A generalized definition of the fractional derivative with applications

• Main Authors: Abu-Shady, M., Kaabar, Mohammed K. A.
• Format: Article
• Published: Mathematical Problems in Engineering 2021
• Subjects: QA Mathematics
• Online Access: http://eprints.um.edu.my/35329/

A generalized fractional derivative (GFD) definition is proposed in this work Fora differentiable function expanded by a Taylor series, we show that (DD beta)-D-alpha f (t) = D alpha+beta f (t); 0 < alpha <= 1; 0 < beta <= 1. GFD is applied for some functions to investigate that the GFD coincides with the results from Caputo and Riemann-Liouville fractional derivatives. The solutions of the Riccati fractional differential equation are obtained via the GFD. A comparison with the Bernstein polynomial method (BPM), enhanced homotopy perturbation method (EHPM), and conformable derivative (Cl)) is also discus sal. Our results show that the proposed definition gives a much better accuracy than the wellknown definition of the conformable derivative. Therefore, GFD has advantages in comparison with other related definitions. This work provides a new path for a simple tool for obtaining analytical solutions of many problems in the context of fractional calculus.