On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

1  Introduction

The analysis and applications of non-integer order derivatives and integrals are known as fractional calculus. Fractional calculus theory has developed rapidly in recent years and has played a number of pivotal roles in science and engineering, helping as a strong and efficient resource for numerous physical phenomena. Over the last two decades, it has been extensively studied by several mathematicians [1–6].

The literature suggests that the Riemann-Liouville fractional (RLF) derivative plays a crucial part in fractional calculus. Researchers are encouraged to broaden the meanings of fractional derivatives due to the variety of applications. Some of the applications are available in [7–12]. Akgül [13] and Atangana et al. [14] investigated the fractional derivative with non-local and non-singular kernel. In [15] Caputo et al. examined the non-local fractional derivative which can work more efficiently with Fourier transformation. Some applications of fractional order operators are available in [16,17]. The existence of solution of Riemann-Liouville fractional integro-differential equations with fractional non-local multi-point boundary conditions and system of Riemann-Liouville fractional boundary value problems with ρ-Laplacian operators are briefly discussed in [18,19]. Currently, Jarad et al. [20] defined the weighted fractional derivatives and fractional integrals. To study fractional calculus and its applications, we refer to the readers [21–27].

Motivated by the recent studies presented in [20] and by combining this idea to extend the RLF operators, we will introduce the generalized weighted (k, s)-RLF operators and study their properties. The weighted Laplace transform to such fractional operators and some applications in mathematical physics will be discussed. Finally, we will finish with some closing remarks.

In the beginning, we recall some related definitions and notions. The integral form of the k-gamma and k-beta functions given in [28] are defined as follows:

Definition 1.1. The k-gamma function is defined by

Γk(ζ)=∫0∞αζ−1e−αkkdα,ℜ(ζ)>0.

Note that: Γ(ζ)=limk→1Γk(ζ) and Γk(ζ)=kζk−1Γ(ζk).

Definition 1.2. For ℜ(ζ),ℜ(η)>0 and k>0, the k-beta function is defined as

Bk(ζ,η)=1k∫01τζk−1(1−τ)ηk−1dτ,

where the Γk and Bk functions are related with an identity Bk(ζ,η)=Γk(ζ)Γk(η)Γk(ζ+η),

Definition 1.3. [29] Suppose that the Ω be a continuous function on interval [a,b]. Then weighted (k, s)-RLF integral of order ζ is given by

(ksJa+,ρζΩ)(α)=(s+1)1−ζkρ−1(α)kΓk(ζ)∫aα(αs+1−ts+1)ζk−1tsρ(t)Ω(t)dt,α∈[a,b],

where ζ,k>0, ρ(α)≠0 and s∈R∖{−1}.

Definition 1.4. [29] Let Ω be a continuous function on [0,∞) and s∈R∖{−1}, with n=[ζ]+1, ζ, ρ(α)≠0, and k>0. Then for all 0<t<α<∞

(ksDa+,ρζΩ)(α)=ρ−1(α)(kα−sddα)nρ(α)(ksJa+,ρnk−ζΩ)(t)=kn−1(s+1)ζ−nk+kkρ−1(α)Γk(nk−ζ)(α−sddα)n×∫aα(αs+1−ts+1)nk−ζk−1tsρ(t)Ω(t)dt.

where ksJa+,ρnk−ζ is a weighted (k, s)-RLF integral.

Jarad et al. [20] defined the generalized weighted Laplace transform as follows:

Definition 1.5. Let ρ, Υ be functions with values in R. Furthermore, Υ(α) is continuous and Υ′(α)>0 on [a,∞). The weighted generalized Laplace transform of Ω is given by

LΥρ{Ω(t)}(u)=∫a∞e−u(Υ(t)−Υ(a))ρ(t)Ω(t)Υ′(t)dt,(1)

and is true for all values of u for which (1) exists.

Theorem 1.1. [20] If Ω∈ACρ[a,α) and of weighted Υ-exponential order. Suppose that the DρΩ be a piecewise continuous function on every interval [a, T], then the weighted generalized Laplace transform of DρΩ exists and

LΥρ{DρΩ}(u)=uLΥρ{Ω(α)}(u)−ρ(a)Ω(a).

