# The Generalized Fractional Proportional Delta Operator and New Generalized Transforms in Discrete Fractional Calculus

Authors: J. Leo Amalraj; M. Maria Susai Manuel; M. Meganathan; M. Syed Ali
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this research work, the aim is to develop the fractional proportional delta operator and present the generalized discrete Laplace transform and its convolution with the newly introduced fractional proportional delta operator. Moreover, this transform is a connection between Sumudu and Laplace transforms, which yields several applications in pure and applied science. The research work also investigates the fractional proportional differences and its sum on Riemann–Liouville and Mittag–Leffler functions. As an application of this research is to find new results and properties of fractional Laplace transform, the comparison of the existing results with this research work is also done. Moreover, we used the two types of solutions, namely, closed and summation forms in Laplace transform and verified with numerical results.


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## Body

## 1. Introduction

Nowadays, the concept of fractional calculus is a tool in mathematical modelling that mainly focuses on differentiation and integration of fractional order. This theory brings union among the concepts of positive integer order integration and differentiation [1–3]. In 1695, L’Hopital’s letter has raised a relevant question, namely, “What does ∂mfx/∂xm mean if m=1/2?” This is considered to be the initiation of some other related questions that have been raised by famous mathematicians such as Abel, Erdelyi, Fourier, Grunwald, Laplace, Letnikov, Levy, Liouville, Marchand, Reiszand, and Riemann during the 19th century. These questions are resulted by the creation of fractional calculus.Even though, the authors in [3, 4] have considered Dαf in the field of fractional calculus, discrete fractional calculus received over the years. In 1956, Kuttner [5] mentioned the nth order Δ on a sequence an as(1)Δnan=∑r=0∞−n−1+rmar+n.Further, in 1974, Diaz and Osler [6] have taken the fractional order of delta on fx as(2)Δνfx=∑k=0∞−1kαkfx+α+k.In 1989, Miller and Ross [2] introduced the discrete analogue of the Riemann–Liouville fractional derivative and proved some properties of the fractional difference operator Δ−ν defined by(3)Δ−νft=1Γν∑s=νt−νt−σsν−1fs,σs=s+1.Additionally, in 2009, George [7] defined the Caputo discrete fractional difference as Δνft=Δ−m−νΔmft. Recently, Bastos et al. developed the theory on h-sum and h-difference operators in discrete fractional calculus. For more recent applications of fractional Laplace transform, one can refer [8–11]. These concepts are very well applied in discrete fractional transforms in the field of fractional calculus [12]. As an application of fractional operators in RL and Mittag-Leffler, the readers can refer [13–16].


