Nabla Fractional Derivative and Fractional Integral on Time Scales

In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann-Liouville sense. We also introduce the nabla fractional derivative in Gr\"unwald-Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed.


Introduction
Fractional calculus is a very significant branch in Mathematics whose applications are very useful for engineering students and researchers in both pure and applied field. The concept of "fractional calculus" was developed in the early of 17th century when L'Hopital asked Leibnitz the value of 1 2 th order derivative. After that many mathematicians showed their interest on this topic. Initially the theory was developed mainly as a purely theoretical area. However, in the last decade it has been used in various fields such as mechanics, physics, chemistry, control theory, and many more, for instance one can see [1][2][3][4].
The analysis of time scales calculus is a fairly new topic for researchers. Stefen Hilger and his Ph.D. supervisor, Bernd Aulbach, initiated the topic in the year 1988. After that, Hilger published two more paper on this topic [5,6]. The theory was highly raised after the publication of two book on time scales by Martin Bohner and Allan Peterson [7,8]. It combines the traditional areas of continuous and discrete analysis into one theory, which has various applications in discrete and continuous hybrid phenomena, quantum calculus and in various problems of economics [9].
The inception of the idea of combining the time scales calculus and fractional calculus occurred in the Ph.D. dissertation of N.R.O. Bastos in 2012, where the delta (Hilger), and nabla derivative on time scales were discussed in fractional calculus using the tool of Laplace transform on some specific real and discrete time scales [10,11]. After the inception of the topic, a number of papers were published see [12][13][14][15][16][17][18][19]. Recently, D. F. M. Torres introduced a generalized definition of Hilger derivative and integrals in a pure sense of Riemann-Liouville (RL) derivative [20,21]. Many research works have been completed in a conformable delta and nabla fractional derivative and integrals [22][23][24]. arXiv:2112.13083v1 [math.GM] 24 Dec 2021

Motivation of the Article
On the basis of above work, here we are motivated to study the nabla derivative and integral using a Grünwald-Letnikov (GL) fractional derivative approach and then we arrive to the Riemann-Liouville sense. We introduce nabla fractional derivative and integral in unified approach of discrete and continuous time scales. Then, we generalize the definition of nabla fractional derivative and integral in arbitrary time scales and develop certain properties of nabla fractional derivative and fractional integral.
The paper is organized as follows. In Section 3, we review briefly the essentials of time scales, as well as some basic definitions of nabla fractional derivative and integral which helps the readers to recognize easily our main findings. We assume that the readers are familiar with the basic view of time scales calculus and we refer the reader to go through [7,8]. The paper also assess the Riemann-Liouville and Grünwald-Letnikov fractional derivative and integral. Our main findings are given in Section 4 with some preliminaries definition and then we present fractional integral and fractional derivative in an arbitrary time scale T. After that we prove certain important characteristics of fractional derivative and integral. We end with Section 5 of conclusions.

Definition 1 ([7]).
A time scale T is a closed subset of R, with the subspace topology inherited from the stranded topology of R. The backward jump operator ρ : T → T is defined as ρ(t) = sup{s < t : s ∈ T} for t ∈ T and forward jump operator σ(t) = inf{s > t : s ∈ T}. If ρ(t) < t then t is said to be a left scattered and if ρ(t) = t, then we say t is a left dense point of T, if σ(t) > t and σ(t) = t, then we say t is right scattered and right dense, respectively. Again, if T has a right scattered minimum a, then let T κ = T − {a}, or else set T κ = T. Here we consider the backward graininess ν :

Definition 2 ([8]).
A function h : T → R is said to be a nabla differentiable at t ∈ T, if for any ε > 0 there exists a neighborhood V of t, such that If h ∇ (t) exists for all t ∈ T κ then it is called nabla derivative of h. Theorem 1 ([8]). Let us consider a function h : T → R and let t ∈ T κ . Then we have (i) If h is continuous at a left-scattered t, then h is nabla differentiable at t with (ii) If t is left dense, then h is nabla differentiable at t if and only if the limit exists as a finite number. In this case Definition 3 ([8,11]). (Higher order nabla derivative): Assume a function h : T → R, we first define the second order derivative h ∇∇ provided h ∇ is differentiable on Similarly, proceeding up to n th order, here we obtain h ∇ n : T K n → R, where T K n is a time scales which is obtained by removing n right scattered left end point.

Definition 4 ([3]
). The Riemann-Liouville fractional differentiation of random order α is defined in the following manner: Riemann-Liouville derivative of order α ∈ R is given by for α < n < α + 1.

