Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving
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 and respectively, when , the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.
In this study, we consider the numerical solution of the general second-order initial value problems (IVPs) of the following form:where is a continues vector valued functions, and is the dimension of the system. Problems of form (1) occur in different areas of applied science such as quantum mechanics, classical mechanics, celestial mechanics, circuit theory, astrophysics, control theory, and biological sciences (see Dormand , Hairer et al. , and Kristensson ) which attract the interest of several researchers (see [4–11]). Many well-known researchers have constructed Runge-Kutta-Nyström methods (RKN) for solving (1) by using numerous techniques, for instance, exponentially fitted and trigonometrically fitted techniques. Simos  the first author studied exponentially fitted RKN method in the literature. Franco  derived exponentially fitted RKN methods with three and four orders. Van de Vyver  constructed the embedded pair of exponentially fitted RKN methods to solve orbital problems. Yang et al.  derived trigonometrically fitted adapted Runge-Kutta-Nyström methods (ARKN) with perturbed oscillators. Fang et al.  presented embedded pair trigonometrically fitted ARKN methods. Senu et al.  derived zero-dissipative RKN method with minimal phase-lag for solving special second-order ODEs with oscillatory solutions. Moo et al.  constructed two new phase-fitted and amplification-fitted RKN based on Garcia and Hairer methods of orders four and five, respectively. Recently, Fang et al.  proposed a block hybrid trigonometrically fitted RKN methods for solving problem (1). Demba et al.  derived two trigonometrically fitted RKN methods based on Garcia and Hairer methods of orders four and five, respectively, for periodic IVPs. Demba et al.  constructed a pair of embedded trigonometrically fitted RKN method for solving second-order IVPs where the solutions are oscillatory. Ehigie et al.  derived two implicit TDRKN methods via mixed collocation.
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 three-stage 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.
2. Exponentially Fitted TDRKN Methods
In this section, we will determine the conditions of exponentially fitted two-derivative Runge-Kutta-Nyström methods. We suppose that the third derivative for (1) is known. An -stage TDRKN method for (1) is defined by the following formula (see Chen et al. ):where are real numbers. To construct exponentially fitted TDRKN method it is needed at each stage to integrate exactly the function and ; therefore the following four equations are obtained:and four more equations corresponding to and where , a real number and indicates the frequency of the problem and is the step-length of integration. The relations , will be used in the derivation process. The following conditions are obtained: and four equations corresponding to and :
2.1. Exponentially Fitted TDRKN Method of Order Four
Referring to the following fourth-order two-stage method derived by Chen et al. , , , , , , , solving (8) to (11) yields Next, solve (12) to (15) and use the above coefficients to find , , , and : These lead to our new method, an explicit exponentially fitted TDRKN two-stage fourth-order method denoted as EFTDRKN4. The corresponding Taylor series expansion of the solution is given by As , the coefficients of the new method EFTDRKN4 and reduce to the coefficients of the original method TDRKN4. That is to say , and are identical to , and of TDRKN4 method.
2.2. Exponentially Fitted TDRKN Method of Order Five
Referring to the following fifth-order three-stage method derived by Chen et al. , , , , , , , , , , , . Solving (8) to (11) yieldsNext, solve (12) to (15) and use the above coefficients to find , , and : These lead to our new explicit exponentially fitted explicit TDRKN three-stage fifth-order method denoted as EFTDRKN5. The corresponding Taylor series expansion of the solution is given by As , the coefficients of the new method EFTDRKN5 and , reduce to the coefficients of the original method TDRKN5.
3. Trigonometrically Fitted TDRKN Methods
Exponentially fitted methods lead to trigonometrically fitted methods: when replacing with , we obtainand four equations corresponding to and :
3.1. Trigonometrically Fitted TDRKN Method of Order Four
Consider the same coefficients fourth-order two-stage method developed by Chen et al. , as in Section 2.1. By solving (22) to (25) using the Chen coefficients and letting , , , , and , we obtain the following:Next, solve (26) to (29), and usethe above Chen coefficients to find , , and : These lead to our new trigonometrically fitted explicit TDRKN which is called TFTDRKN4 method. The corresponding Taylor series expansion of the solution is given by
3.2. 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) and letting , , , , , , and as free parameters, we obtain Next, solving (26) to (29) and using the above Chen coefficients to find , ,, and yield These lead to our new explicit trigonometrically fitted which is called TFTDRKN5 method. The corresponding Taylor series expansion of the solution is given by