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Article

Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

by
Musa Ahmed Demba
1,2,†,‡,
Poom Kumam
1,2,3,*,‡,
Wiboonsak Watthayu
3,‡ and
Pawicha Phairatchatniyom
1,2,‡
1
KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
3
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
*
Author to whom correspondence should be addressed.
Current Address: Department of Mathematics, Faculty of Computing and Mathematical Sciences, Kano University of Science and Technology, Wudil, P.M.B 3244 Kano State, Nigeria.
These authors contributed equally to this work.
Computation 2020, 8(2), 32; https://doi.org/10.3390/computation8020032
Submission received: 21 February 2020 / Revised: 4 April 2020 / Accepted: 8 April 2020 / Published: 15 April 2020

Abstract

:
In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

1. Introduction

In this work, we focus on the numerical solution of the special second-order ordinary differential equation of the form:
y = f ( x , y ) , y ( x 0 ) = y 0 , y ( x 0 ) = y 0 ,
whose solution have a notable periodic character, where y d and f : × d d is sufficiently differentiable. Problems of such form occur frequently in the scientific areas such as molecular dynamics, quantum mechanics, chemistry, nuclear physics, and electronics. Due to its applications, many researchers are motivated to study the numerical solution of Equation (1) (see [1,2,3,4,5,6,7]). Senu [8] proposed an embedded explicit RKN method for solving oscillatory problems, Fawzi et al. [9] derived an embedded 6(5) pair of explicit Runge–Kutta methods for periodic ivps, Franco [10] developed two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems, and Anastassi [11] constructed a 6(4) optimized embedded Runge–Kutta–Nyström pair for the numerical solution of periodic problems. Recently, Demba et al. [12,13] constructed two new embedded explicit trigonometrically-fitted RKN methods for solving the problem in Equation (1). A new embedded explicit exponentially-fitted RKN method based on the 5(3) embedded pair of explicit type derived in [14] is constructed in this work for solving Equation (1). This method can integrate exactly the test equation y = w 2 y , and the numerical results show the efficiency of the proposed method in comparison with other existing RKN methods in the scientific literature.
The paper is structured as follows. In Section 2, we explain the fundamental concepts of an explicit RKN pair, the basic definition of exponentially-fitted RKN method, and the derivation of an explicit exponentially-fitted RKN method. Section 3 deals with the construction of the proposed method. In Section 4, we analyze the algebraic order of the constructed method from their local truncation error (LTE) and we present a detailed information about the stability of the constructed method. In Section 5, we give the numerical results. In Section 6, we present a brief discussion about the graphs obtained, and a conclusion is drawn in the last section of the paper.