The generalized form of Theorem 1.1 is stated in the next result.

Theorem 1.2. Let Ω∈ACρn−1[a,α), such that DρkΩ, k=0, 1, 2, …, n−1 are of weighted Υ-exponential order. If DρnΩ is a continuous function on all intervals [a, T], the weighted generalized Laplace transform of DρnΩ exists and

LΥρ{DρnΩ}(u)=unLΥρ{Ω(α)}(u)−∑k=0n−1un−k−1Ωk(a).

Definition 1.6. [20] The generalization of the weighted convolution of Ω and Υ is defined by

(Ω∗Υρh)(s)=ρ−1(s)∫αsρ(Υ−1(Υ(s)+Υ(α)−Υ(t)))×Ω(Υ−1(Υ(s)+Υ(α)−Υ(t)))ρ(t)h(t)Υ′(t)dt.

2  Generalized Weighted (k, s)-Riemann-Liouville Fractional Operators

In this section, we introduce the generalized weighted (k, s)-RLF operators and describe some of their features.

Definition 2.1. Suppose that the Ω be a continuous function on the finite real interval [a,b] and Υ is strictly increasing function. Then the generalized weighted (k, s)-RLF integral of order ζ is defined by

(Υ,ksJa+,ρζΩ)(α)=(s+1)1−ζkρ−1(α)kΓk(ζ)∫aα(Υs+1(α)−Υs+1(t))ζk−1Υs(t)Υ′(t)ρ(t)Ω(t)dt,α>a,(2)

where ζ,k>0, ρ(α)≠0, s∈R∖{−1} and Υs+1(x)=(Υ(x))s+1.

The integral operator defined in 2 cover many fractional integral operators. For instance,

I.   if we set s=0 and k=1 in (2), we get the generalized weighted-RLF integral given in [20].

II.  If we set Υ(α)=α in (2), we get the weighted (k, s)-RLF integral presented in [29].

III. If we set ρ(α)=1 and Υ(α)=α in (2), we get the weighted (k, s)-RLF integral [29].

IV. If we set s=0, Υ(α)=α and ρ(α)=1 in (2), k-RLF integral is obtained [30].

V.  If we set k=1, s=0, Υ(α)=α and ρ(α)=1 in (2), it gives RLF integral [3].

VI. For s→−1+, Υ(α)=α and ρ(α)=1 in (2), we obtain k-Hadamard fractional integral [31].

The corresponding weighted generalized fractional derivative is defined by the following definition.

Definition 2.2. Let Ω be continuous function on [0,∞) and s∈R∖{−1}, n=[ζ]+1, ζ,k>0, and ρ(α)≠0. Then for all 0<t<α<∞, the inverse derivative operator of integral operator 2 is defined by

(Υ,ksDa+,ρζΩ)(α)=ρ−1(α)(kΥs(α)Υ′(α)ddα)nρ(α)(Υ,ksJa+,ρnk−ζΩ)(t)=kn−1(s+1)1−nk−ζkρ−1(α)Γk(nk−ζ)(1Υs(α)Υ′(α)ddα)n×∫aα(Υs+1(α)−Υs+1(t))nk−ζk−1Υs(t)Υ′(t)ρ(t)Ω(t)dt.(3)

where Υ,ksJa+,ρnk−ζ is a generalized weighted (k, s)-RLF integral.

There are many other fractional derivative operators as special cases of the operator (3).

I.   If we choose s=0 and k=1 in (3), we get the weighted (k, s)-RLF derivative presented in [20]

II.  If we choose Υ(α)=α in (3), we get weighted (k, s)-RLF derivative presented in [29].

III. If we choose ρ(α)=1 and Υ(α)=α in (3), we get (k, s)-RLF derivative [32].

IV. If we set s=0, Υ(α)=α and ρ(α)=1 in (3) it gives to k-RLF derivatives [33].

V.  If we set k=1, s=0, Υ(α)=α and ρ(α)=1 in (3), it reduces to RLF derivative [34].

VI. (3) reduces to k-Hadamard fractional derivative for s→−1+, Υ(α)=α and ρ(α)=1 [31].

In the following definition, we define the space where the generalized weighted (k, s)-RLF integral is bounded.

