On a new class of fractional difference-sum operators based on discrete Atangana Baleanu sums

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1 On a new class of fractional difference-sum operators ased on discrete Atangana Baleanu sums Thaet Adeljawad 1 and Arran Fernandez 2,3 arxiv: v1 [math.ca] 24 Jan Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Araia 2 Department of Applied Mathematics and Theoretical Physics, University of Camridge, Wilerforce Road, Camridge, CB3 0WA, Camridge, United Kingdom 3 Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, TRNC, Mersin-10, Turey Astract We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag-Leffler ernels. The iteration process depends on the inomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property and hence the new introduced iterated fractional difference-sum operators have this semigroup property as well. 1 Introduction and preliminaries Discrete fractional calculus is an important emerging ranch of analysis [35, 28, 30, 13, 14, 39, 9], which has een very useful in the analysis of discrete systems with non-local effects. The solution of discrete fractional differential equations and discrete oundary value prolems has discovered many applications [16], and so this is an important field to develop in terms of mathematical theory. There are many different types of fractional calculus which can e defined, in oth the continuous and discrete contexts. The point of such an exercise is to discover new ways of modelling various fractional systems, and to create new framewors which can then e used in a numer of applications [18, 21]. It is important to continue developing these new models, from several points of view. In the last two years, some important new definitions of fractional calculus have een formulated with exponential and Mittag-Leffler ernels [22, 12], and their properties have een explored in a numer of papers [20, 19, 11, 25]. Furthermore, discrete fractional calculus has een theoretically developed more y formulating and analysing discrete versions of these fractional operators [8, 4, 7]. The idea of our approach in this article depends on iterating the fractional sums corresponding to fractional operators with discrete Mittag-Leffler ernels, extending and generalising these operators in such a way as to recover certain desirale traits such as a semigroup property. The idea of considering iterations of functional operators to get fractional ones is not new indeed, it is the very asis of fractional calculus itself [36, 37] ut it has recently een explored as a way of adding further fractionalisation to operators which are already considered as fractional [31, 26, 27]. The present wor can e seen as an extension of these projects into the discrete context. The advantage of the discrete operators compared to the existing ones is the same as in any ranch of discrete calculus: it is useful to have definitions in oth discrete and continuous contexts, since modelling different processes in the real world requires oth discrete and continuous models [28, 30, 24, 17, 29]. The structure of this wor is as follows. In the current Section 1, we review some asic concepts aout discrete fractional calculus in the frame of nala difference analysis, including the well-nown Atangana Baleanu model of discrete fractional calculus and its fundamental properties. In Section 2, we define 1

2 our new family of operators and analyse them, proving some essential facts aout them. In Section 3, we consider some fractional difference equations in this new model. Finally, in Section 4 we conclude the paper. 1.1 Nala discrete calculus Definition 1.1. i For any l N and any numer z, the l rising factorial of z is ii For any µ R, the µ rising function is z µ l 1 z l z +i, z i0 The following fact is straightforward to prove: and this means z µ is an increasing function on N 0. Γz +µ, z R\{..., 2, 1,0}, 0 µ 0 2 Γz z µ µz µ 1, 3 Definition 1.2 Nala fractional sums see [1, 2]. Define the operator ρt t 1, which is called the acwards jump. For any function f : N a : {a,a+1,a+2,..} R, the nala fractional sum of order µ > 0 and of left type starting from a is defined y: a µ fz 1 Γµ z sa+1 z ρs µ 1 fs, z N a+1. Similarly, for a function f : N : {, 1, 2,..} R, the nala fractional sum of order µ > 0 and of right type finishing at is defined y: µ fz 1 1 s ρz µ 1 fs Γµ sz 1 1 σs z µ 1 fs, z 1 N. Γµ Lemma 1.1 [2, 5]. Let α > 0, β > 1, h > 0. Then we have the following identities: sz a α t a β Γβ +1 Γβ +1+α t aα+β 4 α t β Γβ +1 Γβ +1+α tα+β 5 From [28, 2], we recall that the left and right nala fractional sums satisfy the following semigroup property: a α a µ fv a α+µ fv, 6 α µ fv α+µ fv, 7 Definition 1.3 Nala discrete Laplace transforms see [6]. The nala discrete Laplace transform K K 0, applied to a function f defined on N 0, is defined y Kfz 1 z t 1 ft. 8 t1 2

