Exponentially-fitted and trigonometrically-fitted two derivative Runge-Kutta-Nystrom methods for solving y''(x) = f(x, y, y' )

Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions {exp ⁡ (wx),exp ⁡ (-wx)} and {sin ⁡ (wx),cos ⁡ (wx)} respectively, when w ∈ R , the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing

There are no research findings related to exponentially fitted and trigonometrically fitted TDRKN methods in which researchers have not yet explored the advantages or disadvantages in applying exponentially fitting and trigonometrically fitting techniques to TDRKN methods.Hence, in this paper, new two exponentially fitted and trigonometrically fitted TDRKN methods are constructed.In Section 2, explicit exponentially fitted two-derivative Runge-Kutta-Nyström methods with two-stage fourth order and three-stage fifth order are derived.In Section 3, trigonometrically fitted explicit TDRKN methods with two-stage fourth order and threestage fifth order were derived.In Section 4, the convergence and linear stability analysis is analyzed.Numerical illustrations are presented in Section 5 to show the efficiency of the new methods.Finally, the discussion and conclusion are given in Section 6.
and four more equations corresponding to  and where V = ℎ,  a real number and indicates the frequency of the problem and ℎ is the step-length of integration.The relations cosh(V) = ( V +  −V )/2, sinh(V) = ( V −  −V )/2 will be used in the derivation process.The following conditions are obtained: sinh cosh sinh and four equations corresponding to  and   :

Trigonometrically Fitted TDRKN Methods
Exponentially fitted methods lead to trigonometrically fitted methods: when replacing V = ℎ with V, we obtain cos sin sin and four equations corresponding to  and   :

Trigonometrically Fitted TDRKN Method of Order
Five.Consider the same coefficients fifth-order three-stage method developed by Chen as in Section 2.2.By solving (22) to (25) Next, solving (26) to (29) and using the above Chen coefficients to find  1 ,  1 , 1 , and  1 yield These lead to our new explicit trigonometrically fitted which is called TFTDRKN5(3) method.The corresponding Taylor series expansion of the solution is given by )

Analysis of Convergence and Stability of the Method
In this section, the convergence and linear stability of the new methods derived will be discussed.Note that zero-stability and consistency are sufficient conditions for the new method to be convergent.The method ( 2) is said to be consistent if it has at least order two and every one-step method is always zero-stable (see Lambart [ [23]]).Hence, the proposed method is zero stable and consistent; as a result, the proposed method can be said to be convergent.For the linear stability, consider the test equation: is the frequency of the problem.Applying method ( 2) to (36) yields where V = ℎ and  = ℎ.Matrix (V, ) is called stability matrix and  1,1 ,  1,2 ,  2,1 and  2,2 are defined as given in [10].The absolute stability regions of the new methods are presented as follows (see Figures 1-4 for EFTDRKN4(2),

Problems Tested and Numerical Results
In order to examine the effectiveness of the new TFTDRKN and EFTDRKN methods proposed in this paper, we apply them to some test problems.We also employ several integrators from the literature for comparison.The numerical methods we choose for experiments are as follows: (i) EFTDRKN5(3): the three-stage fifth-order exponentially fitted TDRKN method derived in this paper.
(xiii) RKNG5( 6): the classical six-stage fifth-order RKN method which is the limit method of ARKNGV5 as the frequency matrix  → 0 derived in [7].
Problem 1.We consider the linear equation studied in [29]: with the initial values The problem is integrated on the interval [0, 20] with stepsizes ℎ = 1/2  ,  = 2, . . ., 5 and  = 3.The analytical solution of this problem is given by the following: Problem 2. We consider the inhomogeneous linear equation: with the initial values The problem is integrated on the interval [0, 10] with stepsizes ℎ = 1/2  ,  = 3, . . ., 6 and  = 7.The analytical solution of this problem is given by: Problem 3. We consider the inhomogeneous linear system: with the initial values The analytical solution of this problem is given by the following: The problem is integrated on the interval [0, 5] with stepsizes ℎ = 1/2  ,  = 5, . . ., 8 and  = 1.
Problem 4. We consider the inhomogeneous linear system: with the initial values The analytical solution of this problem is given by the following: The problem is integrated on the interval [0, 5] with stepsizes ℎ = 1/2  ,  = 1, . . ., 4 and  = 2.
Problem 5. We consider the harmonic oscillator equation with frequency  and small perturbation  that is studied in [30]: with the initial values The analytical solution of this problem is given by the following: In this experiment we choose the parameters values  = 10 −6 ,  = 1.The problem is integrated on the interval [0, 10000] and the stepsizes are taken as ℎ = 1/2  ,  = 3, . . ., 6 and  = 1.
Problem 6.We consider the famous Van der Pol equation studied in [31]: with the initial values We choose  = 40 and integrate the problem in the interval [0, 100] with the stepsizes ℎ = 1/2  ,  = 5, 6, 7, 8.The reference numerical solution is obtained by method RKNG4 with a very small stepsize.

Discussion and Conclusion
An analysis of the construction of a class of exponentially fitted and trigonometrically fitted two-derivative Runge-Kutta-Nyström methods for solving   () = (, ,   ) has been carried out.
The efficiency of the methods developed is presented in         From Figures 13,15,17,18,19,21,23,25, and 27, we observed that, for Problems 5-9, the TFTDRKN4(2) and TFTDRKN5(3) methods have better accuracy than the other methods and good for oscillatory problems.In Figures 26  and 28, the new methods TFTDRKN4(2) and TFTDRKN5(3) have the same order of accuracy with the classical TDRKN methods.However, the results for Problems 6 and 10 given in Figures 16,22,26,and 28 showed that TFTDRKN4(2) and TFTDRKN5(3) methods can be used to solve accurately problems without exact solutions.Figures 14,16,20,and 22 show the effect of trigonometrically fitting approach over the classical TDRKN methods when certain selected ℎ value is considered and from figures it is clearly shown that the global error increases slowly  compared to original TDRKN which rapidly increases along the integration process.

Mathematical Problems in Engineering
In this paper, two exponentially fitted explicit twoderivative RKN methods which are suitable for problems with exponential solutions and also two trigonometrically fitted explicit two-derivative RKN methods suitable for oscillating solutions especially for long-range integration have been derived.From the numerical results, it can be concluded that the new methods are more promising compared to standard TDRKN and with other existing methods in the literature.

Figures 5 -
28 by plotting the graph of the decimal logarithm of the maximum global error against the logarithm number of function evaluations.Observing from the graph plotted in Figures5-12, we noticed that, for all problems, the proposed methods EFTDRKN4(2) and EFTDRKN5(3) are more efficient compared to TDRKN methods of the same order and other existing methods in the literature.