Definition 2.3. Let f be defined on [a,b] and Xρp(a,b), 1≤p≤∞ be the space of all Lebesgue measurable functions for which ∥Ω∥Xρp<∞, where

∥Ω∥Xρp=[(s+1)∫ab∣ρ(α)Ω(α)∣pΥs(α)Υ′(α)dα]1p,1≤p<∞,

ρ(α)≠0,s∈R∖{−1}and

∥Ω∥Xρ∞=esssupa≤α≤b|ρ(α)Ω(α)|<∞.

Note that Ω∈Xρp(a,b) ⇔ ρ(α)Ω(α)(Υs(α)Υ′(α))1p∈Lp(a,b) for 1≤p<∞ and Ω∈Xρ∞(a,b) ⇔ ρ(α)Ω(α)∈L∞(a,b).

Theorem 2.1. Let ζ>0, k>0, 1≤p≤∞ and Ω∈Xρp(a,b). Then Υ,ksJa+,ρζΩ is bounded in Xρp(a,b) and

∥Υ,ksJa+,ρζΩ∥Xρp≤(s+1)−ζk(Υs+1(b)−Υs+1(a))ζkΓk(ζ+1)∥Ω∥Xρp.

Proof. For 1≤p<∞, we have

∥Υ,ksJa+,ρζΩ∥Xρp=[(s+1)∫ab|ρ(α)(s+1)1−ζkρ−1(α)kΓk(ζ)×∫aα(Υs+1(α)−Υs+1(t))ζk−1Υs(t)Υ′(t)ρ(t)Ω(t)dt|pΥs(α)Υ′(α)dα]1p=(s+1)−ζkkΓk(ζ)[∫ab|∫aα(Υs+1(α)−Υs+1(t))ζk−1(s+1)Υs(t)Υ′(t)ρ(t)Ω(t)dt|p(s+1)−ζkkΓk(ζ)×(s+1)Υs(α)Υ′(α)dα]1p.(4)

Substituting Υs+1(α)=v and Υs+1(t)=u on the right side of (4), we obtain

∥Υ,ksJa+,ρζΩ∥Xρp=(s+1)−ζkkΓk(ζ)[∫Υs+1(a)Υs+1(b)|∫Υs+1(a)Υs+1(α)(v−u)ζk−1ρ(Υ−1(u1s+1))Ω(Υ−1(u1s+1))du|pdv]1p.

By using Minkowski’s inequality, we have

∥Υ,ksJa+,ρζΩ∥Xρp≤(s+1)−ζkkΓk(ζ)∫Υs+1(a)Υs+1(b)|ρ(Υ−1(u1s+1))Ω(Υ−1(u1s+1))|[∫uΥs+1(b)(v−u)(ζk−1)pdv]1pdu≤(s+1)−ζkkΓk(ζ)∫Υs+1(a)Υs+1(b)|ρ(Υ−1(u1s+1))Ω(Υ−1(u1s+1))|[(Υs+1(b)−u)(ζk−1)p+1(ζk−1)p+1]1pdu.

Applying Hölder’s inequality, we get

∥Υ,ksJa+,ρζΩ∥Xρp≤(s+1)−ζkkΓk(ζ)[∫Υs+1(a)Υs+1(b)|ρ(Υ−1(u1s+1))Ω(Υ−1(u1s+1))|pdu]1p×[∫Υs+1(a)Υs+1(b)((Υs+1(b)−u)(ζk−1)p+1(ζk−1)p+1)qpdu]1q,

where 1p+1q=1.

∥Υ,ksJa+,ρζΩ∥Xρp≤(s+1)−ζkkΓk(ζ)[∫ab|ρ(t)Ω(t)|p(s+1)Υs(t)Υ′(t)dt]1p×[∫Υs+1(a)Υs+1(b)((Υs+1(b)−u)(ζk−1)p+1(ζk−1)p+1)qpdu]1q≤(s+1)−ζk(Υs+1(b)−Υs+1(a))ζkΓk(ζ+1)∥Ω∥Xρp.

