A new local non-integer derivative and its application to optimal control problems

Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.