2. Fundamental Concepts

A Runge–Kutta–Nyström method of explicit type is represented generally as:
y n + 1 = y n + h y n + h 2 i = 1 s b i f ( x n + c i h , Y i ) ,
y n + 1 = y n + h i = 1 s d i f ( x n + c i h , Y i ) ,
Y i = y n + c i h y n + h 2 j = 1 i 1 a i j f ( x n + c j h , Y j ) .
where y n + 1 and y n + 1 denote the approximations of y ( x n + 1 ) and y ( x n + 1 ) , respectively, and x n + 1 = x n + h , n = 0 , 1 , . The corresponding Butcher tableau is given by:
c A b d
where A is a matrix ( a i j ) s × s , c = ( c 1 , c 2 , , c s ) T , b = ( b 1 , b 2 , , b s ) , and d = ( d 1 , d 2 , , d s ) .
An embedded m ( n ) pair of RKN methods is based on the method ( c , A , b , d ) of order m and the other RKN method ( c , A , b ^ , d ^ ) of order n ( n < m ). The higher order method yields the approximate solution ( y n + 1 , y n + 1 ) , while the lower order method yields the approximate solution ( y ^ n + 1 , y ^ n + 1 ) , which is only used for the estimation of the local truncation error.
A pair of embedded explicit RKN method is generally represented by the following Butcher tableau:
c A b T d T b ^ T d ^ T
In this study, a variable step-size procedure is utilized. Local error estimation at the point x n + 1 = x n + h is determined by δ n + 1 = y ^ n + 1 y n + 1 and δ n + 1 = y ^ n + 1 y n + 1 . To control the the step size h, we use the local error estimation given by Est n + 1 = max ( δ n + 1 , δ n + 1 ) . We utilize the step-size control procedure in [4] for the numerical solution of Equation (1). That is:
  • if E s t n + 1 < T o l / 100 , h n + 1 = 2 h n ;
  • if T o l / 100 E s t n + 1 < T o l , h n + 1 = h n ; and
  • if E s t n + 1 T o l , h n + 1 = h n / 2 and repeat the step.
Here, T o l is the tolerance. Note that the approximation y n is used as the initial value for the (n+1)th step.
Definition 1.
A Runge–Kutta–Nyström method (Equations (2)–(4)) is said to be exponentially-fitted if it integrates exactly the functions e w x and e w x with w > 0 , the principal frequency of the problem.
When an explicit Runge–Kutta–Nyström method (Equations (2)–(4)) is applied to the test equation y = w 2 y , we obtain the following equations:
y n + 1 = y n + h y n + h 2 i = 1 s b i ( w 2 Y i ) ,
y n + 1 = y n + h i = 1 s d i ( w 2 Y i ) ,
where
Y 1 = y n ,
Y i = y n + c i h y n + h 2 j = 1 i 1 a i j ( w 2 Y j ) , i = 2 , 3 , , s .
Let y n = e w x n , evaluating the value of y n , y n + 1 , y n and y n + 1 and, putting in Equations (5)–(8), we get the system of equations below:
T 1 : = e μ = 1 + μ + μ 2 i = 1 s b i + b i c i μ + b i μ 2 j = 1 i 1 a ij e w x n Y j ,
T 2 : = e μ = 1 + μ i = 1 s d i + d i c i μ + d i μ 2 j = 1 i 1 a ij e w x n Y j .
where μ = w h .

3. Construction of the Proposed Method

In this section, we construct a new embedded explicit exponentially-fitted RKN method.
In this study, the RKN5(3) embedded pair is used as given in [14]. The coefficients of the method are given in Table 1.
To obtain the adapted method in the embedding procedure, we consider firstly the coefficients of the lower-order method (order 3) in the RKN5(3) pair. We solve the system of equations in Equations (9) and (10) considering those coefficients but taking two of them as unknowns, specifically the parameters b ^ 3 , d ^ 3 . We obtain the following solution:
b ^ 3 = 9 280 42000 e μ 42000 70 μ 7 840 μ 5 294 μ 6 11500 μ 3 3250 μ 4 42000 μ 27750 μ 2 7 μ 8 μ 2 7 μ 4 + 1350 + 70 μ 3 + 900 μ + 300 μ 2 , d ^ 3 = 3 70 31500 e μ 31500 7 μ 7 294 μ 5 70 μ 6 3000 μ 3 840 μ 4 21375 μ 9000 μ 2 μ 7 μ 4 + 1350 + 70 μ 3 + 900 μ + 300 μ 2 .
In Taylor series form, we have:
b ^ 3 = 9 56 1 300 μ 3 23 9000 μ 4 + 41 42000 μ 5 157 3024000 μ 6 4051 136080000 μ 7 23299 4082400000 μ 8 + 1087931 134719200000 μ 9 3719141 1616630400000 μ 10 + 59764643 315242928000000 μ 11 283516309 66201014880000000 μ 12 + 2524972693 66201014880000000 μ 13 213318906023 9532946142720000000 μ 14 + 4603543083343 810300422131200000000 μ 15 + , d ^ 3 = 9 28 1 600 μ 3 7 4500 μ 4 + 31 54000 μ 5 + 29 1134000 μ 6 4153 136080000 μ 7 8741 2041200000 μ 8 + 22901 4082400000 μ 9 530767 404157600000 μ 10 + 91271 8083152000000 μ 11 + 74720717 4728643920000000 μ 12 + 5158878497 198603044640000000 μ 13 2900378753 198603044640000000 μ 14 + 147077961917 47664730713600000000 μ 15 8774361379613 36463518995904000000000 μ 16 + .
As μ 0 , the coefficients b ^ 3 and d ^ 3 of the lower-order adapted method reduce to the coefficients of the original lower-order method in the RKN5(3) approach. In a similar way, solving the above system in Equations (9) and (10) using the coefficients of the higher-order method (order 5) taking as unknowns the coefficients b 3 and d 4 , we obtain the following solution:
b 3 = 225 28 168 e μ 168 10 μ 3 μ 4 57 μ 2 168 μ μ 2 1350 + 70 μ 3 + 7 μ 4 + 900 μ + 300 μ 2 , d 4 = 5 16 2400 e μ 2400 6 μ 5 275 μ 3 60 μ 4 950 μ 2 2150 μ μ 42 μ 4 + 10 μ 5 + μ 6 + 375 μ 2 + 750 + 120 μ 3 + 750 μ .
In Taylor series form, we have:
b 3 = 9 56 + 1 1800 μ 4 13 75600 μ 5 + 29 1814400 μ 6 + 41 27216000 μ 7 + 1433 816480000 μ 8 38177 26943840000 μ 9 + 2843 7185024000 μ 10 2999609 63048585600000 μ 11 + 12016247 1471133664000000 μ 12 319437773 39720608928000000 μ 13 + 7553067077 1906589228544000000 μ 14 167041111997 162060084426240000000 μ 15 + , d 4 = 5 48 + 1 16800 μ 6 1 28800 μ 7 + 1 129600 μ 8 + 1 2520000 μ 9 53 26400000 μ 10 + 9403 5443200000 μ 11 16423 19656000000 μ 12 + 2244419 10478160000000 μ 13 + 1078223 44906400000000 μ 14 211491487 3113510400000000 μ 15 + 3036653827 63154728000000000 μ 16 + .
As μ 0 , the coefficients b 3 and d 4 of the higher-order adapted method reduce to the coefficients of the original higher-order method in the RKN5(3) approach.
The obtained coefficients depending on μ together with the rest of coefficients of the original RKN5(3) method form the new adapted embedded method, which is named as EEERKN5(3).

