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

Muhammad Samraiz1, Muhammad Umer1, Thabet Abdeljawad2,3,*, Saima Naheed1, Gauhar Rahman4, Kamal Shah2,5

1 Department of Mathematics, University of Sargodha, P.O. Box 40100, Sargodha, 40100, Pakistan
2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
3 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
4 Department of Mathematics and Statistics, Hazara University Mansehra, Mansehra, 21300, Pakistan
5 Department of Mathematics, University of Malakand, Chakdara Dir (L), KPK, 18000, Pakistan

* Corresponding Author: Thabet Abdeljawad. Email: email

(This article belongs to the Special Issue: Applications of Fractional Operators in Modeling Real-world Problems: Theory, Computation, and Applications)

Computer Modeling in Engineering & Sciences 2023, 136(1), 901-919. https://doi.org/10.32604/cmes.2023.024029

Abstract

In this paper, we establish the new forms of Riemann-type fractional integral and derivative operators. The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases. The properties of new operators like semi-group, inverse and certain others are discussed and its weighted Laplace transform is evaluated. Fractional integro-differential free-electron laser (FEL) and kinetic equations are established. The solutions to these new equations are obtained by using the modified weighted Laplace transform. The Cauchy problem and a growth model are designed as applications along with graphical representation. Finally, the conclusion section indicates future directions to the readers.

Keywords


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 [16].

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 [712]. 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 [2127].

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)=0αζ1eαkkdα,(ζ)>0.

Note that: Γ(ζ)=limk1Γk(ζ) and Γk(ζ)=kζk1Γ(ζk).

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

Bk(ζ,η)=1k01τζk1(1τ)ηk1dτ,

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)1ζkρ1(α)kΓk(ζ)aα(αs+1ts+1)ζk1tsρ(t)Ω(t)dt,α[a,b],

where ζ,k>0, ρ(α)0 and sR{1}.

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

(ksDa+,ρζΩ)(α)=ρ1(α)(kαsddα)nρ(α)(ksJa+,ρnkζΩ)(t)=kn1(s+1)ζnk+kkρ1(α)Γk(nkζ)(αsddα)n×aα(αs+1ts+1)nkζk1tsρ(t)Ω(t)dt.

where ksJa+,ρnkζ 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)=aeu(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 ΩACρ[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 ΩACρn1[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=0n1unk1Ωk(a).

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

(ΩΥρh)(s)=ρ1(s)αsρ(Υ1(Υ(s)+Υ(α)Υ(t)))×Ω(Υ1(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)1ζkρ1(α)kΓk(ζ)aα(Υs+1(α)Υs+1(t))ζk1Υs(t)Υ(t)ρ(t)Ω(t)dt,α>a,(2)

where ζ,k>0, ρ(α)0, sR{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 s1+, 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 sR{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)=ρ1(α)(kΥs(α)Υ(α)ddα)nρ(α)(Υ,ksJa+,ρnkζΩ)(t)=kn1(s+1)1nkζkρ1(α)Γk(nkζ)(1Υs(α)Υ(α)ddα)n×aα(Υs+1(α)Υs+1(t))nkζk1Υs(t)Υ(t)ρ(t)Ω(t)dt.(3)

where Υ,ksJa+,ρnkζ 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 s1+, 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), 1p be the space of all Lebesgue measurable functions for which ΩXρp<, where

ΩXρp=[(s+1)abρ(α)Ω(α)pΥs(α)Υ(α)dα]1p,1p<,

ρ(α)0,sR{1}and

ΩXρ=esssupaαb|ρ(α)Ω(α)|<.

Note that ΩXρp(a,b) ρ(α)Ω(α)(Υs(α)Υ(α))1pLp(a,b) for 1p< and ΩXρ(a,b) ρ(α)Ω(α)L(a,b).

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

Υ,ksJa+,ρζΩXρp(s+1)ζk(Υs+1(b)Υs+1(a))ζkΓk(ζ+1)ΩXρp.

Proof. For 1p<, we have

Υ,ksJa+,ρζΩXρp=[(s+1)ab|ρ(α)(s+1)1ζkρ1(α)kΓk(ζ)×aα(Υs+1(α)Υs+1(t))ζk1Υs(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))ζk1(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+,ρζΩXρp=(s+1)ζkkΓk(ζ)[Υs+1(a)Υs+1(b)|Υs+1(a)Υs+1(α)(vu)ζk1ρ(Υ1(u1s+1))Ω(Υ1(u1s+1))du|pdv]1p.

By using Minkowski’s inequality, we have

Υ,ksJa+,ρζΩXρp(s+1)ζkkΓk(ζ)Υs+1(a)Υs+1(b)|ρ(Υ1(u1s+1))Ω(Υ1(u1s+1))|[uΥs+1(b)(vu)(ζk1)pdv]1pdu(s+1)ζkkΓk(ζ)Υs+1(a)Υs+1(b)|ρ(Υ1(u1s+1))Ω(Υ1(u1s+1))|[(Υs+1(b)u)(ζk1)p+1(ζk1)p+1]1pdu.

