objective in this section is to construct a continuous five-step trigonometrically fitted method. The method has the form β = (.) j= where u = wh β j (u) j = are the coefficients that depend on the step-size and frequency. In order to derive (.) we proceed by seeking to approximate the exact solution y(x) on the interval [xn xnh ] by interpolation function U(x) of the form 8u (q q ) u q 8u q q q (q q ) U(x) = a a x a x a3 x3 a x a x u (q q ) β3 = u q 3u q q 7u (q q ) β = u q 9u cos(u) q 3u (q q ) β = u q q u q (q q ) where variable q = cos u. We remark that by evaluating (.) at other points x = xn x = xn3 x = xn additional methods are derived namely: evaluating at x = xn gives (.) a sin(wx) a7 cos(wx) where a a a a3 a a a and a7 are coefficients that must be uniquely determined. We then impose that interpolating function (.) coincides with the exact solution at the end point xn and xn to obtain the equations yn = αˆ yn αˆ yn h βˆj (u) fn j j= with U(xn ) = yn and U(xn ) = yn. (.3) It is also demanded that the function (.) satisfies the differential equation () at points xn j j = to obtain the following set of six equations: U (xn j ) = fn j j =. (.) j=. Development of the method yn = α yn α yn h β j (u) fn j (.) j= (.) 737 αˆ = 3 7q 3q (q q ) αˆ = 8 9q 8q (q q ) (.7)

3 differential equations with oscillatory solutions 738/73 3q q u q u βˆ = (q q ) and u q u q βˇ =. (q q ) 8q q u q βˆ = (q q ) βˆ = q 9q 8u q (q q ) βˆ3 = Again evaluating (.) at x = xn gives u q 7u (q q ) with (q q ) (q q ) βˆ = u q u (q q ) 8q 9q 8 (q q ) (q q ) (q q ) (.8) j= 9q 8q 8 (q q ) αˇ = 7q q 7 (q q ) 7 7u 9q q βˇ = (q q ) u q u q (q q ) u q u q β =. (q q ) 8u q u q (q q ) In order to incorporate the second initial condition of (.) in the derived methods we differentiate (.) and evaluate at point x = xn x = xn and x = xn to have: u 8 q 88q βˇ = (q q ) yn = α` yn α` yn h [β` (u) fn β` (u) fn (.) β` (u) fn β`3 (u) fn3 β` (u) fn 8u q (q q ) 9 8u 9q 9u q βˇ3 = (q q ) β` (u) fn ] (.) 8u q (q q ) (q q ) u q (q q ) u q q 8 β = (q q ) u 8q 3q 9u q βˇ = (q q ) 8 u 8q q u q 8u q u q (q q ) 3u 7 9q q 8u q β 3 = (q q ) with αˇ = 8u q (q q ) 8u q 9q 8 β = (q q ) yn3 = αˇ yn αˇ yn h βˇj (u) fn j α = u 9q 8q u q β = (q q ) and evaluation of (.) at x = xn3 gives βˇ = 8q q (q q ) u q q u q β = (q q ) u q u q u q u α = q 3q u q βˆ = (q q ) (.9) j= u q u q q u q yn = α yn α yn h β j (u) fn j where 9u q (q q ) 738 α` = qp 7q p 7p 7(q q )hp α` = 7q p qp 7p 7(q q )hp

4 differential equations with oscillatory solutions 739/73 u qp 8p 7uq3 u 7q p β` = 7(q q )hp 3pq 7(q q )hp and 8u 3u p 3uq 7pq β = 7(q q )hp uq u p 7(q q )hp 7u p uq 8uq 7u β = 7(q q )hp 8pq 7(q q )hp 9u p u q p 97u qp 9p β` = 7(q q )hp 8pq 88q p 3uq 7(q q )hp 88pq u q p 38w qp 9p 7(q q )hp uq 88uq3 8u 7(q q )hp 8u 7uq 78u p uq β = 7(q q )hp 8u p 9u qp u q p β` = 7(q q )hp 8p 7pq 3q p 3uq 7(q q )hp 7u 8uq 38u p uq β 3 = 7(q q )hp 7pq 7(q q )hp u p 79u q p u qp β`3 = 7(q q )hp 8p 7pq 88q p uq 7(q q )hp 88pq 38u q p 88u qu p 7(q q )hp 8p 7(q q )hp 88uq3 uq 8u 7(q q )hp u p u qp 9u q p β` = 7(q q )hp 8u 7uq 9u p 3uq β = 7(q q )hp 9p 7uq3 8pq7q p 7(q q )hp uq uq 7u 7(q q )hp 7pq 8pq 7(q q )hp 33u qu p 7u q p 9p 7(q q )hp 97u qp 8p 3pq 3uq β` = 7(q q )hp 7pq 3pq 7(q q )hp 3u qp 8u q p 8p 7(q q )hp uq 3uq3 7u 7(q q )hp 3pq 33u qp 8p 7(q q )hp 8uq 8u p 3pq 38u qp β = 7(q q )hp 8p. 7(q q )hp 8u 3u p 7(q q )hp where in the above p = sin u. Again differentiating and evaluating at x = xn we obtain Finally differentiating and evaluating at x = xn gives yn = α yn α yn h [β (u) fn β (u) fn β (u) fn β 3 (u) fn3 β (u) fn (.) yn = α yn α yn h [β (u) fn β (u) fn β (u) fn β 3 (u) fn3 β (u) fn β (u) fn ] β (u) fn ] where with 7q p qp 7p (q q )hp qp 7q p 7p α = 7 (q q )hp 7u p 7q u p qu p α = 7(q q )hp α = α = 7 7u p 7q u p u qp 7(q q )hp 739 (.3)

