DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2015/217578 217578 Research Article Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems Senu N. 1 Ismail F. 1 Ahmad S. Z. 2 Suleiman M. 1 Ding Xiaohua 1 Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor Malaysia upm.edu.my 2 Department of Mathematics Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor Malaysia upm.edu.my 2015 2712015 2015 25 10 2014 06 01 2015 2712015 2015 Copyright © 2015 N. Senu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

1. Introduction

Many differential equations which appear in practice are system of second-order initial value problem (IVP), in which the first derivative does not appear explicitly: (1) y ′′ = f x , y , y x 0 = y 0 , y x 0 = y 0 . This type of ordinary differential equations (ODEs) often appears in many scientific areas of engineering and applied sciences such as celestial mechanics, molecular dynamics, quantum mechanics, and theoretical physics. Quite often the solution of (1) exhibits a pronounced oscillatory character. Oscillatory problems are usually harder to solve than the nonoscillatory problems. Coleman  in his paper developed the order conditions of hybrid method up to order nine. With the order condition developed by Coleman , Franco  constructed explicit two-step hybrid methods of fourth, fifth, and sixth order for solving second order IVPs.

To obtain a more efficient process for solving oscillatory problems, numerical methods are constructed by taking into account the nature of the problem. This results in methods in which the coefficients depend on the problem. Some important classes of numerical methods, such as exponentially fitted or phase-fitted methods, can be obtained if a good estimate of the period is known.

Phase-lag or dispersion error is the angle between the true and the approximated solutions. The analysis of phase-lag or dispersion error was first introduced by Brusa and Nigro . Several authors such as van der Houwen and Sommeijer [4, 5] and Thomas  studied in detail the phase-lag of numerical methods for solving (1). Several authors in their papers  have developed hybrid methods with the purpose of making the phase-lag of the method smaller. The technique of vanishing the phase-lag, the amplification error, and their first integral of phase-lag was introduced by Papadopoulos and Simos . Numerical results indicated that such technique produced methods which are very efficient for solving Schrodinger equation. Kosti et al.  in their work developed an optimized explicit Runge-Kutta Nyström method with four stages and fifth algebraic order for the numerical solution of orbital and related periodic initial value problems. Other than phase-lag much research is also focused on methods having high dissipative order; dissipation is the distance of the computed solution from the standard cyclic solution. Hence, for solving oscillatory problems, it is the aim of every researcher to derive numerical method which has high algebraic order, phase-lag order, and dissipative order. Methods having dissipative order infinity are called a zero-dissipative methods.

In this work, we will derive methods in which the coefficients depend on the problems. The first two methods are called optimized methods which are developed using the same approach introduced by Kosti et al. . They are optimized by imposing the phase-lag, dissipative or amplification error, and the first derivative of the phase-lag relation. The second two methods derived are called phase-fitted methods which are obtained by minimizing the phase-lag. The new optimized and phase-fitted hybrid methods are based on the nonzero-dissipative, four-stage fifth-order and five-stage sixth-order hybrid methods developed by Franco .

The comparison of the new methods with other methods in the scientific literature has shown that the optimized methods are more efficient for solving problems of oscillating in nature.

2. Analysis Phase-Lag of the Methods

An s -stage two-step hybrid method for numerical integration of the IVP (1) is in the form (2) Y i = 1 + c i y n - c i y n - 1 + h 2 j = 1 s a i j f t n + c j h , Y j , i = 1 , , s , y n + 1 = 2 y n - y n - 1 + h 2 i = 1 s b i f t n + c i h , Y i , where b i , c i , and a i j can be represented in Butcher notation by the table of coefficients.

s -stage hybrid methods are as follows: (3) c A b T = c 1 a 1,1 a 1 , s c s a s , 1 a s , s b 1 b s

