Browsing by Author "Abdel-Gawad, Hamdy I."

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Citation Count: Abdel-Gawad, Hamdy I...et al. (2021). "Exact solutions of the fractional time-derivative Fokker–Planck equation: A novel approach", Mathematical Methods in the Applied Sciences.
Exact solutions of the fractional time-derivative Fokker–Planck equation: A novel approach
(2021) Abdel-Gawad, Hamdy I.; Sweilam, Nasser H.; Al-Mekhlafi, Seham M.; Baleanu, Dumitru; 56389
In the present article, an approach to find the exact solution of the fractional Fokker–Planck equation (FFPE) is presented. It is based on transforming it to a system of first-order partial differential equation via Hopf transformation, together with implementing the extended unified method. On the other hand, a theorem provides the reduction of the fractional derivatives to non-autonomous ordinary derivative is given. Thus, the FFPE is reduced to non-autonomous classical ones. Some explicit solutions of the classical, fractional time-derivative Fokker–Planck equation are obtained. It is shown that the solution of the Fokker–Planck equation is bi-Gaussian's, which was not found up to date. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found that the effect of the presence of the fractional derivative prevails that of the fractal derivative. Here, the most interesting result found is that mixed-Gaussian solution is obtained. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning and also in the distribution of loads in networks. Further, varying the order of the fractional time derivatives results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly. © 2021 John Wiley & Sons, Ltd.
Citation Count: Abdel-Gawad, Hamdy I.; Baleanu, Dumitru; Abdel-Gawad, Ahmed H. (2021). "Unification of the different fractional time derivatives: An application to the epidemic-antivirus dynamical system in computer networks," Chaos, Solitons & Fractals, Elsevier, Vol. 142.
Unification of the different fractional time derivatives: An application to the epidemic-antivirus dynamical system in computer networks
(2021) Abdel-Gawad, Hamdy I.; Baleanu, Dumitru; Abdel-Gawad, Ahmed H.; 56389
Different versions of the fractional derivative have been proposed in the literature. One of the objectives of the present work is to unify these versions. Which is done by reducing a fractional derivative to nonautonomous ordinary ones. Thus, fractional ODEs are transformed to non-autonomous ODEs. A second objective is to find the exact solutions of the fractional model equations of the dynamics between the epidemic and antivirus in computer networks. The model considered here, is an extension to the SIR model by accounting for the antivirus dynamics that results to the antidotal state (A), which is abbreviated SIRA. The study is carried here by reducing the fractional SIRA model to non-autonomous ordinary SIRA equations. A novel approach is proposed for solving the reduced system by implementing the extended unified method. Numerical evaluation of the exact solutions of susceptible, infected, recovered and antidotal species, are carried. The cases of Caputo and Caputo–Fabrizio-fractional derivatives are considered. It is observed that, in view of the model considered, the numbers of the suspected and infected computers decrease, with time, to zero in the two case. In both two cases, the number of recovered computers increases rapidly to an asymptotic state, while the variation of the antidotal, against time, is not significant. It is also, remarked that the highest values of SIR correspond to the smallest fractional order in both two cases. We think that the results, obtained here, are consistent with those expected in studying an epidemic-antivirus NLDS.