## 2. The Delta Fractional Sums and Differences

We give here the necessary preliminaries and notations which will be applied for further discussion.Definition 1.
(see [17]). Letℓ>0andu:ℝ⟶ℝ. The forwardℓ−delta operator onuis(4)Δℓuℓℤ=uℓℤ+ℓ−uℓℤ,and the nabla delta operator on u is(5)∇ℓuℓℤ=uℓℤ−uℓℤ−ℓ.
Forℓ=1, we get Δuk=uk+1−uk and ∇uk=uk−uk−1.
Here we takeσℓℓℤ=ℓℤ+ℓ and ρℓ as ρℓℓℤ=ℓℤ−ℓ. For a,b∈R and ℓ>0, we assume the notations ℕa,ℓ=a,a+ℓ,a+2ℓ,…, and b,ℓℕ=b,b−ℓ,b−2ℓ,… and k takes the value ℓℤ, where ℤ is an integer.Definition 2.
(see [17]). Consider the two functionsu,v:ℝ⟶ℝandΔℓvk=uk. Then, we have(6)∑r=1muk−rℓ=vk−vk−mℓ,m∈ℤ+.
Also, we have(7)Δℓ−1uk=∑r=0∞uk+rℓ.Definition 3.
(see [18]). Letℓ>0andν∈−∞,∞. Ifk/ℓ+1−ν∉−ℕ0, the falling factorial is given by(8)kℓν=ℓνΓk/ℓ+1Γk/ℓ+1−ν.
Forℓ=1, we get kν=Γk+1/Γk+1−ν. The division by a pole yields zero.Remark 1.
We get the following identities by applying Definition 1:(9)iΔℓkℓμ=μkℓμ−1,iiΔℓ−1kℓμ=kℓμ+1μ+1.
The difference equationΔℓvk=uk has two kinds of solutions mainly, the closed-form solution of the form vk=Δℓ−1uk and the summation form solution given by vk=∑i=a/ℓk/ℓ−1uiℓ. In this work, we verify whether the findings are equal numerically in both kinds of solutions. For getting the closed-form solution for the product of two function, we need the following discrete version of integration by parts.Lemma 1.
(see [19]). Letℓ>0andu,wbe the real-valued bounded functions. Then,(10)iΔℓ−1ukvk=ukΔℓ−1vk−Δℓ−1Δℓ−1vσℓkΔℓuk,(11)iiΔℓ−1vσℓkΔℓuk=ukvk−ukΔℓvk.
In view [1], (10) and (11) are the discrete versions of the following versions of integration by parts.Definition 4.
(integration by parts).Letu,v:ℕa,ℓ⟶Rbe two functions andb,c∈ℕa,ℓ,b<c. Then,(12)i∫bcukΔℓvkΔℓk=ukvk|bc−∫bcvσℓkΔℓukΔℓk,ii∫bcσℓkΔℓvkΔℓk=ukvk|bc−∫bcvkΔℓukΔℓk.
In caseℓ=1, we obtain the existing integration by parts as in [1].Lemma 2.
Letℓ>0,s∈Randℓs≠−1. Then,(13)aΔℓ−111+ℓsk−a/ℓ+1=1/1+ℓsk−a/ℓ+1−ℓs/1+ℓs.Proof.
By takinguk=1/1+ℓsk−a/ℓ+1 in (4) and taking aΔℓ−1 on both sides, we get (14).Corollary 1.
Letℓ>0,s∈R, andℓs≠−1. Then,(14)1/1+ℓsk−mℓ−a/ℓs−1/1+ℓsk−a/ℓs=∑r=1m11+ℓsk−rℓ−a/ℓ+1.Proof.