The methods of the form (2) are defined by (4) Y 1 = y n - 1 , Y 2 = y n , Y i = 1 + c i y n - c i y n - 1 + h 2 j = 1 i a i j f t n + c j h , Y j , i = 3 , , s , y n + 1 = 2 y n - y n - 1 + h 2 b 1 f n - 1 + b 2 f n + i = 3 s b i f t n + c i h , Y i , where f n - 1 = f t n - 1 , y n - 1 , f n = f t n , y n , and h = Δ t = t n + 1 - t n and the first two nodes are c 1 = - 1 and c 2 = 0 . The method only requires to evaluate f t n , y n , f t n + c 3 h , Y 3 , , f t n + c s h , Y s for each step after starting procedure. This method is considered as two-step hybrid method with s - 1 stages per step and can be written in Butcher tableau;   s -stage explicit hybrid methods are as follows: (5) - 1 0 c 3 a 3,1 a 3,2 c s a s , 1 a s , 2 a s , s - 1 b 1 b 2 b s - 1 b s

In order to construct the new method, we use the test equation (6) y ′′ x = - v 2 y x for    v > 0 .

By replacing f ( x , y ) = - v 2 y into (2) we have (7) Y i = 1 + c i y n - c i y n - 1 - h 2 j = 1 s a i j λ 2 y , i = 1 , , s y n + 1 = 2 y n - y n - 1 - h 2 i = 1 s b i λ 2 y .

Define z = v h , so (7) can also be written as (8) Y i = 1 + c i y n - c i y n - 1 - z 2 j = 1 s a i j y , i = 1 , , s y n + 1 = 2 y n - y n - 1 - z 2 i = 1 s b i y .

Alternatively (8) can be written in vector form as follows: (9) Y = e + c y n - c y n - 1 - z 2 A Y (10) y n + 1 = 2 y n - y n - 1 - z 2 b T Y , where Y = ( Y 1 , , Y s ) T ,   c = ( c 1 , , c s ) T , e = ( 1 , , 1 ) T , (11) A = a 11 a 1 s a s 1 a s s , b = b 1 b s .

From (9) we obtain (12) Y = I + z 2 A - 1 e + c y n - I + z 2 A - 1 c y n - 1 .

Substituting (12) into (10) gives (13) y n + 1 = 2 - z 2 b T I + z 2 A - 1 e + c y n - 1 - z 2 b T I + z 2 A - 1 c y n - 1 .

Rewrite (13) and then the following recursion is obtained (14) P ξ , z = ξ 2 - S z 2 ξ + T z 2 = 0 , where (15) S z 2 = 2 - z 2 b T I + z 2 A - 1 e + c , T z 2 = 1 - z 2 b T I + z 2 A - 1 c .

The following definition is given by van der Houwen and Sommeijer  for Runge-Kutta Nyström method and has been used by Franco  for hybrid method.

Definition 1.

Apply the hybrid method (2) to (6). Next we define the phase-lag, φ z and dissipation, and d ( z ) (16) φ z = z - co s - 1 S z 2 2 T z 2 , d z = 1 - T z 2 . If φ ( z ) = O ( z q + 1 ) , then the hybrid method is said to have phase-lag order q . If d ( z ) = O ( z r + 1 ) , then the hybrid method is said to have dissipation order r .

3. Derivation of the New Hybrid Methods

In this section, we construct optimized hybrid methods of four-stage fifth order and five-stage sixth order. First, let us denote (15) in polynomials form as follows: (17) S z 2 = 2 - α 1 z 2 + α 2 z 4 - α 3 z 6 + + α i z 2 i , T z 2 = 1 - β 1 z 2 + β 2 z 4 - β 3 z 6 + + β i z 2 i .

The relations between phase-lag (dispersion), dissipation (amplification error), and the derivative of phase-lag are taken into consideration throughout obtaining the new methods. On the other hand, phase-fitted methods which are based on the same hybrid methods are also developed. The effectiveness of the new optimized methods is compared with that of the phase-fitted method as well as other existing methods in the literature.

