A NEW IMPROVED RUNGE-KUTTA FORMULA FOR DIRECTLY SOLVING

This paper deals with the derivation of an explicit two-stage thirdorder Improved Runge-Kutta Nyström (IRKNG) method for directly solving general second order ordinary differential equations (ODE). This method is twostep and the number of functions to be evaluated per step is less via comparsion to the existing RK methods. Numerical outcomes are offered to show the validity and competency of the newly IRKNG method as comparison with the general RKN and RK techniques. AMS Subject Classification: 65L05, 65l06


Introduction
This paper is devoted to the construction of numerical method for solving general second order ordinary differential equations (ODEs) of the form This problem often occurs in several fields of engineering and applied science,

The formulation of the newly IRKNG methods
Motivated by the Rabiei approach [13] to derive the algebraic order conditions for the IRKN formula, we extended the IRKN method to solve problem (1).
The following form of IRKNG formula is taken for solving problem (1) where and for 2 ≤ i ≤ s, Inserting (8) into (6) as well as (9) into (7), we get By using the row condition c i = i−1 j=1 a ij for Runge-Kutta method and after some algebraic manipulation, equations (10) and (11) become whereā ij = s k=1 a ik a kj , for i = 2, 3, . . . , s. j = 1, 2, . . . , i − 1. By inserting equations (4), (8) and (9) into equation (2), we obtain Substituting the following algebraic order conditions given in [14] into equation we obtain where d i = s j=1 b i a ij , i = 2, . . . , s. Replacing u i with q i and u − i with q −i in formulas (3) and (15), also replacing u 1 with q 1 and u −1 with q −1 in formula (5). Therefore, the general s-stage IRKNG method for solving (1) can be presented as follows: where IRKNG method (16)-(21) is expressed using the following Butcher table (see Table 1).

Order conditions for IRKNG method
Taylor series expansion is used to expand the IRKNG method (16)-(21) to get their parameters. After implementing some algebraic simplifications this extension equal to the real solution. Consequently, a nonlinear equations (called the order conditions) have found using Maple algebra package.
The Taylor series expansion of z(t n + h) and z ′ (t n + h) for the method order p are written respectively as follows: Therefore, the algebraic order conditions of s-stage IRKNG formula up to fourth order can be given below.
For z: order three: order four: For z ′ : order three: order four:

Construction of the newly IRKNG method
We proceed to derive two-stage IRKNG method of order three, denoted as IRKNG3. The following simplifying conditions are used From (35) we have The order conditions of IRKNG3 method must satisfy the following equations Solving equations (38) to (40) simultaneously yields a solution depend on c 2 as follows: The global truncation error for the fourth order IRKNG3 formula is given by where τ (4) and τ ′(4) are the truncation error of z and z ′ , respectively. Hence, we have By using the Minimize command in optimization package in Maple software, we obtain c 2 = 3 5 , which gives Finally, the parameters of IRKNG method of two-stage third-order indicated by IRKNG3 can be written as follows (see Table 2).

Problems tested
Some problems are solved to show the efficiency of IRKNG method in this section. We are comparing the numerical result with the currently RKN and RK formulas. In the comparison we have chosen the following methods: • IRKNG3: the new two-stage IRKNG method of third-order constructed in this paper.
• RKN3C: the RKN method of third-order with three-stage given by Chawla et al. [6].
• RK3B: the RK method of third-order with three-stage given by Butcher [4].
• IRK3F: the third order IRK method with three-stage derived by Rabiei and Ismail [14].
The exact solution is The exact solution is  The exact solution is  From Figures 1 to 4, we observed that IRKNG3 performs better efficiency than the existing RKN3C, RK3B and IRK3F methods of the same order.

Conclusion
A new third order two stage IRKNG3 method for direct solution of general second-order ODEs has constructed. The order conditions for IRKNG3 method are derived by using the approach of Taylor series expansion. From Numerical results given in all figures, we observed that the IRKNG3 method is better than the existing methods in terms of maximum error and number of function evaluations.