By applying (6) in (14), we get the proof of (15).Example 1.
For the particular valuesa=2,ℓ=3,s=4,k=8, and m=3, Corollary 1 is verified by MATLAB numerically and the final value on both sides is found to be 3.2485.Lemma 3.
Lets∈T=ℕa,ℓ; then, for allk, we have(15)Δℓ;kk−sℓt+1t+1!=k−sttt!.Proof.
The proof follows by using definitions and direct calculations, whereΔℓ;kuk,s=uk+ℓ,s−uk,s/ℓ.Lemma 4.
[20]). For the time scaleT=ℕa,ℓ, the delta Taylor polynomial is(16)hnk,s=k−shnn!,n∈ℕ0.Definition 5.
(see [21]). Letλ∈−∞,∞,λ<1andz,β,α∈ℂwith the conditionReα>0, and the discrete delta Mittag-Leffler factorial series is defined by(17)∑k=0∞λkz+k−1α−1kaz+kα−1β−1Γαk+β=Eα,βλ,z.
Forβ=1, we have(18)Eαλ,z=∑k=0∞λkz+k−1α−1kαΓαk+1.Definition 6.
(see [20]) (delta h-discrete Mittag-Leffler). Let λ∈−∞,∞,λhα<1, and z,β,α∈ℂ with Reα>0, and the delta discrete Mittag-Leffler factorial function is(19)hEα,βλ,z=∑k=0∞λkz+k−1α−1hkαz+kα−1hβ−1Γαk+β.
Forβ=1, we have(20)hEαλ,z=∑k=0∞λkz+k−1α−1hkαΓαk+1.Definition 7.
(see [22]) (delta ℓ− fractional sums). For a function u:ℕa,ℓ⟶R and ν>0, the delta left νth order ℓ− fractional sum is defined as(21)aΔℓ−νuk=1Γν∫aσk−νℓk−σℓsℓν−1usΔℓs=1Γν∑t=a/ℓk/h−νk−σℓtℓℓν−1utℓ.
The delta rightℓ−fractional sum of orderν>0foru:b,ℓℕ⟶Ris given by(22)ℓΔb−νuk=1Γν∫ρk+νℓbρℓs−kℓν−1us∇ℓs=1Γν∑r=k/ℓ+νb/hrℓ−σℓkℓv−1urℓ.Definition 8.
(see [20]) (delta ℓ−RL fractional differences). The νth order delta left ℓ−fractional operator has the form(23)aΔℓνuk=ΔℓnaΔℓ−n−νuk=ΔℓnΓn−ν∑r=a/ℓk/ℓ−n−νk−σℓrℓℓn−ν−1urℓ,k∈ℕa+n−νℓ,ℓ,and the νth order delta right ℓ−fractional difference is given as(24)ℓΔbνuk=−1n∇ℓnℓΔb−n−νuk=−1n∇ℓnΓn−ν∑r=k/ℓ+n−νb/ℓrℓ−σℓtℓn−ν−1urℓℓk∈b−n−νℓ,ℓℕ.Definition 9.
(see [21]) (Caputo ℓ−fractional differences). Let u be a function defined on ℕa,ℓ and b,ℓℕ. The νth order left as well as right delta Caputo fractional ℓ−delta differences are, respectively, defined as(25)aCΔℓνuk=ΔℓnaΔℓ−n−νuk,k∈ℕa+n−νℓ,ℓ,ℓCΔbνuk=−1n∇ℓnℓΔb−n−νuk,k∈b−n−νℓ,ℓℕ,where n=ν+1.Lemma 5.
(see [20]). Letν,μ,ℓ>0. Then,(26)a+μℓΔℓ−νk−aℓμ−1=ΓμΓμ+νk−aℓν+μ−1,ℓΔb−μℓ−νb−kℓμ−1=ΓμΓμ+νb−kℓν+μ−1.


