Properties of Karcı’s Fractional Order Derivative

The derivative concept was defined by Newton and Leipzig. After these scientific, there are many approaches about the order of derivative, since derivative defined by Newton and Leipzig considered as order of 1. Many scientists such as Caputo, Riemann, etc. defined the fractional order derivative. Karcı is one of them who defined fractional order derivative. KKKK(tt) αα was defined by Karcı, and it is not a linear derivative operator; it is a non-linear derivative operator. In this paper, we verified the most important properties of KKKK(tt) αα . KKKK(tt) αα has got an α parameter and this parameter can be any complex number. The properties of KKKK(tt) αα , which are derivative of product, derivative of quotient, the chain rule, the relationship between KKKK(tt) αα and complex numbers, etc., were verified in this paper. The most of these properties were not satisfied by other definitions for fractional order derivatives such as Caputo, Riemann-Lioville, Euler, etc. Khallil and his friends also defined fractional order derivative in a special case. This derivative satisfies these properties for special functions; in general, this definition also does not satisfy these properties.


Introduction
The fractional calculus (variational calculus) is a three centuries old concept and one of the branch of the fractional calculus is fractional order derivatives. The fractional order derivative concept was defined by many scientists such as Euler, Caputo, Riemman-Lioville, etc. (Das, 2011). There is an idea such that the fractional calculus may depict the behaviours of nature almost in real behaviours of nature (Das, 2011).
Riemann-Liouville and Caputo do not satisfy the quotient derivative property such as Riemann-Liouville and Caputo derivatives do not satisfy the chain rule. The Caputo definition assumes that the function f is differentiable.
Khallil and his friends gave a new definition for fractional order derivative for sake of satisfying the mentioned problems of fractional order derivatives (Khallil et al, 2014). In order to get rid of these deficiencies Khallil and his friends defined fractional order derivative as follow. In this study, it was focused on the shortcomings and wrong points involved in the methods of Euler, Riemann-Liouville and Caputo for FODs. Especially, the FODs of constant and identity functions will be obtained for Euler, Riemann-Liouville and Caputo methods. Euler and Riemann-Liouville methods were yielded shortcomings and errors in results for constant functions, and on contrary, Caputo method was yielded in correct result for constant function. All methods have not provided accurate results for identity function. We studied on fractional order derivatives which defined by Karcı, since that definition satisfies all conditions for general functions. Assume that f(t) is a differentiable function and fractional order derivative (defined by Karcı) is denoted by K ∂ α .

Karcı's Fractional Order Derivatives and Its Properties
Euler and Riemann-Liouville derivatives of f(x)=cx 0 : (summarised from Karcı, 2013b) These three methods for fractional order derivatives are mostly used methods. First of all, the results of Euler and Riemann-Liouville methods for f(x)=cx 0 are illustrated in Eq.7 and Eq.8 where c is a constant. For n=1 and α=1/4 Riemann-Liouville method: n=1 and α=2/3 The obtained result is inconsistent, since the result is a function of x. However, initial function is a constant function and its derivative is zero, since there is no change in the dependent variable.
The Euler and Riemann-Liouville methods do not work for constant functions as seen in Eq.7 and Eq.8.
Caputo derivative of f(x)=x: It can be seen from definition, the ratio of the change in dependent variable over to the change in independent is always 1 (one) for identity function. In this case, the derivative must be 1 in any fractional order derivative. However, fractional order derivatives of identity function with respect to Caputo method is different from 1. This means that all methods yielded inconsistent results. Due to this case, there is a need to redefine the fractional order derivative as in Karcı definition (Definition 1).

Definition1
: Assume that f(t):R→R is a function, α∈R and L(.) be a L'Hospital process. The K The fractional order derivative was derived due to the deficiencies of definitions for fractional order derivatives. The Assume that f(t)=c and c∈R, c is a constant (summarised from Karcı, 2013a;Karcı, 2013b).
Assume that f(t), g(t):R→R are continuous functions, α∈R and h(t)=f(t)g(t). The fractional order derivative of h(t) is as follows (summarised from Karcı, 2015c).
Assume that f(t) and g(t) are real functions and, so, h(t) is also a real function. The fractional order derivative of h(t) can be obtained as follow. ) ( f(t) and g(t) are real functions, so, h(t) is also a real function. The fractional order derivative of h(t) is The chain rule for fractional order derivative with respect to Definition 1 is (summarised from Karcı, 2015d) dt where α∈R, y=f(u), and u=g(t). y and u are not absolutely independent variables; however, t is an absolutely independent variable. The derivative means that the derivative is the response of dependent variable to changes in independent variables. So, the change in y with respect to u is not absolutely derivative, since u is not independent variable. The change in y with respect to t through u must be taken in care. So, The definition for fractional order derivative with respect to Definition 1 is as follow: The chain rule result is Assume that In order to prove this assumption, the left-hand side and right hand side must be obtained. The left hand side is The last equation verifies that all orders of derivative for chain rule must be equal for the commutative property of ( ) Assume that f(t):R→R, and α1, α2∈R. K