4. Algebraic Order and Error Analysis

In this part, we carry out the local truncation error and orders of convergence analysis based on the Taylor series expansion as given below:
L T E = y n + 1 y ( x n + h ) , L T E d e r = y n + 1 y ( x n + h ) .
The L T E and L T E d e r of the lower-order method (order 3) are:
L T E = h 4 24 ( f x x + 2 y f x y + ( y ) 2 f y y + f y y ) + O ( h 5 ) , L T E d e r = h 4 24 ( f x x x + 3 y f y x x + 3 y f x y + 3 ( y ) 2 f x y y + 3 y f y y y + ( y ) 3 f y y y + f y f x + ( f y ) 2 y ) + O ( h 5 ) .
From Equation (16), we can observe that the algebraic order of the lower-order method is 3 because all of the coefficients up to h 3 turns to zero. Similarly, the L T E and L T E d e r of the higher-order method (order 5) are:
L T E = h 6 21600 ( 4 y 3 + 3 y 2 f y y + 6 y f y x x + 6 y 2 f x x y y + y 4 f y y y y + 4 y f x x x y + 12 f y f x x + 12 f y 2 y + 6 y 2 f y y y y + 12 y 2 f y y f y + 12 y f x y y y + 24 f y y f x y + f x x x x 12 w 4 y ) + O ( h 7 ) , L T E d e r = h 6 720 ( f x x x x x + 18 y f y y f y y + 15 y 2 f x y y + 10 y f x x x y + 10 y f x x x y + 10 y f x y 2 + f y 2 f x + 5 f x x f x y + f y f x x x + y 4 f x y y y y + 5 y f x x x x y + 10 f y x x f x + 10 y 2 f x x x y y + 10 y 3 f x x y y y + 5 y 3 f y y 2 + y 5 f y y y y y + 15 y f y y y y 2 + 11 y 3 f y y y f y + 30 y f x x y y y + 30 y 2 f x y y y y + 8 f y y f x y + 10 y f y y f x + 10 y 3 f y y y y y + 10 y 2 f y y y f x + 23 y 2 f y f x y y + 15 y 2 f y y f x y + 20 y f x y y f x + 13 f y y f y x x + 5 y f y y f x x ) + O ( h 7 ) .
From Equation (17), the higher-order method has order 5 because all of the coefficients up to h 5 turns to zero.

