A new embedded phase-fitted modified Runge-Kutta method for the numerical solution of oscillatory problems

Abstract

In this work, a new way for constructing an efficiently modified Runge-Kutta (RK) method to solve first-order ordinary differential equations with oscillatory solutions is provided. The proposed method solves the first-order ODEs by first converting the second order ODEs to an equivalent first-order ODEs. The method of the embedded has algebraic orders five and four. The numerical results of the new method have been compared with those of existing methods and showed that the new method is more efficient.

Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving y′′(x)=fx,y

“…(xii) PFAFRKF4(5): the five-stage fourth-order phasefitted and amplification-fitted modified RK method given in [26].…”

Solving Oscillating Problems Using Modifying Runge-Kutta Methods

“…A fourth-order MRK method of five-stage is given as follows [10]: To construct a modified RK method, we set 𝑑 2 , 𝑑 3 , and 𝑑 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 𝑑 2 , 𝑑 3 and 𝑑 4 as follow: Now, solving equations ( 6), (7), and (8), we get the values of free parameters in terms of 𝑧, and the expressions are too complicated, we use the following Taylor series expressions,…”

“… 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].…”

“…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).…”

Phase‐Fitted and Amplification‐Fitted Higher Order Two‐Derivative Runge‐Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

“…In the last few years, Senu et al [5] constructed zero dissipative explicit RK method for solving second-order ODEs with periodical solutions which has algebraic order three with dissipation of order infinity. Recently, Fawzi et al in their papers [6,7] developed fourth-algebraic-order phase-fitted and amplification-fitted modified RK method and fourth-order seven-stage phase-fitted and amplificationfitted RK methods, respectively.…”