5 differential equations with oscillatory solutions 7/73 3uq 7pq 3pq (q q )hp 33u qp 8u 3u p (q q )hp 7uq3 uq uq β = 7 (q q )hp β = 7 88pq (q q )hp 8pq 9p u q p (q q )hp 3. Local truncation error Following Simos [] Taylor series technique shall be used to estimate the local truncation error of the derived fivestep method. Thus for small values of the parameter u the coefficients in equation (.) can be expressed as 8 α = β = u u u8 u u u u β = u u u8 u u u u β = u u u β3 = u u u u u u u β = u u u β = u u 8 78 u8 3 9 u u 789 The Local Truncation Error (LTE) for the five-step method α = 8p 9u qp 7u u p (q q )hp uq 88uq3 uq β = 7 (q q )hp 7pq (q q )hp 3pq 8p 39u q p (q q )hp 3u qp 8u 78u p (q q )hp 3uq3 3uq uq β3 = 7 (q q )hp 7pq 88pq (q q )hp 8p u q p 9u q (q q )hp u p 7u 838u p (q q )hp 88uq3 uq 3uq β = 7 (q q )hp 7pq (q q )hp 8pq 33u qp 9p (q q )hp 8u q p 8u 9u p (q q )hp 7uq3 uq 3pq 8p β = 7 (q q )hp 9u qp 3u p. (q q )hp The methods derived in equations (.) - (.) above will be combined and implemented as a block in solving numerical examples. 3. Error and Stability Analysis of the Scheme 7

6 differential equations with oscillatory solutions 7/73. Implementation of the Scheme described by equation (.) is obtained as 9 (3) LTE((.)) = h w y (xn ) y() (xn ) (3.) 8 The strategy adopted for the implementation of the methods is such that all the discrete methods obtained from the continuous method as well as their derivatives which have the same order of accuracy with very low error constants for fixed h are combined as simultaneous integrators. The absolute errors calculated in the code are defined as where y(i) denotes the ith derivative of y with respect to the independent variable x. 3. Zero stability The block method is zero stable if the roots zs s = of the first characteristic polynomial ρ(z) which is defined by ρ(z) = det[zin E] Error = yexact ycomputed (3.) satisfies zs and every root with zs = has multiplicity not exceeding two in the limit as h. where I is the identity matrix and α α β β β β3 β β αˆ αˆ βˆ βˆ βˆ βˆ3 βˆ βˆ αˇ αˇ βˇ βˇ βˇ βˇ βˇ βˇ 3 α α β β β β 3 β β E = α` α` β` β` β` β`3 β` β` α α β β β β 3 β β α α β β β β 3 β β E = lim (E) h E = 3 where yexact is the exact solution ycomputed is the computed result and Error is the absolute error. The value of w that produce an optimal solution in terms of accuracy are considered. All computations were carried out using Maple 7. Numerical Examples Example We consider the following problem: y = y 99 sin(x) y() = y () = whose exact solution is given by y(x) = cos(x) sin(x) sin(x). Example We consider a highly oscillatory test problem y λ y = y() = y () =. (.) (.) For λ = the exact solution is known to be y(x) = cos(x) sin(x). Example 3 We consider the non-linear initial value problem (y ) π 3 π y (x) = y y( ) = and y ( ) = (.3) y Thus = z ( z ) ρ(z) = det[zi E] solving for z in whose exact solution is given by y = sin x. z ( z ) = gives z = or z =. Hence the block method is zero stable. 7. Conclusion. Convergence of the method According to Gurjinder et.al (3) Definition.. A block method is said to be consistent if it has an order of convergence p with p. The block method derived in this article are all consistent as all the methods are of order p >. It is a standard result that The necessary and sufficient condition for a linear multistep method to be convergent is for it to be consistent and zero stable see Dahlquist [Henrici[]]. Thus the block methods derived in this article are convergent. 7 We have proposed a five-step block trigonometrically fitted methods for the direct solution of general second order initial value problems with oscillatory solutions. The method has the advantage of being self starting having good accuracy with order consistent and zero stable. The methods are implemented without the need for the development of predictors nor requiring any other method to generate starting values. Implementation of the method with numerical examples on linear non-linear and problems with oscillatory solution showed that the methods are superior to most of the existing multistep methods available for approximating similar class of problems.