Analysis of Stability

The linear stability of the RKN method in Equations (2)–(4) is obtained by applying it to the test equation y = w 2 y . In particular, for the method given in Table 1, setting H = ( w h ) 2 , the numerical solution satisfies the following recurrence system:
G n + 1 = E ( H ) G n ,
where
G n + 1 = y n + 1 h y n + 1 , G n = y n h y n , E ( H ) = 1 + H b T N 1 e w h ( 1 + H b T N 1 c ) w h d T N 1 e 1 + H d T N 1 c , N = I H A ,
A = [ a i j ] 4 × 4 is the corresponding matrix of coefficients and I is the identity matrix of fourth order,
b = [ b 1 , b 2 , b 3 , b 4 ] T , d = [ d 1 , d 2 , d 3 , d 4 ] T , e = [ 1 , 1 , 1 , 1 , 1 , 1 ] T , c = [ c 1 , c 2 , c 3 ] T .
It is considered that E ( H ) has complex conjugate eigenvalues for sufficiently small values of μ [15]. With this consideration, a periodic numerical solution is obtained. The periodic behavior depends on the eigenvalues of E ( H ) , which is called the stability matrix and its characteristic equation can be written as:
λ 2 t r ( E ( H ) ) λ + d e t ( E ( H ) ) = 0
.
Definition 2.
An interval ( H b , 0 ) corresponding to the RKN method in Equations (2)–(4) is said to be an interval of absolute stability if, for all H ( H b , 0 ) , it holds that | λ 1 , 2 | < 1 , where λ 1 , 2 are the roots of the above characteristic equation.
Definition 3.
An interval ( H p , 0 ) corresponding to the RKN method in Equations (2)–(4) is said to be periodic if, for every H ( H p , 0 ) , | λ 1 , 2 | = 1 , with λ 1 λ 2 , where λ 1 , 2 are the roots of the above characteristic equation.
Using Maple package, as well as the definitions in Equations (2) and (3), we find that the higher-order method of our new embedded pair (EEERKN5(3)) has a non-vanishing interval of absolute stability, while the lower-order method of our new embedded pair (EEERKN5(3)) has a non-vanishing interval of periodicity. Therefore, the higher-order method of our new embedded pair (EEERKN5(3)) has ( 9.48 , 0 ) as the interval of absolute stability, while the lower-order method of our new embedded pair (EEERKN5(3)) has ( 458.42 , 0 ) as the interval of periodicity.

