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 α D β f t = D α + β f t ; 0 < α ≤ 1 ; 0 < β ≤ 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 CD 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.
Highlights
Fractional calculus theory is a natural extension of the ordinary derivative which has become an attractive topic of research due to its applications in various fields of science and engineering. e integral inequalities in fractional models play an important role in different fields
generalized fractional derivative (GFD) has been suggested to provide more advantages than other classical Caputo and Riemann–Liouville definitions such as the derivative of two functions, the derivative of the quotient of two functions, Rolle’s theorem, and the mean value theorem which have been satisfied in the GFD. e present definition satisfies DαDβf(t) Dα+βf(t) for a differentiable function f(t) expanded by Taylor series. e fractional integral is introduced
Some fractional differential equations can be solved analytically in a simple way with the help of our proposed definition which exactly agrees with the classical Caputo and Riemann–Liouville derivatives’ results
Summary
Fractional calculus theory is a natural extension of the ordinary derivative which has become an attractive topic of research due to its applications in various fields of science and engineering. e integral inequalities in fractional models play an important role in different fields. Mathematical Problems in Engineering function is a constant at the origin, its fractional derivation has a singularity at the origin for instant exponential and Mittag-Leffler functions. Due to these drawbacks, the applicability range of Riemann–Liouville fractional derivatives is limited. Caputo derivatives are defined only for differentiable functions, while the functions that do not have first-order derivative may have fractional derivatives of all orders less than one in the Riemann–Liouville sense (see [13]). E conformable derivative satisfies some important properties that cannot be satisfied in Riemann–Liouville and Caputo definitions.
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