The need for the fractional operators | ||
| Journal of the Egyptian Mathematical Society | ||
| Volume 29, Issue 1, 2021, Pages 1-7 PDF (1.23 M) | ||
| Document Type: Review article | ||
| DOI: 10.1186/s42787-021-00134-7 | ||
| Author | ||
| E. A. Abdel‑Rehim* | ||
| Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt. | ||
| Abstract | ||
| In this review paper, I focus on presenting the reasons of extending the partial diferen‑ tial equations to space-time fractional diferential equations. I believe that extending any partial diferential equations or any system of equations to fractional systems with‑ out giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the frst order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, difusion processes and the continuous time random walk. Some partial diferential equations that are successfully extended to space-timefractional diferential equations are also presented. | ||
| Keywords | ||
| Space-fractional Fokker–Planck operator; Caputo-time-fractional operator; Riesz–Feller space-fractional operator Mittag–Lefer function; Continuous time random walk | ||
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