## 3. Generalized and Convolution Delta Discreteℓ−Laplace Transform

We have the following definition for the discrete delta Laplace transform onℕa,ℓ.Definition 10.
Assume thatukis defined onℕa,ℓ. Then, the generalized delta discrete Laplace transform ofuis defined by(27)ℒa,ℓuks=∫a∞ℓe˜⊖sσk,aukΔℓk,ℓΔb−μℓ−νb−kℓμ−1=ΓμΓμ+νb−kℓν+μ−1.ℓΔb−μℓ−νb−kℓμ−1=ΓμΓμ+νb−kℓν+μ−1.
Now (26) is the same as(28)ℒa,ℓuks=aΔℓ−1uk11+ℓsk−a/ℓ+1|a∞=∑i=a/ℓ∞uiℓ11+ℓsi−a/ℓ+1.Remark 1.
(i)
In casea=0, we get(29)ℒℓuks=Δℓ−1uk11+ℓsk/ℓ+1|0∞=∑i=0∞uiℓ11+ℓsi+1.(ii)
In a special caseℓ=1, we have(30)ℒauks=aΔ−1uk11+sk−a+1|a∞=∑i=a∞ui11+si−a+1,which is the existing result in [1].Theorem 1.
Letk∈ℕa,ℓ,ℓ,ν>0, andℓs≠−1; then, we have(31)aΔℓ−1kℓμ11+ℓsk−a/ℓ+1=∑i=1μ+1−1μi−1kℓμ+1−i1/1+ℓsk−a/ℓsi.Proof.
Takinguk=kℓ1 and vk=1/1+ℓsk−a/ℓ+1 in (10) and using (9 and 14), we get(32)aΔℓ−1kℓ111+ℓsk−a/ℓ+1=−kℓ11/1+ℓsk−a/ℓs−1/1+ℓsk−a/ℓs2.
Again takinguk=kℓ2 and vk=1/1+ℓsk−a/ℓ+1 in (10) gives(33)aΔℓ−1kℓ211+ℓsk−a/ℓ+1=−kℓ21/1+ℓsk−a/ℓs+2saΔℓ−1kℓ11/1+ℓsk−a/ℓ+1.
Now applying (35) and simplifying, we get(34)aΔℓ−1kℓ211+ℓsk−a/ℓ+1=−kℓ21/1+ℓsk−a/ℓs−2kℓ11/1+ℓsk−a/ℓs2−2kℓ11/1+ℓsk−a/ℓs3,which can be rewritten as(35)aΔℓ−1kℓ211+ℓsk−a/ℓ+1=∑i=13−12i−1kℓ3−i1/1+ℓsk−a/ℓsi.
By continuing the above process up toμ times, we get (34).Lemma 6.
Letμ,ℓ>0ands≠0; then,(36)ℒℓkℓμs=μ!sμ+1.Proof.
By applying the limits 0 to∞ in (34), we get (39) from (32).Example 2.
By equating (32) and (39), we get(37)ℒℓkℓμs=∑i=0∞iℓℓμ11+ℓsi+1=μ!sμ+1.
The MATLB verification by takingℓ=3,μ=2, and s=10 yields 0.002 as the solution numerically in both forms. The solution of input function (signal) is analyzed graphically as follows.
Figure1 represents the input for function (signal) k33. For frequency domain for μ=2, Figure 2 shows the output. From this analysis, we conclude that the condition for convergence is not equal to 0.Figure 1
Time domain (t).Figure 2
Frequency (s).Remark 2.
By virtue of (39), we can write the Laplace transform of kℓν−1 for fraction ν>0 as follows:(38)ℒℓkℓν−1s=Γνsν.Definition 11.
(see [20]). Lets∈ℝ,0<ν<1, andu,v:ℕa,ℓ⟶ℝbe the real-valued functions. Then, the deltaℓ−discrete convolution ofuandvis defined by(39)u∗vk=∫ak∫akvk−σℓs+ausΔℓs=∑i=a/ℓk/ℓ−1uiℓvk−σℓiℓ+a,k∈ℕa,ℓ.Theorem 2.
(theℓ−convolution theorem). Fors∈ℝandu,vdefined onℕa,ℓ, we have(40)ℒa,ℓu∗vks=ℒa,ℓuksℒa,ℓvks.Proof.
By the definition, we have(41)ℒa,ℓu∗vks=h∑i=a/ℓ∞11+ℓsi−a/ℓ+1∑r=a/ℓi−1urℓviℓ−σℓrℓ+a=∑r=a/ℓ∞∑i=r+1∞urℓvi−r−1+a/ℓℓ1+ℓsi−a/ℓ+1=∑r=a/ℓ∞∑j=a/ℓ∞urℓvrℓ1+ℓsj+r+1−a/ℓ−a/ℓ+1=∑r=a/ℓ∞urℓ1+ℓsr−a/ℓ+1∑j=a/ℓ∞vjℓ1+ℓsj−a/ℓ+1=ℒa,ℓuksℒa,ℓvks,where the change of variable j=i−r−1+a/ℓ.
The following lemma is a generalization of the result in [1] to hℤ.Lemma 7.
Letube defined onℕa,ℓ. Then,(42)ℒa,ℓΔℓuks=sℱas−ua,where ℱas=ℒa,ℓuks.Proof.
By Definition1 and using (13), we see that(43)ℒa,ℓΔℓuks=∫a∞ℓe˜⊖sσk,aΔℓukΔℓk=ℓe˜⊖sσk,auk|a∞−∫a∞⊖she˜⊖sk,aukΔℓk=−ua+s∫a∞ℓe˜⊖sσk,aukΔℓk=sℱas−ua.