5. Numerical Experiments

To show the robustness of the constructed method, we consider the following standard embedded RKN methods for the numerical comparison:
  • EEERKN5(3): The new embedded pair constructed in this paper;
  • RKN5(3): A 5(3) pair of explicit RKN methods given by Van de Vyver in [14];
  • ARKN5(3): A 5(3) pair of explicit ARKN methods derived by Franco in [16];
  • RKN6(4)6ER-PFAF: A 6(4) optimized embedded RKN pair obtained by Anastassi and Kosti in [11]; and
  • FRKN4: A Runge–Kutta–Nyström pair obtained by Van de Vyver in [17],
They are used to integrate the following periodic initial value problems:
Problem 1.
(Almost Periodic Problem) in [18]
y 1 = y 1 + 0.001 cos ( x ) , y 1 ( 0 ) = 1 , y 1 ( 0 ) = 0 , y 2 = y 2 + 0.001 sin ( x ) , y 2 ( 0 ) = 0 , y 2 ( 0 ) = 0.9995 , x [ 0 , 100 ] .
The exact solution is
y 1 ( x ) = cos ( x ) + 0.0005 x cos ( x ) ,
y 2 ( x ) = sin ( x ) 0.0005 x sin ( x ) ,
We take w = 1.0 to apply our method and the adapted methods in [11,16,17].
Problem 2.
(Two-Body Problem) in [19]
y 1 = y 1 ( y 1 2 + y 2 2 ) 3 2 , y 1 ( 0 ) = 1 , y 1 ( 0 ) = 0 , y 2 = y 2 ( y 1 2 + y 2 2 ) 3 2 , y 2 ( 0 ) = 0 , y 2 ( 0 ) = 1 .
The exact solution is
y 1 ( x ) = cos x ,
y 2 ( x ) = sin x ,
We solve this problem in [ 0 , 100 ] taking w = 1 for the adapted methods considered.
Problem 3.
(Almost Periodic Problem) Van de Vyver in [17]
y 1 = y 1 + ϵ cos ( Ψ x ) , y 1 ( 0 ) = 1 , y 1 ( 0 ) = 0 , y 2 = y 2 + ϵ sin ( Ψ x ) , y 2 ( 0 ) = 0 , y 2 ( 0 ) = 1 , x [ 0 , 100 ] .
The exact solution is
y 1 ( x ) = ( 1 ϵ Ψ 2 ) ( 1 Ψ 2 ) cos ( x ) + ϵ ( 1 Ψ 2 ) cos ( Ψ x ) ,
y 2 ( x ) = ( 1 ϵ Ψ Ψ 2 ) ( 1 Ψ 2 ) sin ( x ) + ϵ ( 1 Ψ 2 ) sin ( Ψ x ) ,
where ϵ = 0.001 and Ψ = 0.1 .
For the application of the adapted method developed in this paper and the methods by Anastassi and Kosti in [11], Franco in [16], and Van de Vvyver in [17], we consider w = 1 .
Problem 4.
(Nonlinear Problem) in [20]
y + y + y 3 = B cos ( Ω x ) , y ( 0 ) = 1 , y ( 0 ) = 0 ,
with B = 0.002 and Ω = 1.01 , the exact solution is
y ( x ) = 0.200179477536 cos ( Ω x ) + 0.246946143 × 10 3 cos ( 3 Ω x ) + 0.304016 × 10 6 cos ( 5 Ω x ) + 0.374 × 10 9 cos ( 7 Ω x )
.
We solve this problem in [ 0 , 100 ] taking w = 1 for the adapted methods considered.
The numerical results are shown in Table 2, Table 3, Table 4 and Table 5.
To further show the efficacy of the constructed method (EEERKN5(3)), we use the graphical approach to display the performance of EEERKN5(3) in comparison with other existing methods in the literature, as shown in Figure 1, Figure 2, Figure 3 and Figure 4. Tol = 10 2 i , i = 1 , 2 , 3 , 4 , 5 .

6. Discussion

Our proposed method (EEERKN5(3)) has the least error norm and least computational time, signifying that it is highly efficient and accurate for solving Equation (1), as shown in Table 2, Table 3, Table 4 and Table 5 and Figure 1, Figure 2, Figure 3 and Figure 4. The graphs show the accuracy, measured in log 10 ( M a x g l o b a l e r r o r ) versus the log 10 ( N u m b e r o f f u n c t i o n e v a l u a t i o n s ) . Therefore, we can deduce that (EEERKN5(3)) is more suitable for solving Equation (1) than the other existing methods in the scientific literature.

7. Conclusions

In this work, we construct a new efficient embedded explicit exponentially-fitted RKN method for solving periodic initial value problems. The constructed method contains four variable coefficients that depend on a parameter which is given by the product of the parameter of the method w and the step-length h [21,22]. The numerical experiment performed show clearly that EEERKN5(3) is more efficient for solving problem in Equation (1) than the other existing methods used for comparison.

Author Contributions

Conceptualization, M.A.D. and P.K.; methodology, M.A.D.; software, P.P.; validation, M.A.D., P.K., and W.W.; formal analysis, M.A.D.; investigation, P.P.; resources, P.K.; data curation, M.A.D.; writing—original draft preparation, M.A.D.; writing—review and editing, P.K.; visualization, M.A.D.; supervision, P.K.; project administration, W.W.; and funding acquisition, P.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), King Mongkut’s University of Technology, Thonburi.

Acknowledgments

The authors appreciate the efforts made by the reviewers of this manuscript for their constructive comments and also appreciate the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), King Mongkut’s University of Technology, Thonburi. The first author with Grant No.: 15/2562 was supported by the Petchra Pra Jom Klao PhD Research Scholarship from King Mongkut’s University of Technology, Thonburi.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
RKNRunge–Kutta–Nyström
IVPInitial value problem
LTELocal Truncation error

