A new definition of conformable fractional derivative on arbitrary time scales

Rahmat, Mohamad Rafi Segi

doi:10.1186/s13662-019-2294-y

Cited by 19 publications

(14 citation statements)

References 10 publications

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“…Several variants of conformable (fractional) derivative have been defined on time scales (see [22,28,30]). In this section we give the definition of conformable nabla derivative (strictly following [20]) depending on the function G 1-γ (t, a) (as defined in Sect.…”

Section: Conformable Nabla Derivativementioning

confidence: 99%

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

Rahmat

Noorani

2021

Adv Differ Equ

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In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function $\widehat{\mathcal{G}}_{\eta }(t, a)$ G ˆ η ( t , a ) for $t,a\in \mathbb{T}$ t , a ∈ T . The time scale power function takes the form $(t-a)^{\eta }$ ( t − a ) η for $\mathbb{T}=\mathbb{R}$ T = 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 $(t-a)^{(\eta )}$ ( t − a ) ( η ) 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 $\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))$ ▽ a γ 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.

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Section: Conformable Nabla Derivativementioning

confidence: 99%

“…Further development on conformable fractional derivative and its applications on arbitrary time scales can be seen through the articles [24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

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

Rahmat

Noorani

2021

Adv Differ Equ

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“…Theorem 9 For t 0 ∈ I and vector x 0 , the conformable linear state equation (28) with continuous A(t) has the unique and continuously differentiable solution…”

Section: Theorem 8 Consider Fractional Nonhomogeneous Systemmentioning

confidence: 99%

“…They have also established various properties like the product, quotient, and chain rules, and additionally mean value theorem of conformable fractional derivative. Some other new ideas on conformable derivative can be seen in [1,2,9,11,28,30,31,33]. However, the conformable fractional derivative is not considered to be the same as a fractional order derivative, it is a first-order derivative multiplied by an additional simple factor.…”

Section: Introductionmentioning

confidence: 99%

Linear conformable differential system and its controllability

et al. 2020

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This article deals with the sequential conformable linear equations. We have focused on the solution techniques of these equations and particularly on the controllability conditions of the time-invariant system. For the controllability conditions and results, we have defined the conformable controllability Gramian matrix, the conformable fundamental matrix, and the conformable controllability matrix.

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“…This is described and interpreted theoretically as the nonlocality of fractional derivative [9,25]. Although there have been several attempts to devise local versions of fractional derivative, for instance, conformal local derivative [18], local fractional derivative of KG-type [4] (cf. pp.…”

Section: The Case Of Caputomentioning

confidence: 99%

Principal parts of a vector bundle on projective line and the fractional derivative

Husain¹,

Sultana²

2019

Turk J Math

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This work is an exposition on computational aspects of principal parts of a vector bundle on projective line over the field of characteristic zero. Principal parts help determine the possibility of algebraically formalizing infinitesimal-neighborhoods of subschemes inside some ambient scheme. The purpose of this study is to look for the possibility of formalizing the algebraic geometric interpretation of fractional derivative. For the latter, this study follows the approach proposed by Vasily Tarasov. The difference is that Tarasov proposed a geometric interpretation using finiteorder jet bundles from differential geometry. Present study proposes finite-order principal parts of the structure-sheaf of real projective line as its formal algebraic geometric parallel.

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A new definition of conformable fractional derivative on arbitrary time scales

Cited by 19 publications

References 10 publications

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

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

Linear conformable differential system and its controllability

Principal parts of a vector bundle on projective line and the fractional derivative

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