3.1. Optimized Hybrid Methods

Polynomials (17) for hybrid method which satisfied algebraic order conditions up to order six can be written in these expressions:

for s = 5 , (18) S z 2 = 2 + - b 1 - b 1 c 1 - b 3 - b 3 c 3 0000000 - b 2 - b 2 c 2 - b 4 - b 4 c 4 z 2 + b 3 a 31 + b 4 a 41 + c 1 b 3 a 31 + c 1 b 4 a 41 00000 + b 4 a 43 + b 4 a 43 c 3 + b 3 a 32 + b 4 a 42 00000 + c 2 b 3 a 32 + c 2 b 4 a 42 z 4 + - b 4 a 32 a 43 - b 4 a 32 a 43 c 2 - b 4 a 31 a 43 00000 - b 4 a 31 a 43 c 1 z 6 T z 2 = 1 + - b 1 c 1 - b 3 c 3 - b 2 c 2 - b 4 c 4 z 2 + c 1 b 3 a 31 + c 1 b 4 a 41 + b 4 a 43 c 3 00000 + c 2 b 3 a 32 + c 2 b 4 a 42 z 4 + - b 4 a 32 a 43 c 2 - b 4 a 31 a 43 c 1 z 6

for s = 6 , (19) S z 2 = 2 + - b 2 - b 4 - b 4 c 4 - b 3 0000000 - b 3 c 3 - b 5 - b 5 c 5 z 2 + b 3 a 32 + b 4 a 42 + b 5 a 54 + b 5 a 54 c 4 00000 + b 5 a 52 + b 4 a 43 + b 5 a 53 00000 + c 3 b 4 a 43 + c 3 b 5 a 53 z 4 + - b 5 a 53 a 32 - b 5 a 54 a 42 - b 5 a 43 a 54 00000 - b 5 a 43 a 54 c 3 - b 4 a 43 a 32 z 6 + b 5 a 54 a 32 a 43 z 8 T z 2 = 1 + b 1 - b 4 c 4 - b 3 c 3 - b 5 c 5 z 2 + - b 3 a 31 - b 4 a 41 + b 5 a 54 c 4 - b 5 a 51 00000 + c 3 b 4 a 43 + c 3 b 5 a 53 z 4 + b 5 a 53 a 31 + b 5 a 54 a 41 - b 5 a 43 a 54 c 3 00000 + b 4 a 43 a 31 z 6 - b 5 a 54 a 31 a 43 z 8 .

In order to develop optimized hybrid method the following relations must hold:

the phase-lag condition, (20) φ z = z - cos - 1 S z 2 2 T z 2 = 0 ,

dissipation condition, (21) d z = 1 - T z 2 = 0 ,

and the first derivative of (20), (22) φ z = 0 .

The hybrid method in Franco (2006) , which is four-stage fifth order with dispersion order eight and dissipation order five or which we called nonzero dissipative method since the order of the dissipation is finite, is then substituted into the nullifying equations (20)–(22), and the equations are solved numerically. The coefficients of the method with a 41 , a 42 , and a 43 taken as free parameters can be written in Butcher tableau as follows (see (26)).

The optimized hybrid method is obtained with a 41 , a 42 , and a 43 given by (23) a 41 = 74646684 - 102396 cos z 2 z 6 000 + 33746796 cos z - 911163492 z 4 000 + 33746796 cos z sin z 0000000 - 911163492 sin z z 3 000 + cos z 2 - 2781245376 cos z - 23312031984 0000000 + 134987184 cos z 2 z 2 000 + 998395776 cos z sin z 000000 - 26956685952 sin z z 000 + 107826743808 - 111820326912 cos z 000 + 3993583104 cos z 2 74646684 - 102396    cos z 2 · 6625 z 6 sin z 2 + 728 - 1 . (24) a 42 = 950688900 - 1304100 cos z 2 z 6 000 + 15918300 cos z 2 - 23208881400 0000000 + 429794100 cos z cos z 2 z 4 000 + - 11604440700 sin z 0000000 + 429794100 cos z sin z z 3 000 + 31836600 cos z 3 - 72344216376 cos z 000000 + 1687339800 cos z 2 + 96583330152 z 2 000 + 26956685952 sin z 00000 - 998395776 cos z sin z z 000 - 107826743808 - 3993583104 cos z 2 000 + 111820326912 cos z 950688900 - 1304100    cos z 2 · 3125 z 6 sin z 2 + 728 - 1 (25) a 43 = 27955081728 · sin z h - 4 + z 2 + 4 cos z cos z - 27 · 165625 z 6 sin z 2 + 728 - 1 .