3 More generally, for any a, if f is a function on N a, the nala discrete Laplace transform K a is defined y K a fz ta+1 1 z t a 1 ft. 9 Lemma 1.2. For any µ R \ {..., 2, 1,0}, we have the following results on nala discrete Laplace transforms. i Kt µ 1 z Γµ z µ, 1 z < 1, ii Kt µ 1 t z µ 1 Γµ z+ 1 µ, 1 z <. Proof. See [15]. Remar 1.1. We can extend i of Lemma 1.2 to the more general statement that K a t a µ 1 s Γµ s µ. Definition 1.4. [Nala Discrete Mittag-Leffler see [1, 2, 10, 6]] For λ R with λ < 1 and α,β,ρ,v C with Reα > 0, the nala discrete Mittag-Leffler functions with one, two, and three parameters are defined respectively y: E α λ,v E α,β λ,v λ v α Γα +1 ; 10 λ vα+β 1 Γα +β ; 11 E ρ λ,v v ρ α,β!γα +β. 12 Note that we have E α λ,v E α,1 λ,v and E α,β λ,v E 1 λ,v, just as in the continuous case [34]. α,β Proposition 1.1. For any λ,α,β,ρ,v C as in Definition 1.4 and γ C with Reγ > 0, we have the following difference and summation properties of discrete Mittag-Leffler functions. v ta+1 v E α λ,v λe α,α λ,v; v E ρ α,β λ,v Eρ α,β 1 λ,v; E α,β λ,t a E α,β+1 λ,v a; a γ E ρ λ,v a α,β Eρ λ,v a. α,β+γ Proof. The proof is straightforward see [6]. In the proof of the last part, the assertion [5, 1, 2, 28] that a γ t a ν 1 Γν Γν +γ t aγ+ν 1 has een used. Also, notice that the second part is the particular case γ 1 of the last part. The remainder of this section is dedicated to summarising some nown results aout discrete Laplace transforms for Mittag-Leffler functions and functions of convolution type. More details can e found in [10]. 3

4 Definition 1.5 Nala discrete convolutions see [10, 28]. Let a R and consider two functions f,g : N a R. Their nala discrete convolution is f gv v sa+1 gv ρs+afs. 13 Proposition 1.2 [10, 28]. For any a R and functions f,g defined on N a, we have the following convolution property of Laplace transforms in the nala discrete context: K a f gs K a fsk a gs. 14 Lemma 1.3 [10]. Let a R and let f e a function defined on N a. Then Lemma 1.4. [15] For any a R and ν R +, we have K a fts sk a fs fa. 15 K a a ν fs s ν K a fs. Lemma 1.5. [10] Let 0 < α 1, a R, and f e a function defined on N a. Then: i K a E α λ,t az zα 1 z α λ. ii K a E α,α λ,t az 1 z α λ. 1.2 Discrete Atangana Baleanu fractional differences Let us review the asic theory of fractional sums and differences defined using discrete Mittag-Leffler ernels, as presented in [6] ased on the original ideas in [34, 32, 12]. Definition 1.6 [6]. Let α [0,1] and a < in R. For a function f defined on N a, its nala discrete left fractional difference is defined in Caputo type y and in Riemann-Liouville type y Ca α f t Bα Ra α f t Bα t t sa+1 α s fse α,t ρs, 16 t sa+1 α fse α,t ρs. 17 Similarly, for a function f defined on N, its nala discrete right fractional difference is defined in Caputo type y C α f t Bα 1 α s fse α,s ρt. 18 and in Riemann Liouville type y st R α f t Bα 1 α t fse α,s ρt. 19 The notations R and C are used to denote fractional differences of Riemann Liouville and Caputo type respectively. Note that since the discrete Mittag-Leffler ernel 10 converges for λ < 1, and in this case λ α, the ernels in all four of the aove definitions are convergent for 0 < α < 1 2. st 4

5 We now define the fractional sums corresponding to the fractional difference defined in Definition 1.6. Again this is analogous to the definition found in [12] for the continuous case. Definition 1.7. [6] For 0 < α < 1 and a function f defined on N a, the left fractional sum of type is a α f t Bα ft+ α a Bα α f t. 20 Similarly, for a function f defined on N, the right fractional sum of type is α f t Bα ft+ α α f t. 21 Bα Theorem 1.1. For any α 0, 1 2 and any function f defined on N a N, we have the following relations etween the fractional differences of Caputo and Riemann Liouville type and the associated fractional sum. Ra α a α f t ft; a α R a α f t ft; R α α f t ft; Proof. See [6]. α R α f t ft; Ca α f t R a α f t fa Bα E αλ,t a; C α f t R α f t f Bα E αλ, t. The following lemma, the discrete analogue of a result proved for the continuous model in [19], is essential to proceed in confirming our representations. Lemma 1.6 [4]. For any 0 < α < 1 α 2, with λ : and f eing a function defined on N a, we have Ra α f [ ] t Bα ft+ λ a α ft Iterated fractional difference-sum operators 2.1 Definitions In this section, y iterating the fractional sums we shall formulate a new class of fractional difference-sum operators, which has a semigroup property in one of its two parameters. These operators are the discrete analogue of the iterated differintegrals defined in [26]. Consider the left fractional sum defined in 20. If we iterate this operator n times using the inomial 1 5

