Solving Oscillating Problems Using Modifying Runge-Kutta Methods

This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge-Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and ampliﬁcation error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientiﬁc literature.


Introduction
This essay dealing with the system of special first-order ordinary differential equation in the following form: Such problems are often observed in various applied sciences, such as quantum chemistry, astronomy, quantum mechanics, electronics, elastics, and chemical physics (see [1,2]). Conventionally equation (1) is solved by using Runge-Kutta (RK) methods or two-step methods [3]. Several authors presented optimized numerical methods based on the phase lag characteristics (such as [4,5,6]). Exact and numerical solutions produce an angle between them is known as Phase-lag, and the distance of the numerical solution from the periodic solution is called amplification error. A five-stage four order Runge-Kutta method with phase-fitted and amplification-fitted proposed by Adel et al. [7]. Hussain et al. [8] proposed an optimized Runge-Kutta method to solve problem (1). The phase lag has the feature to develop more accurate numerical methods. This motivates us to propose the Runge-Kutta methods by using phase lag properties. Based on the previous work, we present a modified approach for the Runge-Kutta methods by combining the annulment of phase lag, phase lag's derivative, and amplification factor. The new MRK methods have been used to solve oscillation problems. The paper is orderly as follows: In Section 2, the phase-lag properties of the MRK method are given. In Section 3, we present how the new methods are derived. In section 4, we offer numerical results to illustrate the efficiency of the new MRK method. Section 5 is dedicated to Conclusions.

Phase Lag Properties of MRK Methods
We are concerned with the numerical solution of the ODEs of the form (1) with an oscillating solution. The general form of explicit MRK method for solving (1) is defined as follows: Constructing the new method depends on phase-lag and dissipation analysis proposed in [9].

Derivation of new MRK methods
This section proceeds to derive the modified RK method by abolishing phase lag, phase lag's derivative, and amplification error.

Five-Stage Fourth-Order Method
A fourth-order MRK method of five-stage is given as follows [10]: To construct a modified RK method, we set 2 , 3 , and 4 as free parameters and the other parameters are the same as in Table 2. Motivated by the approach in [8], we obtain the phase lag, phase lag derivative and the amplification factor, which depend on 2 , 3 and 4 as follow: It can observe that for → 0, the new method reduces to an original RK method given in Dormand [10].

Six-Stage Five-Order Method
Consider the following six-stage modified RK method which can be expressed in Butcher tableau [3]. According to the method in Table 3, the phase lag, phase lag derivative, and amplification factor become  It can demonstrate that for → 0, the new method same as the original RK method given in Butcher [3].

Numerical Results
To evaluate the performance of the new modified Runge-Kutta methods suggested in this paper, we apply them to five oscillatory problems and then compared the numerical results with the several well-known efficient methods. We use the criteria of absolute error to measure the accuracy of the method, which is given by Absolute error = max (| ( ) − |). Where ( ) is the true solution and is the numerical solution. Figures 1-5 demonstrate the efficiency graphs of Log 10 (Max Error) versus step size ℎ. Integration interval is [0, 1000] for all problems with step sizes ℎ = 0.1 2 , = 1,2,3,4 ⁄ . The following numerical methods are used in the comparison.
(i)  MRK4: modified five-stage fourth-order RK method presented in Table 1 and Eqs. (9) in Section 3 in this paper.  RK4D: the classical five-stage RK method of order four given in [10].  RK4PF: the phase-fitted and amplification-fitted RK method given in [7].  RK5S: six-stage RK method of order five given in [11].  RK4MS: the modified four-order RK method given in [9].  RK4B: the five-stage RK method of fourth-order given in [3].

Conclusion
Classical Runge-Kutta methods are adjusted for solving the system of first-order ordinary differential equations whose solutions have a marked periodic style in this paper. The new methods are based on a fourth and fifth algebraic order given in [10] and [3], respectively. The proposed methods have nullified the phase-lag, amplification error, and phase lag's derivative. The results demonstrate the competence and effectiveness of the newly constructed methods, while they are compared to the standard methods and efficient RK methods. It is clear that the new MRK methods are the most accurate, and especially other than existing RK methods in the scientific literature.