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

: 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.


Introduction
In this work, we focus on the numerical solution of the special second-order ordinary differential equation of the form: 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.

Fundamental Concepts
A Runge-Kutta-Nyström method of explicit type is represented generally as: 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: 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 (ŷ n+1 ,ŷ n+1 ), which is only used for the estimation of the local truncation error.
Here, Tol 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 wx and e −wx 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: where Let y n = e wx 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: where µ = wh.

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 parametersb 3 ,d 3 . We obtain the following solution:  In Taylor series form, we have: As µ → 0, the coefficientsb 3 andd 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: In Taylor series form, we have: 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).

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: The LTE and LTE der of the lower-order method (order 3) are: 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 LTE and LTE der of the higher-order method (order 5) are: 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 = −(wh) 2 , the numerical solution satisfies the following recurrence system: 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: 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.

Numerical Experiments
To show the robustness of the constructed method, we consider the following standard embedded RKN methods for the numerical comparison: The exact solution is y 1 (x) = cos(x) + 0.0005 x cos(x), We take w = 1.0 to apply our method and the adapted methods in [11,16,17].

Problem 4. (Nonlinear Problem) in
We solve this problem in [0, 100] taking w = 1 for the adapted methods considered.
The numerical results are shown in Tables 2-5.  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 Figures 1-4. Tol = 10 −2i , i = 1, 2, 3, 4, 5.

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 Tables 2-5 and Figures 1-4. The graphs show the accuracy, measured in log 10 (Max global error) versus the log 10 (Number o f f unction evaluations). Therefore, we can deduce that (EEERKN5 (3)) is more suitable for solving Equation (1) than the other existing methods in the scientific literature.

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)

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: RKN Runge-Kutta-Nyström IVP Initial value problem LTE Local Truncation error