6 theorem, and mae use of the semigroup property 6 and the fact that a 0 ft ft, then we have: a α [ n ft Bα + α ] n Bα a α ft n n n α Bα n a α ft n ft Bα n n n α t + Bα n t ρs α 1 fs Γα 1 sa+1 [ t n fs δt ρs Bα sa+1 n n n α + Bα n t ρs ], α 1 Γα 1 where δt ρs is the Dirac delta function, namely the function defined on the time scale N y { 0 if t s, δt ρs 1 if t s. 23 More generally, if T is an aritrary time scale, then the Dirac delta function is defined y { 0 if t s, δ T t ρs 1 t ρ T t if t s, 24 where ρ T t is the acward jumping operator [21] on the time scale T. In particular, if T R, we have t ρ R t 0 and hence we otain the classical Dirac delta function. Now, for α [0,1], the aove formulae for iteration of discrete operators can e fractionalised y replacing the index n with a general µ R and replacing the finite sum y an infinite sum lie the fractional inomial theorem. This idea is formalised in the following definition. Definition 2.1. Let α [0,1], µ R, and f e a function defined on N a. The iterated left fractional difference-sum of f with order α,µ, denoted y a α,µ ft, is defined as: µ a α,µ µ α ft Bα µ a α ft [ t µ fs δt ρs+ Bα sa+1 1 µ ] µ α Bα µ t ρs α Γα Since µ can e negative, the left iterated difference of order α, µ can e defined in exactly the same way, i.e.: a α, µ ft µ Bα µ α µ+ a α ft [ t Bα µ fs δt ρs+ sa+1 1 µ ] Bα µ α µ+ Γα t ρsα The iterated right fractional difference-sum and difference operators can e defined in an exactly analogous way to Definition 2.1, namely as follows. 6

7 Definition 2.2. Let α [0,1], µ R, and f e a function defined on N. The iterated right fractional difference-sum of f with order α,µ, denoted y α,µ ft, is defined as: α,µ ft µ µ α Bα µ α ft [ 1 µ fs δs ρt+ Bα st 1 µ ] µ α Bα µ s ρt α Γα Since µ can e negative, the right iterated difference of order α, µ can e defined in exactly the same way, i.e.: α, µ ft µ Bα µ α α µ+ ft [ 1 Bα µ fs δs ρt+ st 1 µ ] Bα µ α µ+ Γα s ρtα Notice that on the time scale N we have s ρt σs t, and therefore the Dirac delta function in the sense of delta time scale analysis δ σs t is the same as δs ρt. Remar 2.1. Below are some special cases of the aove Definitions 2.1 and 2.2, which reflect their appropriateness as definitions of a two-parameter model of fractional calculus. When µ n N, we recover the expression derived aove for iterating the fractional sum n times. In particular, when µ 1 we have When α 0, we recover the function ft itself: When µ 1 we have More generally, for µ n we have For a proof of this, see Theorem 2.1 elow. a α,1 ft a α ft; α,1 ft α ft. a 0,µ ft 0,µ ft ft. In the case α 1 we have the following conventions: a α, 1 ft R a α ft; 29 α, 1 ft R α ft. 30 a α, n ft R a α n ft; 31 α, n ft R α nft. 32 a 1,µ ft a 1 ft µ a µ ft; a 1, µ ft a µ ft. Theorem 2.1. For any a < in R, α 0,1, and n N, the identities 31 and 32 are valid. 7

