Phase Fitted And Amplification Fitted Of Runge-Kutta-Fehlberg Method Of Order 4(5) For Solving Oscillatory Problems

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.


Introduction:
Many efforts have been carried out for the resolution of second order ordinary differential equations (ODEs) with oscillatory solutions. Special second order ordinary differential equations can be written as follows: This kind of equations appear in numerous scientific fields like physical chemistry, quantitative chemistry, theoretical physics, and molecular physics. Differential equations of oscillatory in nature cannot be solved efficiently using traditional methods. These problems need to be integrated over a period of the oscillation, so oscillatory problem require larger steps and can be extended over a long-time step.
Several researchers such (1), (2) and (3) have proposed an application of phase-lag to solve oscillatory problems. Many of projects are also concentrated with methods containing high dissipative order. In their paper, Nazari and Mohammadian (2) analyzed and introduced highorder low-dissipation low-dispersion diagonally implicit Runge-Kutta schemes for solving oscillatory problems. Van  Another work related to phase fitting can be seen in Hussain et al.  (10). The objective of the phase-fitted method is to develop a method that has variable coefficients, which rely on the produce of hesitation "v" and step-size "h". The creative method presides for the original method as "v" approaches to zero.
In this article basing on Runge-Kutta-Fehlberg method of order 4(5); we develop phase-fitted and amplification fitted method for solving problems with oscillatory solutions. The prospective method is utilized to solve the second order ODEs that are oscillatory in nature. To do that the equations of second order are first lowered to first-order ODEs system.

Derivation of the Method
The s-stage Runge-Kutta-Fehlberg method can be defined: This method is applied for the approximate computation of 1  n y when n y is recognized.
Also, the method above can be shown using Butcher Table 1: .
The coefficients An appropriate style to obtain an assured algebraic arrangement is to fulfill equations number obtained from tree theory (11). Those equations can be exhibited during the prospective method structure.
The phase-lag assessment of Runge-Kutta methods is dependent on trial equation: (3) is applied to equation (5), we get the following:

Definition:
In s-stage Runge-Kutta-Fehlberg method, defined in equations (2) and (3), the equations: are known as the dispersion error and the dissipative error respectively (5). If then the method becomes the dispersive of order q and the dissipative of order r.
The Runge-Kutta-Fehlberg method of 4(5) order is considered, given in following Table 2: To improve the proposed method, we put 4 b and 5 b as free parameters while the other parameters are the same as the current method. We intend to apply the phase-lag (PL) and the dissipation (DS) of the current method above to be

By using Taylor expansion series
, ω denotes the prevailing hesitance and h indicates the step size.

The Test Problems and The Numerical Results
The proposed method will be applied to some problems with oscillatory solutions and the results are compared with the Runge-Kutta-Fehlberg method of order 4(5).
These problems are first decreased to first-order ODEs system. For the following test problems, the interval of integration is [0, 100].
The Tables 3,4 The following notations are used: TOL the tolerance chosen

Conclusion:
The number of function evaluations for the prospective method with phase-fitting and amplification-fitting is less than the number for the standard method. Also, the maximum errors of the potential method are lower than the present method depending on the tolerance chosen.