An optimized four stage fifth-order hybrid method is as follows: (26) - 1 0 25 28 1325 43904 1 2 1 2 1 35775 43904 - 23 5 a 41 1 2 1 2 1 a 42 a 43 173 1908 1 2 1 2 1 2791 3450 307328 3056775 - 125 636732 and the free parameters in Taylor expansion are given by (27) a 41 = 16744 33125 - 17687 74200 z 2 + 14096539 519400000 z 4 - 977401307 8638660800000 z 6 - 22049905351 571722278400000 z 8 + 1535705806157 1760904617472000000 z 10 + O z 12 (28) a 42 = 383111 15625 - 53061 875000 z 2 - 11620359 245000000 z 4 + 17015477671 814968000000 z 6 - 646287207193 988827840000000 z 8 - 12921642015637 830615385600000000 z 10 + O z 12 , (29) a 43 = - 13866608 828125 + 247618 828125 z 2 - 336053 16562500 z 4 - 44765797 55093500000 z 6 - 3154671007 200540340000000 z 8 + 3137160877 2079677600000000 z 10 + O z 12 .

For the construction of five-stage sixth-order optimized hybrid methods, we substitute (18) into (20)–(22) and using the coefficients of the five-stage sixth-order method which is nonzero dissipative given in Franco (2006)  (see (30)).

An optimized five-stage sixth-order hybrid method is as follows: (30) - 1 0 - 1 5 - 4 125 - 6 125 1 2 1 2 1 - 2 5 - 113 3000 - 13 750 - 7 120 1 2 1 2 1 2 3 5200 6561 1 2 1 2 1 a 52 a 53 a 54 1 60 23 24 1 1 2 1 - 125 156 125 192 729 4160

Choosing a 52 , a 53 , and a 54 as the free parameters, the optimized sixth-order hybrid method is obtained: (31) a 52 = 5 - 31528 sin z 2 - 39410 z 10 0000 + sin z 2 - 337800 cos z + 756895 00000000 + 200156 sin z 2 z 8 0000 + 30085590 cos z + 6920388 sin z 2 00000000 + 1677312 cos z sin z 2 00000000 - 33332940 sin z 2 z 6 0000 + 22089600 cos z sin z 0000000 - 33134400 sin z z 5 0000 + 20966400 cos z sin z 2 00000000 - 546624000 00000000 + 265824000 cos z 00000000 - 149760000 sin z 2 z 4 0000 + - 673920000 sin z 00000000 + 449280000 cos z sin z z 3 0000 + sin z 2 - 2021760000 cos z 0000000 - 1864512000 sin z 2 0000000 - 44928000 cos z sin z 2 0000000 + 617760000 sin z 2 z 2 0000 + 2808000000 cos z sin z 0000000 - 4212000000 sin z z 0000 - 28080000000 cos z 0000 - 11232000000 sin z 2 0000 + 28080000000 - 31528 sin z 2 - 39410 · 13122 z 6 56 sin z 2 + 70 z 4 0000000000000 + - 25 + 700 sin z 2 0000000000000000 + 600 cos z sin z 2 z 2 0000000000000 - 7500 + 3750 cos z 0000000000000 - 1500 sin z 2 - 1 (32) a 53 = - 1625 - 490 - 392 sin z 2 z 10 00000000 + - 5425 - 9380 sin z 2 - 4200 cos z z 8 00000000 + sin z 2 - 479340 + 272310 cos z 000000000000 - 56700 sin z 2 z 6 00000000 + - 483840 sin z + 322560 cos z sin z z 5 00000000 + - 11289600 - 1290240 sin z 2 000000000000 + 6451200 cos z sin z 2 z 4 00000000 + 9676800 cos z sin z 000000000000 - 14515200 sin z z 3 00000000 + - 38707200 sin z 2 - 6912000 0000000000000 - 27648000 cos z sin z 2 z 2 00000000 + - 103680000 sin z 0000000000000 + 69120000 cos z sin z z 00000000 + 691200000 - 276480000 sin z 2 00000000 - 691200000 cos z - 490 - 392 sin z 2 · 52488 z 6 56 sin z 2 + 70 z 4 0000000000000 + - 25 + 700 sin z 2 00000000000000000 + 600 cos z sin z 2 z 2 0000000000000 - 7500 + 3750 cos z 0000000000000 - 1500 sin z 2 56    sin z 2 + 70 - 1 (33) a 54 = 650000 - 5 - 4 sin z 2 z 6 000000000 + - 864 + 576 cos z z 4 000000000 + - 864 sin z + 576 cos z sin z z 3 000000000 + 360 - 2160 cos z - 2304 sin z 2 z 2 000000000 + 3600 cos z sin z - 5400 sin z z 000000000 - 4400 sin z 2 + 36000 - 36000 cos ( z ) · 2187 z 6 56 sin z 2 + 70 z 4 000000000000 + - 25 + 700 sin z 2 + 600 cos z z 2 000000000000 - 7500 + 3750 cos z 000000000000 - 1500 sin z 2 - 1 and the free parameters in Taylor expansion are given as follows: (34) a 52 = 4175 4374 + 23530 137781 z 2 + 95173 4133430 z 4 - 645749 1364031900 z 6 - 270131383 358058373750 z 8 - 234666525343 1718680194000000 z 10 + O z 12 (35) a 53 = - 2275 1944 - 47060 137781 z 2 - 15977 413343 z 4 + 82927 48715425 z 6 + 261389377 179029186875 z 8 + 2697629171 11160261000000 z 10 + O z 12 , (36) a 54 = 5200 6561 + 23530 137781 z 2 + 2249 118098 z 4 - 1666769 1364031900 z 6 - 280717231 358058373750 z 8 - 210021906751 1718680194000000 z 10 + O z 12 .

