Browsing by Author "El-Zahar, Essam R."

Now showing 1 - 2 of 2

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: 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.