A new conformable nabla derivative and its application on arbitrary time scales

Mohamad Rafi Segi Rahmat,M Salmi M Noorani

doi:10.1186/s13662-021-03385-x

Mohamad Rafi Segi Rahmat, M Salmi M Noorani

Open Access

https://doi.org/10.1186/s13662-021-03385-x

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Abstract

In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function widehat{mathcal{G}}_{eta }(t, a) for t,ain mathbb{T}. The time scale power function takes the form (t-a)^{eta } for mathbb{T}=mathbb{R} which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form (t-a)^{(eta )} which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form bigtriangledown _{a}^{gamma }y(t)=y(t, f(t)), and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.

Highlights

Fractional calculus becomes an important area of research in mathematical analysis and applications
We show that the conformable exponential function is the unique solution to the given dynamic equation
7 Concluding remarks In this paper we propose a new conformable nabla derivative and integral on arbitrary time scales which generalize the conformable fractional derivative and integral introduced in [5]

Summary

Introduction

Fractional calculus becomes an important area of research in mathematical analysis and applications. In [23], the authors have studied a version of the nabla conformable fractional derivative on arbitrary time scales. For a function f : T → R, the nabla conformable fractional derivative, T∇,αf (t) ∈ R of order α ∈ (iii) If f is left-dense, f is nabla differentiable at t if and only if the limit f (t) – f (s) t–s exists as a finite number In this case f ∇ (t) = f (t) – f (s) .

Conformable nabla derivative

Conformal dynamic equations and inequalities