For p=∞, we obtain

|ρ(α)Υ,ksJa+,ρζΩ(α)|=(s+1)−ζk(Υs+1(b)−Υs+1(a))ζkΓk(ζ+1)∥Ω∥Xρ∞.

Hence the proof is done.

Theorem 2.2. Let Ω be continuous on [0,∞) and s∈R∖{−1} and ρ(α)≠0, n=[ζ]+1. Then for all 0<a<α.

Υ,ksDa+,ρζ(Υ,ksJa+,ρζΩ)(α)=Ω(α),

where ζ,k>0 and nk−ζ>0.

Proof. Consider

Υ,ksDa+,ρζ(Υ,ksJa+,ρζΩ)(α)=(s+1)ζ−nk+kkρ−1(α)kΓk(nk−ζ)(1Υs(α)Υ′(α)ddα)nkn×∫aα(Υs+1(α)−Υs+1(y))nk−ζk−1Υs(y)Υ′(y)ρ(y)(Υ,ksJa+,ρζΩ)(y)dy=(s+1)ζ−nk+kkρ−1(α)kΓk(nk−ζ)(1Υs(α)Υ′(α)ddα)nkn∫aα[(Υs+1(α)−Υs+1(y))nk−ζk−1×Υs(y)Υ′(y)ρ(y)(s+1)1−ζkρ−1(α)kΓk(ζ)∫ay(Υs+1(y)−Υs+1(t))ζk−1Υs(t)Υ′(t)ρ(t)Ω(t)dt]dy=(s+1)2−nρ−1(α)k2Γk(ζ)Γk(nk−ζ)(1Υs(α)Υ′(α)ddα)nkn∫aαΥs(t)Υ′(t)ρ(t)Ω(t)×[∫tα(Υs+1(y)−Υs+1(t))ζk−1(Υs+1(α)−Υs+1(y))nk−ζk−1Υs(y)Υ′(y)dy]dt.(5)

Substitute z=Υs+1(y)−Υs+1(t)Υs+1(α)−Υs+1(t) on the right side of (5), we get

Υ,ksDa+,ρζ(Υ,ksJa+,ρζΩ)(α)=(s+1)1−nρ−1(α)k2Γk(ζ)Γk(nk−ζ)(1Υs(α)Υ′(α)ddα)nkn×∫aαΥs(t)Υ′(t)ρ(t)Ω(t)(Υs+1(α)−Υs+1(t))n−1[∫tα(1−z)ζk−1(z)nk−ζk−1dz]dt=(s+1)1−nρ−1(α)k2Γk(ζ)Γk(nk−ζ)(1Υs(α)Υ′(α)ddα)nkn×∫aαΥs(t)Υ′(t)ρ(t)Ω(t)(Υs+1(α)−Υs+1(t))n−1[kBk(ζ,nk−ζ)]dt=(s+1)1−nρ−1(α)kΓk(nk)(1Υs(α)Υ′(α)ddα)nkn×∫aαΥs(t)Υ′(t)ρ(t)Ω(t)(Υs+1(α)−Υs+1(t))n−1dt=(s+1)1−nρ−1(α)knΓ(n)(1Υs(α)Υ′(α)ddα)nkn×∫aαΥs(t)Υ′(t)ρ(t)Ω(t)(Υs+1(α)−Υs+1(t))n−1dt,

which gives

Υ,ksDa+,ρζ(Υ,ksJa+,ρζΩ)(α)=Ω(α).

This proved the inverse property.

Corollary 2.1. Let the function Ω be continuous on [0,∞) and s∈R∖{−1} and ρ(α)≠0, m=[η]+1, n=[ζ]+1. Then for all 0<a<α

Υ,ksDa+,ρζ(Υ,ksJa+,ρηΩ)(α)=(Υ,ksDa+,ρζ−ηΩ)(α),

where ζ,η,k>0.

Corollary 2.2. Let the function Ω be continuous on [0,∞) and s∈R∖{−1}, ρ(α)≠0, n=[ζ]+1, m=[η]+1 and ζ+η<nk. Then for all 0<a<α

Υ,ksDa+,ρζ(Υ,ksDa+,ρηΩ)(α)=(Υ,ksDa+,ρζ+ηΩ)(α),

where ζ,η,k>0.

