Abstract
In this paper, we made improvement on the conformable fractional derivative. Compared to the original one, the improved conformable fractional derivative can be a better replacement of the classical Riemann-Liouville and Caputo fractional derivative in terms of physical meaning. We also gave the definition of the corresponding fractional integral and illustrated the applications of the improved conformable derivative to fractional differential equations by some examples.
Highlights
The fractional order derivative has always been an interesting research topic in the theory of functional space for many years [1,2,3,4,5,6,7,8,9,10,11]
If we look at the RiemannLiouville and Caputo fractional derivative definition, we have limRLDαð f ÞðtÞ
We can see if the Riemann-Liouville and Caputo fractional derivatives are replaced by conformable derivative, a large error will occur
Summary
The fractional order derivative has always been an interesting research topic in the theory of functional space for many years [1,2,3,4,5,6,7,8,9,10,11]. For α ∈ 1⁄2n − 1, nÞ, the α derivative of f is Both Riemann-Liouville definition and Caputo definition are defined via fractional integrals. The conformable fractional derivative definition is natural and it satisfies most of the properties which the classical integral derivative has such as linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, Rolle’s theorem, and mean value theorem. We can see if the Riemann-Liouville and Caputo fractional derivatives are replaced by conformable derivative, a large error will occur It is a better choice to replace the classical Riemann-Liouville and Caputo fractional derivative with the improved conformable fractional derivative, especially when α ∈ ðn − 1, n and close to n − 1, n = 1, 2, ⋯
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