8 Proof. We prove the result in the special case n 1, i.e. the equations 29 and 30. The general result follows from these two equations together with the semigroup property proved in the next section Theorem 2.3. By symmetry, we consider only 29. Sustituting µ 1 in the representation 26, we find: 1 a α, 1 Bα 1 α ft 1+ a α ft. Since we now that 1 1 and a 0 ft ft, this expression ecomes: [ a α, 1 ft Bα ft+ 1 α a α f ] t. And Lemma 1.6tellsus that the right-handsideofthe final equationisprecisely R a α ft, asrequired. Remar 2.2. For the sae of comparisons in this sequel, we invite the reader to chec [9] aout fractional sums and differences with inomial coefficients. Example 2.1. As an illustrative example, we apply the operators introduced in Definitions 2.1 and 2.2 to some simple functions, as follows: µ a α,µ t a γ 1 µ α Bα µ a α t a γ 1 µ µ α Γγ Bα µ Γγ +α t aγ+α 1 ; µ α,µ t γ 1 µ α Bα µ α t γ 1 µ µ α Bα µ Γγ Γγ +α tγ+α 1. Note that we have used here the following facts [5, Lemma 3.3], [2, Proposition 3.8]: 2.2 Fundamental properties a α t a γ 1 Γγ Γγ +α t aα+γ 1 ; α t γ 1 Γγ Γγ +α tγ+α 1. In this section we prove some important properties of the definition proposed in the previous section, which demonstrate its naturality and usefulness. Theorem 2.2 Nala discrete Laplace transforms in the iterated model. Let α, µ and f e as in Definition 2.1. Then, we have K a a α,µ f z Bα + α µ Bα z α K a fz. 33 Proof. We use the Definition 2.1 of the iterated fractional difference, and the Lemma 1.4 concerning the nala discrete Laplace transform of fractional difference operators. 8

9 K a a α,µ f z µ µ where in the last step the inomial theorem is applied. µ µ α Bα µ K a a α fz µ α Bα µ z α K a fz µ z α α Bα µ K a fz µ K a fz, Bα + α Bα z α Theorem 2.3 The semigroup property. Let α [0,1] and µ,ν R and f e a function defined on N a. Then the left and right iterated fractional difference operators each have the semigroup property in µ. Namely, for any t N a we have and for any t N we have a α,µ a α,ν ft a α,µ+ν ft, 34 α,µ α,ν ft α,µ+ν ft, 35 Proof. Using the fact 6 that the left fractional sums have the semigroup property, and y the help of the identity m µ ν µ+ν, m m we have a α,µ a α,ν ft µ [ µ α ν Bα µ a α n ν n α n Bα ν n0 µ ν n µ+ν n α +n Bα µ+ν a α a nα ft,n m µ ν m µ+ν m α m Bα µ+ν a mα ft m0 µ+ν m µ+ν m α m Bα µ+ν a mα ft a α,µ+ν ft. m0 a nα ft This proves the left semigroup property 34. The proof for 35 is similar, starting from 7 and using the same inomial identity. Theorem 2.4 Integration y parts. Let α [0,1] and µ R and a, R with a modulo 1. For any function f defined on N a and g defined on N, we have the following integration y parts identity: 1 sa+1 gs a α,µ fs 1 sa+1 fs α,µ gs. 36 ] 9

10 Proof. We now from [5, Theorem 4.1] the following integration y parts identity for standard fractional difference-sum operators: 1 sa+1 gs a α fs 1 sa+1 fs α gs. 37 Using respectively Definition 2.1, equation, and Definition 2.2, we have 1 1 µ gs a α,µ µ α fs gs Bα µ a α fs sa+1 sa+1 µ 1 sa+1 1 sa+1 µ α Bα µ µ µ α fs Bα µ µ 1 sa+1 1 sa+1 µ α Bα µ fs α,µ gs. gs a α fs fs α gs α gs Integration y parts identities as in Theorem 2.4 are the main tool used to study discrete variational prolems in the frame of iterated difference-sums [7, 3]. 3 Fractional difference equations and applications Let α 0,1] and µ R +. Consider a general fractional ordinary difference equation of the form 0 α, µ xt Axt+t, A R We search for a series solution, i.e. one of the form xt c s t αs, s0 where we assume the given function can e written as t s t αs. First, we sustitute xt into the left-hand side of 38 and utilise Lemma 1.1 to get 0 α, µ xt i0 m0 s0 µ µ Bα µ µ 0 α c i t iα i0 µ Bα µ c i Γiα+1 Γ +iα+1 ti+α t mα m Bα µ Γmα+1 µ c m µ Γm α+1, 10