Definition 5 ([2]
). Let α > 0. The Grünwald-Letnikov derivative of fractional order α of a function h is defined by Referring Definition 5 as the Grünwald-Letnikov fractional derivative is quite common in literature (see [25]). Moreover, once a starting point t 0 has been assigned, for practical reason then the following (see [26]) is often preferred, since it can be applied to function not defined (or simply not known) in (−∞, t 0 ).

Nabla Fractional Derivative and Nabla Fractional Integral
Definition 9. For any time scale T, a function h : T → R is said to be nabla fractional differentiable of order µ at t ∈ T κ , where 0 < µ ≤ 1, if for any ε > 0, there exists a neighborhood V of t, such that for all u ∈ V. If for all t ∈ T κ , h holds the Equation (1), then we call h (µ) ∇ (t) the nabla fractional derivative of order µ. Theorem 2. Nabla fractional derivative is not well defined in T, but in T κ .

Proof. Let h (µ)
∇ (t) be defined at a point t on a time scale T, and assume that t / ∈ T κ . Then, t ∈ T \ T κ . From Definition 1, t must be unique which is equal to a, later, for any ε > 0 there Thus for ζ ∈ R and µ ∈ (0, 1] we have Here, Equation (2) is true for each ζ ∈ R, which means for each ζ is the nabla derivative of h of order µ if t / ∈ T κ , which cannot be true, so h (µ) ∇ is well defined only on T κ .

Theorem 3.
For any time scale T, let h : T κ → R. Then, for µ ∈ (0, 1] we have the following: (i) If t is left dense and h is nabla differentiable of order µ at t, then h is continuous at t; (ii) If h is continuous at t and t is left scattered, then h is nabla differentiable at t of order µ with h (µ) exists as a finite number. In this case h (µ) Proof. (i) Given that h is nabla fractional differentiable at t, then for ε > 0 there exists a neighborhood V of t, such that It follows the continuity of h at t.
(ii) Given that h is continuous and t is left scattered, by continuity Hence, there exists a neighborhood V of t, such that From Definition 9, we obtain our result: (iii) Given that t is left dense, then we obtain ρ(t) = t, so there exists a neighborhood V of t, such that (t−u) µ exists as a finite number, say L, and t is left dense. Then, for from which we conclude that h is fractional differentiable of order µ at t and h so left hand side =⇒ right hand side.

Case 2:
If t is left scattered, then ρ(t) < t, and by using Theorem 3 (ii), we obtain The proof is complete.
which completes the proof.
Proof. (i) Let µ ∈ (0, 1]. Given that h and g are nabla differentiable at t ∈ T κ of order µ, for any ε > 0 there exist neighborhoods V 1 and V 2 of t, thus for all u ∈ V 1 also for all u ∈ V 2 By using the Equations (3) and (4), we obtain that ∇ is a nabla differentiable at t ∈ T κ of order µ. (ii) If t is left dense, i.e., ρ(t) = t for t ∈ T κ , then (hg) (µ) Other part of the proof is very similar to this. (iii) Using the above result and Proposition 1, we obtain Since h(t)h(ρ(t)) = 0, so we obtain .
(iv) Using the result of Theorem 4 (ii) and (iii) we obtain the following: .
This completes the proof.
Theorem 5. Let k be a constant, n ∈ N. Then, for 0 < µ ≤ 1, we obtain the following: Proof. (i) Here we prove this result by using the method of induction. If n = 1, then h(t) = t − k hence h  (ρ(t) − k) j (t − k) n−1−j holds for h(t) = (t − k) n . We shall prove the result is true for By using Theorem 4 (ii), we obtain h (µ) Using Theorem 4 (iii), we obtain g (µ) (i) If g(t) = t 2 , then from Theorem 5, we obtain g (µ) By using Theorem 5, we obtain the following results: (ii) If g(t) = t 3 , then g Corollary 1. Nabla fractional derivative in some specific time scales T.
(i) If we consider the real time scale T = R, then all the elements of T are dense. So, by using Theorem 3 (iii), we have that h (µ) , which is similar to the ordinary derivative. (ii) If T = Z, for t ∈ T one has ρ(t) = t − 1 and then ν(t) = t − (t − 1) = 1. Now, by using Theorem 3 (ii), we obtain h (µ) , which is similar as the usual backward operator; (iii) Let T = hZ, where h > 0. Then we obtain ρ(t) = sup{u ∈ T : u < t} = sup{t − nh : for n ∈ N} = t − h and then the function From Definition 3, the second order nabla derivative is In general, the m th derivative for t ∈ hZ and m ∈ N, g (m) where the binomial coefficient ( m r ) is defined as follows: Since the binomial coefficient vanish when r > m, so no contribution in the summation is given from the presence of terms with r > m, the upper limit of the formula can be raised to any value greater than m and hence, the finite summation in this formula can be replaced with the infinite series, i.e., g (m) Letting h tend to zero, then all points of the time scale become dense, and the time scale becomes the continuous time scale. If the value of m is replaced by an arbitrary real number µ ∈ R, µ > 0, and changing the factorial function with a Euler gamma function using the recurrence relation (n − 1)! = Γ(n), then without losing the generality, if we replace m by any arbitrary real number µ ∈ R, then the nabla fractional derivative, from Definition 3 and Theorem 3, is Moreover, once a starting point a assign as nh = t − a for t > a, such that Since for any continuous function g(t) Grünwald-Letnikov derivative and Riemann-Liouville derivative coincide with positive non integer order derivative, so we have where RL a D µ t g(t) denote the Riemann-Liouville fractional derivative defined on time scales, which is most useful in the study of fractional calculus. If µ < 0, then we have i.e., when µ = −µ, then from Equations (6) and (9) we obtain For any ld-continuous function and for µ = −µ, then from Let us take

