This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions  and   for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency  is used. The novel method is more accurate than the conventional Runge-Ku

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

  In this paper we shall generalize fifth explicit Runge-Kutta Feldberg(ERKF(5)) and Continuous explicit Runge-Kutta (CERK) method using shooting method to solve second order boundary value problem  which can be reduced to order one.These methods we shall call them as shooting Continuous Explicit Runge-Kutta method, the results are computed using matlab program.

In this paper Volterra Runge-Kutta methods which include: method of order two and four will be applied to general nonlinear Volterra integral equations of the second kind. Moreover we study the convergent of the algorithms of Volterra Runge-Kutta methods. Finally, programs for each method are written in MATLAB language and a comparison between the two types has been made depending on the least square errors.

     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.

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.

      This paper presents an alternative method for developing effective embedded optimized Runge-Kutta (RK) algorithms to solve oscillatory problems numerically.   The embedded scheme approach has algebraic orders of 5 and 4. By transforming second-order ordinary differential equations (ODEs) into their first-order counterpart, the suggested approach solves first-order ODEs. The amplification error, phase-lag, and first derivative of the phase-lag are all nil in the embedded pair. The alternative method’s absolute stability is demonstrated. The numerical tests are conducted to demonstrate the effectiveness of the developed approach in comparison to other RK approaches. The alternative approach outperforms the current RK methods

The derivation of 5^th order diagonal implicit type Runge Kutta methods (DITRKM5) for solving 3^rd special order ordinary differential equations (ODEs) is introduced in the present study. The DITRKM5 techniques are the name of the approach. This approach has three equivalent non-zero diagonal elements. To investigate the current study, a variety of tests for five various initial value problems (IVPs) with different step sizes h were implemented. Then, a comparison was made with the methods indicated in the other literature of the implicit RK techniques. The numerical techniques are elucidated as the qualification regarding the efficiency and number of function evaluations compared with another literature of the implic