11 where we have set m +i in the last line. On the other hand, the right hand side of 38 can e written as [ Ac m + m ]t αm. Now, if we equate the coefficients in these two infinite series, we reach the identity 1 Γmα+1 m c m µ m0 Bα µ α Γm α+1 µ+ Ac m + m, m N Solving for m 0, we have For positive m, the identity 39 will yield c 0 A+ 0 Bα µ. c m A+ m Bα µ 1 c µ m α Bα µ Γm α+1 µ+ Γmα+1Γmα+1 A+ µ. 40 Bα Hence, we otain the following solution: xt A+ t Bα µ m t mα m1 1 c µ m α Bα µ Γm α+1 µ+ Γmα+1 A+ µ. 41 Bα Actually, the coefficients c i, i 1,2,3,... can e calculated recursively from 40. Remar 3.1. Of special interest is the particular case µ 1 of the aove prolem. In this case, the solution representation 41 ecomes xt A+ t Bα m t mα c m 1 α BαΓm α+1, 1+ Γmα+1 m1 1 A+ Bα this eing a solution of the fractional difference equation where t s0 s t αs and c 0 µ 1. R 0 α xt Axt+t, 0 A+ Bα and the coefficients c i, i N, can e determined from 40 with Remar 3.2. It is worth noting that the semigroup property of Theorem 2.3 will e invaluale in the further study of fractional difference equations in the discrete iterated model. There are many classes of difference equations which are easy to solve when we have a semigroup property ut difficult or impossile when we do not [28, 1, 2]. Equipped with a semigroup property, we can easily cancel operators, in order to simplify and solve an equation, y applying new difference-sum operators to oth left and right sides of the equation. 4 Conclusions In this paper, we have introduced a new ind of discrete fractional calculus, whose advantages over existing models can e summarised as follows. Atangana Baleanu fractional calculus, in the continuous case, has discovered many applications in modelling fractional systems with non-local and non-singular dynamics, ehaviour that cannot e modelled using the classical Riemann Liouville ernels. In a short space of time, the formula has estalished itself as a major competitor among the many different approaches to fractional calculus. 11

12 Discrete fractional calculus DFC is a whole different field of research from continuous fractional calculus. Lie the original discrete calculus and difference equations, DFC can e used to study many real-world processes whose ehaviour is too discrete to e well modelled y continuous fractional calculus. In any type of calculus discrete or continuous, integer-order or fractional an important question is whether or not a semigroup property is satisfied. If we apply the operator twice with order α, do we get the same result as y applying it once with order 2α? This fundamental issue has given rise to much deate aout the validity of certain fractional models, and several new models have een proposed purely in order to regain a semigroup property. Our definition gives a unique way of comining the particular structure of the formula, the discrete ehaviour of DFC, and the semigroup property thans to the introduction of a second parameter. Previous literature covers the comination of any two of these three the discrete model, the continuous iterated model, and discrete models with semigroup properties, ut this is the first time that all three have een put together. In this paper, we covered some asic properties and examples of our new fractional difference-sum operator. By analysing the effect on it of the discrete Laplace transform, we also demonstrated how it can e used to solve a certain family of fractional difference equations. Numerical methods have een an important part of the recent development of fractional calculus, including in the Atangana Baleanu model [23, 38, 41, 40]. Discrete calculus is often etter suited to numerical schemes than the continuous case, due to the finite structure of computation there [33]. Thus, we expect that any new progress in discrete fractional calculus of type should have ramifications in numerical analysis, although to explore such ramifications would e eyond the scope of the current paper. Acnowledgements The first author would lie to than Prince Sultan University for funding this wor through research group Nonlinear Analysis Methods in Applied Mathematics NAMAM group numer RG-DES The second author would lie to than the Engineering and Physical Sciences Research Council EPSRC for their support in the form of a research student grant. References [1] T. Adeljawad. Discrete Dyn Nat Soc 2013, , 12 pages. [2] T. Adeljawad. Adv Differ Equ, 2013, 2013:36. [3] T. Adeljawad. International Journal of Mathematics and Computation , pp [4] T. Adeljawad, Q. M. Al-Mdallal. J Comp Appl Math , , doi.org/ , j.cam [5] T. Adeljawad, F. Atici. Astr Appl Anal , Article ID , 13 pages, doi: /2012/ [6] T. Adeljawad, D. Baleanu. Adv Differ Equ :232, DOI /s [7] T. Adeljawad, D. Baleanu. Adv Differ Equ :232, DOI /s [8] T. Adeljawad, D. Baleanu. Reports on Mathematical Physics , pp [9] T. Adeljawad, D. Baleanu, F. Jarad, R. Agarwal. Discrete Dyn Nat Soc 2013, , 6 pages. [10] T. Adeljawad, F. Jarad, D. Baleanu. Adv Differ Equ, 2012/1/72. [11] M. Al-Refai. Electronic Journal of Differential Equations ,

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