Applying Hölder’s inequality, we get

Υ,ksJa+,ρζΩXρp(s+1)ζkkΓk(ζ)[Υs+1(a)Υs+1(b)|ρ(Υ1(u1s+1))Ω(Υ1(u1s+1))|pdu]1p×[Υs+1(a)Υs+1(b)((Υs+1(b)u)(ζk1)p+1(ζk1)p+1)qpdu]1q,

where 1p+1q=1.

Υ,ksJa+,ρζΩXρp(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)(ζk1)p+1(ζk1)p+1)qpdu]1q(s+1)ζk(Υs+1(b)Υs+1(a))ζkΓk(ζ+1)ΩXρp.

For p=, we obtain

|ρ(α)Υ,ksJa+,ρζΩ(α)|=(s+1)ζk(Υs+1(b)Υs+1(a))ζkΓk(ζ+1)ΩXρ.

Hence the proof is done.

Theorem 2.2. Let Ω be continuous on [0,) and sR{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 nkζ>0.

Proof. Consider

Υ,ksDa+,ρζ(Υ,ksJa+,ρζΩ)(α)=(s+1)ζnk+kkρ1(α)kΓk(nkζ)(1Υs(α)Υ(α)ddα)nkn×aα(Υs+1(α)Υs+1(y))nkζk1Υs(y)Υ(y)ρ(y)(Υ,ksJa+,ρζΩ)(y)dy=(s+1)ζnk+kkρ1(α)kΓk(nkζ)(1Υs(α)Υ(α)ddα)nknaα[(Υs+1(α)Υs+1(y))nkζk1×Υs(y)Υ(y)ρ(y)(s+1)1ζkρ1(α)kΓk(ζ)ay(Υs+1(y)Υs+1(t))ζk1Υs(t)Υ(t)ρ(t)Ω(t)dt]dy=(s+1)2nρ1(α)k2Γk(ζ)Γk(nkζ)(1Υs(α)Υ(α)ddα)nknaαΥs(t)Υ(t)ρ(t)Ω(t)×[tα(Υs+1(y)Υs+1(t))ζk1(Υs+1(α)Υs+1(y))nkζk1Υs(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)1nρ1(α)k2Γk(ζ)Γk(nkζ)(1Υs(α)Υ(α)ddα)nkn×aαΥs(t)Υ(t)ρ(t)Ω(t)(Υs+1(α)Υs+1(t))n1[tα(1z)ζk1(z)nkζk1dz]dt=(s+1)1nρ1(α)k2Γk(ζ)Γk(nkζ)(1Υs(α)Υ(α)ddα)nkn×aαΥs(t)Υ(t)ρ(t)Ω(t)(Υs+1(α)Υs+1(t))n1[kBk(ζ,nkζ)]dt=(s+1)1nρ1(α)kΓk(nk)(1Υs(α)Υ(α)ddα)nkn×aαΥs(t)Υ(t)ρ(t)Ω(t)(Υs+1(α)Υs+1(t))n1dt=(s+1)1nρ1(α)knΓ(n)(1Υs(α)Υ(α)ddα)nkn×aαΥs(t)Υ(t)ρ(t)Ω(t)(Υs+1(α)Υs+1(t))n1dt,

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 sR{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+,ρζηΩ)(α),

where ζ,η,k>0.

Corollary 2.2. Let the function Ω be continuous on [0,) and sR{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)=ρ1(α)(kΥs(α)Υ(α)ddα)nρ(α)(Υ,ksJa+,ρnkζ)(Υ,ksDa+,ρηΩ)(α)=ρ1(α)(kΥs(α)Υ(α)ddα)nρ(α)(Υ,ksJa+,ρnkζ)(Υ,ksDa+,ρη(Υ,ksJa+,ρηΥ,ksJa+,ρηΩ)(α).

By using Theorem 2.2, we have

Υ,ksDa+,ρζ(Υ,ksDa+,ρηΩ)(α)=ρ1(α)(kΥs(α)Υ(α)ddα)nρ(α)(Υ,ksJa+,ρnkζ)(Υ,ksJa+,ρηΩ)(α)=ρ1(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,ηR+, ρ(α)0 and sR{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 sR{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)1ζkρ1(α)kΓk(ζ)aα(Υs+1(α)Υs+1(t))ζk1Υs(t)Υ(t)ρ(t)(Υ,ksJa+,ρηΩ)(t)dt

=(s+1)1ζkρ1(α)kΓk(ζ)aα(Υs+1(α)Υs+1(t))ζk1Υs(t)Υ(t)ρ(t)×[(s+1)1ηkρ1(t)kΓk(η)at(Υs+1(t)Υs+1(τ))ηk1Υs(τ)Υ(τ)ρ(τ)Ω(τ)dτ]dt=(s+1)2ζ+ηkρ1(α)k2Γk(ζ)Γk(η)aαΥs(τ)Υ(τ)ρ(τ)Ω(τ)×τα(Υs+1(α)Υs+1(t))ζk1(Υs+1(t)Υs+1(τ))ηk1Υs(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)2ζ+ηkρ1(α)k2Γk(ζ)Γk(η)×aα(Υs+1(α)Υs+1(τ))ζ+ηk1(s+1)Υs(τ)Υ(τ)ρ(τ)Ω(τ)kBk(ζ,η)dτ=(s+1)1ζ+ηkρ1(α)kΓk(ζ+η)aα(Υs+1(α)Υs+1(τ))ζ+ηk1Υs(τ)Υ(τ)ρ(τ)Ω(τ)dτ=Υ,ksJa+,ρζ+ηΩ(α).

