A new definition of conformable fractional derivative on arbitrary time scales

Mohamad Rafi Segi Rahmat

doi:10.1186/s13662-019-2294-y

Mohamad Rafi Segi Rahmat

Open Access

https://doi.org/10.1186/s13662-019-2294-y

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Abstract

In this paper, a new kind of conformable fractional derivative on arbitrary time scales is introduced. The basic conformable derivative rules are proved. We introduce a new definition of exponential functions, and their potential uses in the definition of conformable integrations are explored. Linear first-order conformable differential equations with constant coefficients are investigated, as well as the conformable analogue of Gronwall’s inequality.

Highlights

A new definition (Definition 1.1) of conformable derivative was introduced and explored in [1]
The basic principle of this new definition is the use of proportional-derivative (PD) controller in control theory where u and time t with two tuning parameters have the algorithm d u(t) = κpE(t) + κd dt E(t), where κp and κd are the proportional gain and derivative gain, respectively, and E is the error between the state variable and the process variable
2 Conformable delta derivative we introduce the generalized conformable derivative on arbitrary time scales T which includes Definition 1.1

Summary

Introduction

A new definition (Definition 1.1) of conformable derivative was introduced and explored in [1]. The basic principle of this new definition is the use of proportional-derivative (PD) controller in control theory where u and time t with two tuning parameters have the algorithm d u(t) = κpE(t) + κd dt E(t), where κp and κd are the proportional gain and derivative gain, respectively, and E is the error between the state variable and the process variable. Definition 1.1 ([1]) Let α ∈ [0, 1], and let the functions κ0, κ1 : [0, 1] × R → R+0 (where R+0 = [0, ∞)) be continuous such that lim α→0+.

Segi Rahmat Advances in Difference Equations

It follows that