7 differential equations with oscillatory solutions 7/73 w =. Table. The exact solutions computed results and error from the proposed methods for Example with h = 3 x yexact ycomputed Error Error in Adeniran et.al[] Table. The exact solutions computed results and error from the proposed methods for Example h =. w =. x yexact ycomputed Error Table 3. Comparison of errors for Example. Abhulimen and Okunuga (8)-ABHUK x Proposed method p = k = Adeniran et.al [] p = 3 k = ABHUK[] p= Awoyemi et. al.[] p=k=

8 differential equations with oscillatory solutions 73/73 Table. The exact solutions computed results and error from the proposed methods for Example 3 h =. w =. x yexact ycomputed Acknowledgment [8] The authors wish to thank the anonymous referees for their critical analysis of the work as well as the editorial team of the Malaya Journal of Matematik for the opportunity to publish this work. [9] [] References [] [] [3] [] [] [] [7] Abhulimen C. E. and Okunuga S. A Exponentially fitted second derivative multistep methods for stiff initial value problems in ordinary differential equations Journal of applied mathematics and bio-informatics ()(8) 7 8. Adeniran A. O. Odejide S. A and Ogundare B. S One Step Hybrid Numerical Scheme For the Direct Solution of General Second Order Ordinary Differential Equations International Journal of Applied Mathematics 8()() 97-. Alabi M. O Oladipo A. T. and Adesanya A. O Initial Value solvers for Second Order Ordinary Differential Equations Using Chebyshev Polynomial as Basis Functions Journal of Modern Mathematics ()(8) 8 7. Awoyemi D. O A Class of Continuous Stormer-Cowell Type Methods for Special Second Order Ordinary Differential Equations Spectrum Journal ( & )(998) 8. Awoyemi D. O. Adebile E. A Adesanya A.O. and Anake T.A Modified block method for the direct solution of second order ordinary differential equations International Journal of Applied and Computational Mathematics 3(3)() Chawla M. M. and Sharma S. R Families of Fifth Order Nystrom Methods for y = f (x y) and Intervals of Periodicity Computing (98) 7. Dormand J. R. and Prince P.J Runge-Kutta-Nystrom triples Computers and Mathematics with Applications 3(987) [] [] [3] [] [] [] [7] [8] Error Error in Alabi et.al.[3] Gear C. W Numerical IVPs in ODEs Prentice Hall Englewood cliffs NI 97. Gurjinder Singh Kanwar V. and Saurabh Bhatia Exponentially fitted variants of the two-step Adams-Bashforth method for the numerical integration of initial value problem Journal of Application and Applied Mathematics 8()(3) 7 7. Hairer E. and Wanner G A Theory for Nystrom Methods Numerische Mathematik (97) 383. Hairer E Unconditionally Stable Methods for Second Order Differential Equations Numerische Mathematik 3(979) Henrici P Discrete Variable Methods in ODE. New York: John Wiley and Sons 9. Jeltsch R A Note on A-stability of Multi-step Multiderivative Method BIT (97) Psihoyios G. and Simos T. E Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions Journal of Computational and Applied Mathematics 8(3) 3. Simos T. E. (). Dissipative Trigonometrically-fitted Methods for Second Order IVPs with Oscillating Solutions Int. J. Mod. Phys 3()() Twizel E. H. and Khaliq A. Q. M Multi-derivative Methods for Periodic IVPs SIAM Journal of Numerical Analysis (98). Vigo-Aguilar J. and Ramos H On the choice of the frequency in trigonometrically-fitted methods for periodic problems Journal of Computational and Applied Mathematics 77() 9. Yusuph Y. and Onumanyi P New Multiple FDMs through Multistep Collocation for y = f(x y) National Mathematical Center Abuja Nigeria.????????? ISSN(P): Malaya Journal of Matematik ISSN(O):3?????????