3.2. Phase-Fitted Hybrid Methods

In this paper, we also develop the phase-fitted the original hybrid methods. To develop phase-fitted method, (20) must hold. Equation (18) is substituted into (20), choosing a 31 as the free parameter and using the same coefficients as in (26), together with (37) a 41 = 16744 33125 , a 42 = 383111 15625 , a 43 = - 13866608 828125 .

We obtain a phase-fitted hybrid method of four-stage fifth-order with the solution of a 31 which is given by (38) a 31 = - 265 34726809600 cos z 2 00000000 + 34726809600 z 2 - 11575603200 z 4 00000000 + 105369600 z 4 cos z 2 - 34726809600 00000000 + 3875760 z 10 - 62289 z 12 + 1539968640 z 6 00000000 - 106798720 z 8 cos z 2 · 39337984 z 4 cos z 2 - 23520 + 769 z 2 - 1 and the Taylor expansion of the free parameter is (39) a 31 = 1325 43904 + 529841 14869757952 z 6 + 67815450029 3847103777341440 z 8 + 8656887188933513 1176290450959918694400 z 10 + O z 12 .

For the construction of five-stage sixth-order phase-fitted hybrid method, we substitute (19) into (20) and set a 52 as the free parameter. We use the coefficients given in (30) together with (40) a 53 = - 2275 1944 , a 54 = 5200 6561 .

Choosing a 52 as the free parameter, we get a phase-fitted hybrid method of five-stage sixth-order and a 52 is given by (41) a 52 = 810000 - 210 z 8 - 3744000 + 1872000 z 2 + 157125 z 4 0000 + 5200 z 6 - 728 z 8 0000 + 4160 810000 - 210 z 8 cos z · 328050 z 4 - 1 and the Taylor expansion of a 52 is (42) a 52 = 4175 4374 - 2353 688905 z 4 + 15223 20667150 z 6 - 168103 2728063800 z 8 + 130789 63654822000 z 10 + O z 12 .

