Browsing by Author "Ebaid, Abdelhalim"

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Citation Count: El-Zahar, Essam R...et al. (2020). "Absolutely stable difference scheme for a general class of singular perturbation problems", Advances in Difference Equations, vol. 2020, No. 1.
Absolutely stable difference scheme for a general class of singular perturbation problems
(2020) El-Zahar, Essam R.; Alotaibi, A. M.; Ebaid, Abdelhalim; Baleanu, Dumitru; Machado, Jose Tenreiro; Hamed, Y. S.; 56389
This paper presents an absolutely stable noniterative difference scheme for solving a general class of singular perturbation problems having left, right, internal, or twin boundary layers. The original two-point second-order singular perturbation problem is approximated by a first-order delay differential equation with a variable deviating argument. This delay differential equation is transformed into a three-term difference equation that can be solved using the Thomas algorithm. The uniqueness and stability analysis are discussed, showing that the method is absolutely stable. An optimal estimate for the deviating argument is obtained to take advantage of the second-order accuracy of the central finite difference method in addition to the absolute stability property. Several problems having left, right, interior, or twin boundary layers are considered to validate and illustrate the method. The numerical results confirm that the deviating argument can stabilize the unstable discretized differential equation and that the new approach is effective in solving the considered class of singular perturbation problems.
Citation Count: Alharbi, Fahad M.; Baleanu, Dumitru; Ebaid, Abdelhalim, "Physical properties of the projectile motion using the conformable derivative", Chinese Journal of Physics, Vol. 58, pp. 18-28, (2019).
Physical properties of the projectile motion using the conformable derivative
(Elsevier Science BV, 2019) Alharbi, Fahad M.; Baleanu, Dumitru; Ebaid, Abdelhalim; 56389
In this paper, the projectile motion in a resisting medium has been investigated by using the conformable derivative. In order to preserve the dimensionality of the physical quantities, an auxiliary parameter sigma, which has a dimension of seconds, was imposed in the fractional derivative. The converted FDEs have been analytically solved. In the literature, some authors have suggested some relations between the auxiliary parameter sigma and the resistant parameter k. Their procedure is a special case in view of the current results. So, it has been proved in this paper that the dimensions of the physical quantities are always correct without any further assumptions that relate sigma with k. Moreover, it is shown in this paper that the fractional order has no effect neither on the trajectory nor on the range of the projectile, i.e., unlike the corresponding previous results. However, the flight time of the projectile depends on the non-integer order a of the conformable derivative. The impacts of the involved parameters on the projectile properties are discussed through tables and several graphs. The values of the range and the flight time are tabulated for the purpose of comparisons with a previous work in the literature and also with the experimental data. Hence, we give some light on the difference between the conformable derivative and the other definitions when applied on the projectile problem.
Citation Count: El-Zahar, Essam R...et al. (2020). "Re-Evaluating the Classical Falling Body Problem", Mathematics, Vol. 8, No. 4.
Re-Evaluating the Classical Falling Body Problem
(2020) El-Zahar, Essam R.; Ebaid, Abdelhalim; Aljohani, Abdulrahman E.; Machado, Jose Tenreiro; Baleanu, Dumitru; 56389
This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth's rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.