## 4. Delta Fractional Proportional Differences onℓℤ

From the time scale calculus in view of [23, 24], the delta exponential function e˜pk,a=1+pℓk−a/ℓ, where pk=ρ−1/ρ on ℓℤ satisfies the difference equation(44)Δℓuk=ρ−1ρuk,ua=1,where ρ≠ℓ/ℓ+1. Notice that ℓ=1 implies that the proportional factor ρ≠1/2.The delta ℓ−discrete proportional difference of order 0<ρ≤1 for function u on ℕa,ℓ is given by(45)Δℓρuk=1−ρuk+ρΔℓuk,k∈ℕa+ℓ,ℓ,where the regressivity condition insists that ρ≠ℓ/ℓ+1.Remark 3 (see [23]).
The first-order proportional type difference equation(46)Δℓρvk=1−ρvk+ρΔℓvk=uk,va=vahas the solution vk=vae˜pk,a+1/ρ∫ake˜pk−s−ℓ,0usΔℓs, where p=ρ−1/ρ and ∫akusΔℓs=∑i=a/ℓk/ℓ−1uiℓ.Definition 12.
For0<ρ≤1, the first-order proportional sum withΔρis defined by(47)aΔℓ−1,ρuk=1ρ∫ake˜pk−s−ℓ,0usΔℓs,p=ρ−1ρ.Remark 4.
By using the definition of convolution, the above relation can be written asaΔℓ−1,ρuk=1/ρu∗e˜pk−a,0.Proposition 1.
Letube defined onℕa,ℓ; then, we have(48)aΔℓ−1,ρΔℓρuk=uk−e˜pk−a,0ua=uk−ρ+ρ−1ℓρk−aℓua.Proof.
Applyingℓ−Laplace transform on aΔℓ−1,ρΔℓρuk, we see that(49)ℒa,ℓaΔℓ−1,ρΔℓρuks=ℒa,ℓ1ρΔℓρuk∗e˜pk−a,0s=1ρℒa,ℓ1−ρuk+ρΔℓutsℒa,ℓe˜pk−a,0s=1−ρρℱas+sℱas−uaρsρ−ρ+1=ℱas−uaρsρ−ρ+1.
Applying inverse Laplace transformℒa,ℓ−1, we obtain (48).
To generalize the first-order proportional difference to fractional sum, we see that(50)aΔℓ−n,ρuk=1ρ∫ake˜pk−ψ1−ℓ,0Δℓψ1,1ρ∫aψ1e˜pψ1−ψ2−ℓ,0Δℓψ2,…1ρ∫aψn−1e˜pψn−1−ψn−ℓ,0usΔℓψn,aΔℓ−n,ρuk=1ρnΓn∫ake˜pk−ψ−nℓ,0k−σℓψℓn−1uψΔℓψ,where Lemma 3 is used. By virtue of the above result, we now present the following generalized type fractional proportional sum.Definition 12.
For0<ρ≤1andν>0, we define the left proportional fractional sum ofuby(51)aΔℓ−ν,ρuk=1ρνΓν∫ake˜pk−ψ−νℓ,0k−σℓψℓv−1uψΔℓψ=ℓρνΓν∑i=a/ℓk/ℓ−νe˜pk−i+νℓ,0k−σℓiℓℓv−1uiℓ,
and the right proportional fractional sum as(52)ℓΔb−ν,ρuk=1ρνΓv∫tbe˜pψ−νℓ−k,0ρℓψ−kℓν−1uψΔℓψ=ℓρνΓν∑i=k/ℓ+νb/ℓe˜pi−νℓ−k,0ρℓiℓ−kℓv−1uiℓ.Remark 2.
Using convolution, we can express the left proportional fractional sum as(53)aΔℓ−ν,ρuk=1ρνΓνρ+ρ−1ℓρ1−νe˜pk,0kℓν−1∗uk.
In particular, whenℓ=1, we have(54)aΔ−ν,ρuk=1ρνΓν2ρ−1ρ1−νe˜pk,0kν−1∗uk.Definition 13.
(RL-fractional proportional). For0<ρ≤1andν>0, we define the left proportional fractional sum ofuby(55)aΔℓν,ρuk=Δℓn,ρaΔℓ−n−ν,ρuk=Δℓn,ρρn−νΓn−ν∫ake˜pk−ψ−n−νℓ,0k−σℓψℓn−ν−1uψΔℓψ,and the right proportional fractional sum as(56)ℓΔbν,ρuk=−1n∇ℓn,ρℓΔb−n−ν,ρuk=−1n∇ℓn,ρρn−νΓn−ν∫kbe˜pψ−n−νℓ−k,0ρℓψ−kℓn−ν−1uψΔℓψ.Remark 3.