4. Problems Tested and Numerical Results

In this section, we compare the optimized methods of order five and order six with the phase-fitted methods together with a few existing methods in the literature in order to determine the accuracy of the new optimized methods. The existing methods are the original hybrid method by Franco , optimized Runge-Kutta Nyström method order five developed by Kosti et al. , phase-fitted Runge-Kutta Nyström method of order four developed by Papadopoulos et al. , Runge-Kutta Nyström method of order five by Hairer et al. , Runge-Kutta method of order six given in Butcher , and phase-fitted and amplification-fitted Runge-Kutta Nyström method of order four developed in Papadopoulos et al. . Efficiency curves of the fifth and sixth order methods are given in Figures 17 and 814, respectively. The test problems are listed below.

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 2 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 3 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 4 with t end = 1 0 4 and h = 1 - 0.125 i , i = 1 , , 5 .

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 5 with t end = 1 0 4 and h = 0.125 / 2 i , i = 2 , , 6 .

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 6 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 7 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized and phase-fitted order five methods and their comparisons for Problem 8 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized order six method and their comparisons for Problem 2 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized order six method and their comparisons for Problem 3 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized order six method and their comparisons for Problem 4 with t end = 1 0 4 and h = 1 - 0.125 i , i = 1 , , 5 .

The efficiency curves of the optimized order six method and their comparisons for Problem 5 with t end = 1 0 4 and h = 0.125 / 2 i , i = 2 , , 6 .

The efficiency curves of the optimized order six method and their comparisons for Problem 6 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized order six method and their comparisons for Problem 7 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

The efficiency curves of the optimized order six method and their comparisons for Problem 8 with t end = 1 0 4 and h = 0.125 / 2 i , i = 1 , , 5 .

Problem 2 (Chawla and Rao [<xref ref-type="bibr" rid="B16">15</xref>]).

Consider (43) y ′′ t = - 100 y t , y 0 = 1 , y 0 = - 2 , and the fitted frequency, v = 10 .

Exact solution is y = - ( 1 / 5 ) sin 10 t + cos 10 t .

Problem 3 (Attili et al. [<xref ref-type="bibr" rid="B17">16</xref>]).

Consider (44) y ′′ t = - 64 y t , y 0 = 1 4 , y 0 = - 1 2 , and the fitted frequency, v = 8 . Exact solution is as follows: y = ( 17 / 16 ) sin ( 8 t + θ ) and θ = π - ta n - 1 ( 4 ) .

Problem 4 (Allen Jr. and Wing [<xref ref-type="bibr" rid="B18">17</xref>]).

Consider (45) y ′′ t = - y t + t , y 0 = 1 , y 0 = 2 , and the fitted frequency, v = 1 . Exact solution is y = sin t + cos t + t .

Problem 5 (Lambert and Watson [<xref ref-type="bibr" rid="B19">18</xref>]).

Consider (46) d 2 y 1 t d t 2 = - v 2 y 1 t + v 2 f t + f ′′ t d 2 y 2 t d t 2 = - v 2 y 2 t + v 2 f t + f ′′ t y 1 0 = a + f 0 , y 1 0 = f 0 , y 2 0 = f 0 , y 2 0 = v a + f 0 . Exact solution is as folllows: y 1 t = a cos v t + f t , y 2 t = a sin v t + f t , f t is chosen to be e - 0.05 t , and parameters v and a are 20 and 0.1, respectively.

Problem 6 (Papadopoulos et al. [<xref ref-type="bibr" rid="B12">19</xref>]).

Consider (47) y ′′ t = - v 2 y t + v 2 - 1 sin t , y 0 = 1 , y 0 = v + 1 , v = 10 .

The analytical solution is y ( t ) = cos ( v t ) + sin ( v t ) + sin ( t ) .

Problem 7 (Stiefel and Bettis [<xref ref-type="bibr" rid="B20">20</xref>]).

Consider (48) y 1 ′′ t + y 1 t = 0.001 cos t , y 1 0 = 1 , y 1 0 = 0 , y 2 ′′ t + y 2 t = 0.001 sin t , y 2 0 = 0 , y 2 0 = 0.9995 , for    v = 1 .

Exact solution is (49) y 1 = cos t + 0.0005 t sin t y 2 = sin t - 0.0005 t cos t .

