A Generalized Definition of the Fractional Derivative with Applications

Abstract

A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function expanded by a Taylor series, we show that $D^{\alpha }{D}^{\beta }f\left(t\right)=D^{\alpha +\beta }f\left(t\right);0<\alpha \le 1;0<\beta \le 1$ . 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 $\left(\text{BPM}\right)$ , enhanced homotopy perturbation method $\left(\text{EHPM}\right)$ , and conformable derivative $\left(\text{CD}\right)$ is also discussed. Our results show that the proposed definition gives a much better accuracy than the well-known 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.