The goal of this paper is to expose a new numerical method for solving initial value time-lag of delay differential equations by employing a high order improving formula of Euler method known as third order Euler method. Stability condition is discussed in detail for the proposed technique. Finally some examples are illustrated to verify the validity, efficiency and accuracy of the method.

     The main objective of this paper is to calculate the perturbations of tide effect on LEO's satellites . In order to achieve this goal, the changes in the orbital elements which include the semi major axis (a) eccentricity (e) inclination , right ascension of ascending nodes ( ), and fifth element argument of perigee ( ) must be employed. In the absence of perturbations, these element remain constant. The results show that the effect of tidal perturbation on the orbital elements depends on the inclination of the satellite orbit. The variation in the ratio  decreases with increasing the inclination of satellite, while it increases with increasing the time.

     In this article, we aim to define a universal set consisting of the subscripts of the fuzzy differential equation (5) except the two elements  and , subsets of that universal set are defined according to certain conditions. Then, we use the constructed universal set with its subsets for suggesting an analytical method which facilitates solving fuzzy initial value problems of any order by using the strongly generalized H-differentiability. Also, valid sets with graphs for solutions of fuzzy initial value problems of higher orders are found.

      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

This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method.

The tracking of satellites motion and their path around the earth is important things in the mechanical of satellites motion. Significant parameters for the determination of time entrance and existence of the satellite could be obtained from the shadow of the earth. In the present work the tracking and time determination for entry and exit from earth shadow have been studied. In the present work we built a software for tracking the motion of satellites in orbit around the earth and determine the change of both distance and speed as a function of time. The perturbations effect on the satellite has been neglected from the earth atmosphere drag and the earth gravity and other effects. The equation for calculating the shadow is solved using num

The transition from low Earth orbit 200-1500 (km) to geostationary Earth orbit 42162 (km) was studied in this work by many methods of transfer. The delta-v requirement (Δv), the time of flight (Δt), the mass ratio of propellant consume (Δm/m) and total mass was calculated for many values altitude in the same plane also when the plane is change. The results from work show that (Δv) that required for transfer when the plane of orbit change is large than (Δv) required when the transfer in coplanar maneuvers while the bi-elliptical transfer method need time of transfer longer than a Hohmann transfer method. The most energy efficiency was determined when the transfer in coaxial between elliptical orbits

This research dealt with choosing the best satellite parking orbit and then the transition of the satellite from the low Earth orbit to the geosynchronous orbit (GEO). The aim of this research is to achieve this transition with the highest possible efficiency (lowest possible energy, time, and fuel consumption with highest accuracy) in the case of two different inclination orbits. This requires choosing a suitable primary parking orbit. All of the methods discussed in previous studies are based on two orbits at the same plane, mostly applying the circular orbit as an initial orbit. This transition required the use of the advanced technique of the Hohmann transfer method for the elliptical orbits, as we did in an earlier research, namely

The perturbed equation of motion can be solved by using many numerical methods. Most of these solutions were inaccurate; the fourth order Adams-Bashforth method is a good numerical integration method, which was used in this research to study the variation of orbital elements under atmospheric drag influence.  A satellite in a Low Earth Orbit (LEO), with altitude form perigee = 200 km, was selected during 1300 revolutions (84.23 days) and A_Sat / M_Sat value of 5.1 m²/ 900 kg. The equations of converting state vectors into orbital elements were applied. Also, various orbital elements were evaluated and analyzed. The results showed that, for the semi-major axis, eccentricity and inclination have a secula