Problem 8 (Franco [<xref ref-type="bibr" rid="B2">2</xref>]).

Consider (50) y ′′ t + 13 - 12 - 12 13 y t = g 1 t g 2 t , y 0 = 1 0 , y 0 = - 4 8 g 1 t = 9 cos 2 t - 12 sin 2 t , g 2 t = - 12 cos 2 t + 9 sin 2 t and whose analytic solution is given by (51) y t = sin t - sin 5 t + cos 2 t sin t + sin 5 t + sin 2 t for    v = 5 .

The following notations are used in Figures 114:

OPHM5: new optimized hybrid method four-stage fifth order derived in this paper;

OPHM6: new optimized hybrid method five-stage sixth order derived in this paper;

PFHM5: new phase-fitted hybrid method four-stage fifth order derived in this paper;

PFHM6: new phase-fitted hybrid method five-stage sixth order derived in this paper;

E-HM5: explicit hybrid method of four-stage fifth-order with dispersion of order eight and dissipation of order five developed by Franco ;

E-HM6: explicit hybrid method of five-stage sixth-order developed by Franco ;

OPRKN5: Optimized Runge-Kutta Nyström method of four-stage fifth-order developed by Kosti et al. ;

PFRKN4: Phase-fitted Runge-Kutta Nyström method of four-stage fourth-order by Papadopoulos et al. ;

RKN5: A classical Runge-Kutta Nyström method order five in Hairer et al. .

RK6: A Runge-Kutta method of order six with seven stages by Butcher .

PAFRKN4(4): A four-stage fourth-order phase-fitted and amplification-fitted Runge-Kutta Nyström method developed by Papadopoulos et al. .

A measure of the accuracy is examined using absolute error which is defined by (52) Absolute    error = max y x n - y n , where y x n is the exact solution and y n is the computed solution.

From the observation on efficiency curves in Figures 15, it is shown that OPHM5 is more accurate compared to other methods in the literature followed by OPRKN5 and then PAFRKN4 ( 4 ) method. However, for Problem 7, OPHM5 is just as good as OPRKN5, but still OPHM5 required less time to integrate the problem till the end of the interval, while for Problem 8, OPHM5 is the most accurate method followed by PFHM5 or E-HM5 and then OPRKN5. From the efficiency curves we noticed that the phase-fitted version of the hybrid method did not improve the accuracy of the method, whereas optimized method improved the accuracy of the method; hence it performed much better than the original hybrid method and other existing methods.

For method of order six the efficiency curves are shown in Figures 814. In Figures 8 and 9, OPHM6 is better in accuracy compared to OPRKN5 followed closely by PAFRKN4 ( 4 ) and PFHM6. In Figures 1012, as the time increases OPHM6 is gradually better in accuracy compared to OPRKN5 followed by PAFRKN4 ( 4 ) , PFHM6, and the rest of the methods. In Figure 13, again OPHM6 is the most efficient method followed by OPRKN5 PFHM6 and PAFRKN4 ( 4 ) and other methods in the literature, while in Figure 4, the most efficient method is OPHM6, followed by PFHM6, E-HM6, and OPRKN5 methods. Consequently, examining the results in Figures 814 the optimized sixth order method is very efficient in solving oscillating IVPs. Phase-fitted method improved the accuracy of the method slightly compared to the original hybrid method but could not compete with the optimized method.

5. Conclusion

In this paper, optimized hybrid methods of order five and order six are constructed based on the existing hybrid methods which have high phase-lag or dispersion order but are not zero-dissipative, originally developed by Franco . Phase-fitted methods based on the same existing methods are also constructed, and numerical results are shown in Figures 114. From the numerical results we conclude that the new optimized methods are more efficient for integrating oscillatory initial value problems of second order ODEs compared to the phase-fitted methods and other well-known existing methods in the scientific literature.

Furthermore, from the numerical results, it is proven that in solving oscillating special second order ODEs, the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods. The results are not quite true for zero-dissipative methods (see Ahmad et al. ) in which it is found that the phase-fitted method is very efficient in solving oscillating problems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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