Proof. By using Definition 2.2, we have

Υ,ksDa+,ρζ(Υ,ksDa+,ρηΩ)(α)=ρ−1(α)(kΥs(α)Υ′(α)ddα)nρ(α)(Υ,ksJa+,ρnk−ζ)(Υ,ksDa+,ρηΩ)(α)=ρ−1(α)(kΥs(α)Υ′(α)ddα)nρ(α)(Υ,ksJa+,ρnk−ζ)(Υ,ksDa+,ρη(Υ,ksJa+,ρηΥ,ksJa+,ρ−ηΩ)(α).

By using Theorem 2.2, we have

Υ,ksDa+,ρζ(Υ,ksDa+,ρηΩ)(α)=ρ−1(α)(kΥs(α)Υ′(α)ddα)nρ(α)(Υ,ksJa+,ρnk−ζ)(Υ,ksJa+,ρ−ηΩ)(α)=ρ−1(α)(kΥs(α)Υ′(α)ddα)nρ(α)(Υ,ksJa+,ρnk−(ζ+η)Ω)(α),

which implies

Υ,ksDa+,ρζ(Υ,ksDa+,ρηΩ)(α)=(Υ,ksDa+,ρζ+ηΩ)(α).

Hence the semi-group property of new derivative operator is proved.

Corollary 2.3. Suppose that the Ω be a continuous function on [0,∞) and ζ,η∈R+, ρ(α)≠0 and s∈R∖{−1}. Then for all 0<a<α

Υ,ksDa+,ρζ(Υ,ksDa+,ρηΩ)(α)=Υ,ksDa+,ρη(Υ,ksDa+,ρζΩ)(α),

where n=[ζ]+1, m=[η]+1 and ζ+η<nk.

Theorem 2.3. Let the function Ω be continuous on [a,b] and k>0, ρ(α)≠0 and s∈R∖{−1}

Υ,ksJa+,ρη[Υ,ksJa+,ρζΩ(α)]=Υ,ksJa+,ρζ[Υ,ksJa+,ρηΩ(α)]=Υ,ksJa+,ρζ+ηΩ(α),

for all ζ,η>0 and α∈[a,b].

Proof. By utilizing the Definition 2.1 and Dirichlet’s formula, we get

Υ,ksJa+,ρζ[Υ,ksJa+,ρηΩ(α)]=(s+1)1−ζkρ−1(α)kΓk(ζ)∫aα(Υs+1(α)−Υs+1(t))ζk−1Υs(t)Υ′(t)ρ(t)(Υ,ksJa+,ρηΩ)(t)dt

=(s+1)1−ζkρ−1(α)kΓk(ζ)∫aα(Υs+1(α)−Υs+1(t))ζk−1Υs(t)Υ′(t)ρ(t)×[(s+1)1−ηkρ−1(t)kΓk(η)∫at(Υs+1(t)−Υs+1(τ))ηk−1Υs(τ)Υ′(τ)ρ(τ)Ω(τ)dτ]dt=(s+1)2−ζ+ηkρ−1(α)k2Γk(ζ)Γk(η)∫aαΥs(τ)Υ′(τ)ρ(τ)Ω(τ)×∫τα(Υs+1(α)−Υs+1(t))ζk−1(Υs+1(t)−Υs+1(τ))ηk−1Υs(t)Υ′(t)dtdτ.(6)

Substitute y=Υs+1(t)−Υs+1(τ)Υs+1(α)−Υs+1(τ) on the right side of (6), we obtain

Υ,ksJa+,ρζ[Υ,ksJa+,ρηΩ(α)]=(s+1)2−ζ+ηkρ−1(α)k2Γk(ζ)Γk(η)×∫aα(Υs+1(α)−Υs+1(τ))ζ+ηk−1(s+1)Υs(τ)Υ′(τ)ρ(τ)Ω(τ)kBk(ζ,η)dτ=(s+1)1−ζ+ηkρ−1(α)kΓk(ζ+η)∫aα(Υs+1(α)−Υs+1(τ))ζ+ηk−1Υs(τ)Υ′(τ)ρ(τ)Ω(τ)dτ=Υ,ksJa+,ρζ+ηΩ(α).