Abstract
Recently, a definition of fractional which refers to classical calculus form called conformable fractional calculus has been introduced. The main idea of the concept of conformable fractional calculus is how to determine the derivative and integral with fractional order either rational numbers or real numbers. One of the most popular definitions of conformable fractional calculus is defined by Katugampola which is used in this study. This definition satisfies in some respects of classical calculus involved conformable fractional derivative and conformable fractional integral. In the branch of conformable fractional derivatives, some of the additional results such as analysis of fractional derivative in quotient property, product property and Rolle theorem are given. An application on classical calculus such as determining monotonicity of function is also given. Then, in the case of fractional integral, this definition showed that the fractional derivative and the fractional integral are inverses of each other. Some of the classical integral properties are also satisfied on conformable fractional integral. Additionally, this study also has shown that fractional integral acts as a limit of a sum. After that, comparison properties on fractional integral are provided. Finally, the mean value theorem and the second mean value theorem are also applicable for fractional integral.
Highlights
Today various types of fractional calculus have been proposed by many researchers
Most of the types of fractional calculus definitions that have been introduced cannot be used for classical properties such as product rules, quotient rules, chain rules, Rolle theorems, and mean value theorems
The definition called conformable fractional calculus is the definition of fractional derivative and integral with ∈ (0,1) order and it satisfies classical properties mentioned above
Summary
The most popular definition is given by the Riemann-Liouville, Caputo, Grünwald Letnikov, Hadamard definition. The information about their definition can be found in [12, 14, 15]. The concept of conformable fractional calculus has gained relevance, namely because they kept some of the properties of ordinary derivatives. Even more, this new subject has been important topics to discuss because there are several applications about this topic [5, 16].
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