Now,
lim n→∞ n ∑ r=0 η n,r = lim n→∞ n ∑ r=0 t − a n r t − a n Here, we obtain a condition (see [28]) that, if By using the Equations (10)-(13), we obtain which represents the nabla integral of any arbitrary order µ in a Riemann-Liouville sense. (14) is not the natural one for arbitrary time scales T. For showing this we take an example. If g(t) = t 2 , then from Example 1, g µ ∇ (t) = t + ρ(t), for µ = 1. If the time scale is the continuous time scale T = R, then ρ(t) = t and, hence, from Corollary 1, we find that g (µ)

Remark 1. The definition of nabla fractional integral defined in Equation
But if we take the discrete time scale, we obtain ρ(t) = t − 1, and the nabla derivative on T = Z means the backward difference of t 2 , i.e., ∇(t 2 ) = g(t) − g(t − 1) = 2t − 1. Again, since every ld-continuous function is nabla integrable, so in this case we can claim that t 0 g which is the generalization of the nabla fractional integral defined on the Equation (14), in a Riemann-Liouville sense.

Proposition 3. The nabla fractional integral for any function g defined on
Proof. By using the generalized definition of nabla derivative of fractional order from the Equation (15), we have By using Definition 7, we obtain This completes the proof.
Next definition uses integration as an anti-derivative process.
Definition 10. (Riemann-Liouville fractional derivative on time scales) For t ∈ T and g : T → R, the (left) Riemann-Liouville fractional derivative of order µ ∈ (0, 1] is defined by Remark 2. If T = R, then Definition 10 gives the classical (left) Riemann-Liouville derivative of fractional order µ. Here, we are only studying the derivative in terms of left operators, the analogous right operators are easily acquired by changing the limit of integration.
A different extension to time scales is obtained by using the nabla fractional derivative in terms of Caputo sense, that will be more effective for integer order initial conditions and are more easy to obtain in real world problems [3,11]. Definition 11. (Nabla derivative on time scales in a Caputo sense) For t, t 0 ∈ T, let us assume a finite time scale interval [t 0 , t] ∩ T κ n = [t 0 , t] T κ n . Then, for any g ∈ AC Proof. Let g : T κ → R be a nabla fractional differentiable function. Then, from Definition 10 and Equation (15), we have RL a D α t g(t) = The proof is complete. This concludes the proof.

Conclusions
In this paper, we discussed the nabla fractional derivative on time scales in a unified approach by using Grünwald-Letnikov and Riemann-Liouville derivative, respectively. Then, we have initiated the generalized definition of nabla derivative in fractional order in a pure sense of Riemann-Liouville and Caputo. We claim that a lot of further work can be completed by using this new idea. The aim of formulating the derivative is to solve fractional dynamic equations, stochastic dynamic equations, fuzzy dynamic equations, and one can think to extend the concept in a complex dynamic setting. About applications, it has great prospect in mathematical modeling, for example in epidemiology, anomalous diffusion in magnetic resonance imaging [29], fractal derivatives modeling [30], and consensus problems in time scales on fractional calculus.