References

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Figure 1. Efficiency curves for Problem 1.
Figure 1. Efficiency curves for Problem 1.
Computation 08 00032 g001
Figure 2. Efficiency curves for Problem 2.
Figure 2. Efficiency curves for Problem 2.
Computation 08 00032 g002
Figure 3. Efficiency curves for Problem 3.
Figure 3. Efficiency curves for Problem 3.
Computation 08 00032 g003
Figure 4. Efficiency curves for Problem 4.
Figure 4. Efficiency curves for Problem 4.
Computation 08 00032 g004
Table 1. RKN5(3) method in [14].
Table 1. RKN5(3) method in [14].
0
1 5 1 50
2 3 1 27 7 27
1 3 10 2 35 9 35
1 24 25 84 9 56 0
1 24 125 336 27 56 5 48
5 24 125 168 9 56 1 8
1 12 25 42 9 28 1 6
Table 2. Numerical results for Problem 1.
Table 2. Numerical results for Problem 1.
TOLMETHODSTEPFCNFSTEPMAXETIME(s)
10 2 EEERKN5(3)12248802.570495(−3)0.053
RKN5(3)12248801.076884(−2)0.094
ARKN5(3)24296809.829659(−1)0.271
RKN6(4)6ER-PFAF242145206.005192(−1)0.075
FRKN4484193913.086156(−1)0.063
10 4 EEERKN5(3)522208804.246848(−7)0.050
RKN5(3)522208807.153723(−6)0.055
ARKN5(3)1044417916.406274(−2)0.062
RKN6(4)6ER-PFAF1044626913.549698(−2)0.370
FRKN4416916,68534.185123(−3)0.102
10 6 EEERKN5(3)1123449204.226820(−9)0.047
RKN5(3)1123449201.541216(−7)0.053
ARKN5(3)449117,97023.460856(−3)0.053
RKN6(4)6ER-PFAF449126,95621.912992(−3)0.218
FRKN435,919143,69155.635773(−5)0.487
10 8 EEERKN5(3)2420968004.243372(−11)0.075
RKN5(3)2420968003.319323(−9)0.096
ARKN5(3)19,34777,39731.863540(−4)0.130
RKN6(4)6ER-PFAF19,347116,09731.030210(−4)0.129
FRKN4309,5391,238,17777.583321(−7)3.248
10 10 EEERKN5(3)10,42241,69421.646495(−11)0.134
RKN5(3)10,42141,68711.664952(−11)0.109
ARKN5(3)83,362333,46041.003239(−5)0.338
RKN6(4)6ER-PFAF83,362500,19245.548769(−6)0.403
FRKN42,667,52410,670,12391.495822(−8)26.747
Table 3. Numerical results for Problem 2.
Table 3. Numerical results for Problem 2.
TOLMETHODSTEPFCNFSTEPMAXETIME(s)
10 2 EEERKN5(3)12248801.227156(−1)0.040
RKN5(3)12248808.478978(−1)0.041
ARKN5(3)270108311.804551(+0)0.044
RKN6(4)6ER-PFAF363218821.815228(+0)0.047
FRKN4484193911.942861(+0)0.043
10 4 EEERKN5(3)522208803.621045(−5)0.041
RKN5(3)522208806.990118(−4)0.047
ARKN5(3)1044417911.480069(−1)0.045
RKN6(4)6ER-PFAF1044626912.961366(−1)0.041
FRKN4416916,68531.130567(−2)0.078
10 6 EEERKN5(3)1123449203.722093(−7)0.054
RKN5(3)1123449201.520229(−5)0.063
ARKN5(3)449117,97025.473843(−3)0.051
RKN6(4)6ER-PFAF449126,95626.609722(−3)0.060
FRKN435,919143,69151.165965(−4)0.361
10 8 EEERKN5(3)2420968003.718493(−9)0.058
RKN5(3)2420968003.282692(−7)0.139
ARKN5(3)19,34777,39734.588825(−4)0.122
RKN6(4)6ER-PFAF19,347116,09732.397332(−4)0.090
FRKN4309,5391,238,17771.515111(−6)2.676
10 10 EEERKN5(3)10,42241,69421.717850(−11)0.120
RKN5(3)10,42141,68712.058225(−10)0.054
ARKN5(3)83,362333,46042.680717(−5)0.254
RKN6(4)6ER-PFAF83,362500,19241.145927(−5)0.247
FRKN42,667,52410,670,12391.557560(−8)22.647
Table 4. Numerical results for Problem 3.
Table 4. Numerical results for Problem 3.
TOLMETHODSTEPFCNFSTEPMAXETIME(s)