(i)limν⟶0aΔℓν,ρuk=uk and (ii) limν⟶1aΔℓν,ρuk=aΔℓρuk.Theorem 3.
(semi-group property for delta fractional proportional). Assume thatukis defined onℕa,ℓ. Then,(57)aΔℓ−μ,ρaΔℓ−ν,ρuk=aΔℓ−ν,ρaΔℓ−μ,ρuk=aΔℓ−μ+ν,ρuk.Proof.
By Definition12 and using Lemma 5, we obtain(58)aΔℓ−μ,ρaΔℓ−ν,ρuk=1ρμ+νΓμΓν∫ak∫awe˜pk−w−μℓ,0e˜pw−ψ−νℓ,0k−σℓwℓμ−1k−σℓψℓν−1uψΔℓψΔℓw=1ρμ+νΓν∫ake˜pk−ψ−(μ+νℓ,0uψ1Γμ∫ψkk−wℓμ−1w−ψℓν−1ΔℓwΔℓψ=ρ−μ+νΓν∫akuψe˜pk−ψ−μ+νℓ,0ψΔℓ−μk−ψℓν−1Δℓψ=ρ−μ+νΓμ+ν∫akuψe˜pk−ψ−μ+νℓ,0k−ψℓμ+ν−1Δℓψ=aΔℓ−μ+ν,ρuk.Corollary 2.
Letube defined onℕa,ℓ. Then,(59)aΔℓμ,ρaΔℓ−ν,ρuk=aΔℓ−ν−μ,ρuk.Proof.
The proof follows by taking the negative index ofμ in Theorem 2.
The following two theorems are the applications of Laplace transform on fractional proportional type difference equation and fractional proportional sum, respectively.Theorem 4.
Assume thatuis defined onℕa,ℓ. Then,(60)ℒa,ℓaΔℓ−n,ρuks=ρs+1−ρnℒa,ℓuts−ρ∑r=0n−1ρs+1−ρn−1−rΔℓr,ρua.Proof.
By applyingℓ−Laplace transform on Remark 2 and using Lemma 7, we arrive at(61)ℒa,ℓaΔℓ1,ρuks=ℒa,ℓ1−ρuk+ρΔℓuks=1−ρℒa,ℓuks+ρℒa,ℓaΔhuks=1−ρℒa,huks+ρsℒa,huks−ua=1−ρ+ρsℒa,ℓuks−ρua.
By applying induction onn, we get (53).Theorem 5.
Forν∈R+,uk, and the function, we have(62)ℒa,ℓaΔℓ−ν,ρuks=ρ+ρ−1ℓρ1−νℒa,ℓuksρs−ρ+1ν.Proof.
By the definition of Laplace transform and convolution, we get(63)ℒa,ℓaΔℓ−ν,ρuks=1ρνΓνρ+ρ−1ℓρ1−νℒa,ℓe˜pk,0kℓν−1∗uks=1ρνΓνρ+ρ−1ℓρ1−νℒa,ℓe˜pk,0kℓν−1sℒa,ℓuks=1ρνΓνρ+ρ−1ℓρ1−νΓνs−ρ−1/ρνℒa,ℓuks,which completes the proof.


## 5. Conclusion

This research work focuses on the generalized fractional proportional sums and differences, by defining a new type of generalized discrete generalized Laplace transform and its convolution. Several properties and results of the generalized discrete Laplace transform are discussed with numerical examples. Also, we investigate the newly proposed delta type fractional proportional sums and differences on discrete RL and Mittag-Leffler functions. The findings are verified and analyzed by MATLAB. The core results of our findings in this research generalize the existing results in the literature.



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