Classical Properties on Conformable Fractional Calculus

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.


Introduction
Today various types of fractional calculus have been proposed by many researchers. 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]. 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. Therefore, Khalil et al. [11] introduced a new modest idea. 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. Moreover, there are several researchers introduced their definition conformable fractional calculus in the other form [1][2][3]10]. Recently, 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]. Also, further results about this subject were developed by [6]. The aim of this paper is to provide additional results of conformable fractional calculus based on the definition introduced by Katugampola [10]. There are several properties of conformable fractional calculus that well functioned, like classical calculus involving fractional derivative and fractional integral, such as properties to determine fractional derivatives, definite fractional integral as the limit of a sum and comparison properties of fractional integrals are also developed. Aditionally, some applications are given.

Conformable Fractional Derivative
Katugampola [10] introduced natural definition of fractional derivative definition which satisfies classical derivative properties.
The following theorem is an important result to prove the next consequences.

Corollary of Product Property
Let ∈ (0,1] and , 4 be − differentiable at a point Then, the function 4 satisfies the conditions of the fractional Rolle's theorem. Hence there exists 5 ∈ ( , ), such that * $ = 1, the result follows.

Conformable Fractional Integral
The conformable fractional integral is discussed as follows.

Definition of Conformable Fractional Integral
Let ≥ 0, and ∈ (0,1). Also, let be a continuous function such that E exists. Then If the Riemann improper integral exists. This following theorem explains that − fractional derivative and −fractional integral are inverse of each other as given in the next result.
Proof. Since is continuous, then E ( ) is certainly differentiable. Using theorem 2.

Conformable Fractional Integral of Conformable Fractional Derivative
Let : ( , ) → I be − differentiable and 0 < ≤ 1 . The following a definition of integration as a limit of a sum is provided. This definition has a number of benefits which are reviewed below.

Conformable Fractional Integral as a Limit of a Sum
If is a function defined for < ≤ , Then the definite fractional integral of from to be b is Where Δ = ( − )/R and N = + SΔ .

Conformable Fractional Integral Properties
If ≥ 0, ∈ (0,1) and , 4: [ , ] → ℝ be a continuous function. Then Proof. It is easy to proof property (i). Since it is known that integral of constant functions is multiplication 5 with integral integral of power function. Property (ii) is proved using definition 3.4 as the following.

Comparison Properties of Conformable Fractional Integral
Let  As it is known, the left side is surely the definition of the conformable fractional integral. Therefore the following is true.
Finally, property (iv) can be proven by using property (ii) on theorem 3.6.

Mean Value Theorem for Fractional Integral
If  and [E ( )] = ( ). Now, from the theorem 2.7 it can be stated that there is a number 5 such that < 5 < and However, it is known that

Second Mean Value Theorem for Fractional Integral
Let and 4 be functions satisfying the following

Conclusion
In this note, further properties of conformable fractional calculus involving fractional derivative and integral are provided. This definition satisfies several properties and applications of classical calculus such as the rules to determine conformable fractional derivative and fractional integral, mean value theorem and comparison properties of conformable fractional integral. Some applications are also given.