10 2 EEERKN5(3)12248802.591319(−3)0.062
RKN5(3)12248801.078825(−2)0.062
ARKN5(3)24296809.806283(−1)0.100
RKN6(4)6ER-PFAF242145205.976002(−1)0.300
FRKN4484193913.076264(−1)0.092
10 4 EEERKN5(3)522208804.299671(−7)0.065
RKN5(3)522208807.172465(−6)0.074
ARKN5(3)1044417916.403390(−2)0.191
RKN6(4)6ER-PFAF1044626913.548404(−2)0.165
FRKN4416916,68534.181655(−3)0.126
10 6 EEERKN5(3)1123449204.355510(−9)0.064
RKN5(3)1123449101.542823(−7)0.066
ARKN5(3)449117,97023.458063(−3)0.152
RKN6(4)6ER-PFAF449126,95621.911456(−3)0.143
FRKN435,919143,69155.631169(−5)0.566
10 8 EEERKN5(3)2420968004.516099(−11)0.081
RKN5(3)2420968003.324929(−9)0.095
ARKN5(3)19,34777,39731.862032(−4)0.243
RKN6(4)6ER-PFAF19,347116,09731.029369(−4)0.276
FRKN4309,5391,238,17777.577131(−7)3.588
10 10 EEERKN5(3)10,42241,69421.643935(−11)0.166
RKN5(3)10,42141,68711.658362(−11)0.153
ARKN5(3)83,362333,46041.002430(−5)0.470
RKN6(4)6ER-PFAF83,362500,19245.544237(−6)0.470
FRKN42,667,52410,670,12391.494338(−8)26.650
Table 5. Numerical results for Problem 4.
Table 5. Numerical results for Problem 4.
TOLMETHODSTEPFCNFSTEPMAXETIME(s)
10 2 EEERKN5(3)12251591.170545(−3)0.055
RKN5(3)122536162.777502(−3)0.066
ARKN5(3)12350442.849535(−1)0.141
RKN6(4)6ER-PFAF12474402.653779(−1)0.062
FRKN44391825235.355312(−2)0.062
10 4 EEERKN5(3)262107281.356514(−5)0.047
RKN5(3)262107597.208088(−5)0.062
ARKN5(3)5102076124.321049(−2)0.078
RKN6(4)6ER-PFAF5133128102.781421(−2)0.094
FRKN418157362342.600772(−3)0.088
10 6 EEERKN5(3)5732337159.010356(−8)0.062
RKN5(3)562226041.659456(−6)0.078
ARKN5(3)20858439331.815951(−3)0.141
RKN6(4)6ER-PFAF214013,005331.236425(−3)0.071
FRKN414,66658,868684.713910(−5)0.266
10 8 EEERKN5(3)19597932329.751267(−10)0.078
RKN5(3)23249392327.596427(−9)0.141
ARKN5(3)909136,487411.005629(−4)0.148
RKN6(4)6ER-PFAF925855,773456.549156(−5)0.187
FRKN4134,843539,6781024.210467(−7)2.129
10 10 EEERKN5(3)521321,140964.741679(−12)0.109
RKN5(3)518320,816284.524522(−11)0.125
ARKN5(3)39,556158,425675.609382(−6)0.281
RKN6(4)6ER-PFAF40,226241,631553.583863(−6)0.299
FRKN41,209,3314,837,7321365.483721(−9)14.829

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Demba, M.A.; Kumam, P.; Watthayu, W.; Phairatchatniyom, P. Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems. Computation 2020, 8, 32. https://doi.org/10.3390/computation8020032

AMA Style

Demba MA, Kumam P, Watthayu W, Phairatchatniyom P. Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems. Computation. 2020; 8(2):32. https://doi.org/10.3390/computation8020032

Chicago/Turabian Style

Demba, Musa Ahmed, Poom Kumam, Wiboonsak Watthayu, and Pawicha Phairatchatniyom. 2020. "Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems" Computation 8, no. 2: 32. https://doi.org/10.3390/computation8020032

APA Style

Demba, M. A., Kumam, P., Watthayu, W., & Phairatchatniyom, P. (2020). Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems. Computation, 8(2), 32. https://doi.org/10.3390/computation8020032

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