This completes the proof.

Theorem 2.4. Let ζ, η, k>0, ρ(α)0 and sR{1}. Then we have

Υ,ksJa+,ρη[ρ1(α)(Υs+1(α)Υs+1(a))ηk1]=Γk(η)(Υs+1(α)Υs+1(a))ζ+ηk1ρ1(α)(s+1)ζkΓk(ζ+η),

where Γk(.) represents the k-Gamma function.

Proof. By Definition 2.1, we get

Υ,ksJa+,ρη[ρ1(α)(Υs+1(α)Υs+1(a))ηk1]=(s+1)1ζkρ1(α)kΓk(ζ)aα(Υs+1(α)Υs+1(t))ζk1Υs(t)Υ(t)dt×(Υs+1(α)Υs+1(a))ηk1Ω(t)dt.(7)

Substitute y=Υs+1(α)Υs+1(t)Υs+1(α)Υs+1(a) on the right side of (7), we get

Υ,ksJa+,ρη[ρ1(α)(Υs+1(α)Υs+1(a))ηk1]=(s+1)ζkρ1(α)(Υs+1(α)Υs+1(a))ζ+ηk1kΓk(ζ)×01(1y)ζk1(y)ηk1dy=(s+1)ζk(Υs+1(α)Υs+1(a))ζ+ηk1ρ1(α)kΓk(ζ)kBk(ζ,η)=Γk(η)(Υs+1(α)Υs+1(a))ζ+ηk1ρ1(α)(s+1)ζkΓk(ζ+η).

The proof is done.

Example 2.1. Corresponding to the choice of the parameters s=0,k=1,η=3,a=0 and ρ(t)=1, we get the following graphs with different choices of the function Υ(t).

images

Figure 1: For ϒ(t) = t the graph in Fig. 1 shows the increasing behaviour with 0 + ≤ t ≤ 5

3  The Generalized Weighted Laplace Transform

In the following section, we use the weighted Laplace transformation to the new fractional operators. Firstly, we present the following definition which is a modified form of the Definition 1.5.

Definition 3.1. Suppose that the Ω be a real valued function defined on Ω[a,) and sR{1}. The weighted generalized Laplace transform of Ω is given by

LΥρΩ(u)=(s+1)aeu(Υs+1(α)Υs+1(a))ρ(α)Υs(α)Υ(α)Ω(α)dα

holds for all values of u.

Proposition 3.1.

LΥρ{ρ1(α)(Υs+1(α)Υs+1(a))ζk1}(u)=Γ(ζk)uζk,u>0.

Proof. By the Definition 3.1, we have

LΥρ{ρ1(α)(Υs+1(α)Υs+1(a))ζk1}(u)=(s+1)aeu(Υs+1(α)Υs+1(a))(Υs+1(α)Υs+1(a))ζk1Υs(α)Υ(α)dα.(8)

Substitute t=(Υs+1(α)Υs+1(a)) on the right side of (8), we get

LΥρ{ρ1(α)(Υs+1(α)Υs+1(a))ζk1}(u)=0euttζk1dt=0eut(ut)ζk1(u)ζk1uudt=1uζk0eut(ut)ζk1udt,

the proof is done.

Theorem 3.1. Let the function Ω be continuous on each interval a,α and of weighted Υs+1-exponential order. Then

LΥρ((Υ,ksJa+,ρζΩ)(α))(u)=((s+1)uk)ζkLΥρ{Ω(α)}(u),

where k>0, ρ(α)0, sR{1}.

Proof. By the Definitions 2.1, 1.6 and Proposition 3.1, we have

LΥρ{(Υ,ksJa+,ρζΩ)(α)}(u)=LΥρ{(s+1)1ζkρ1(α)kΓk(ζ)aα(Υs+1(α)Υs+1(t))ζk1Υs(t)Υ(t)ρ(t)Ω(t)dt}(u)=(s+1)ζkkΓk(ζ)LΥρ{ρ1(α)(Υs+1(α)Υs+1(a))ζk1Υs+1ρΩ(α)}(u)=(s+1)ζkkΓk(ζ)LΥρ{ρ1(α)(Υs+1(α)Υs+1(a))ζk1}(u)LΥρ{Ω(α)}(u)=(s+1)ζkkΓk(ζ)Γ(ζk)uζkLΥρ{Ω(α)}(u)=((s+1)uk)ζkLΥρ{Ω(α)}(u).

This completes the proof.

Theorem 3.2. The generalized weighted Laplace transform of the novel derivative is

LΥρ{(Υ,ksDa+,ρζΩ)(α)}(u)=(s+1)nkζk(ku)ζkLΥρ{Ω(α)}(u)knm=0n1unm1(Υ,ksJa+,ρnkζΩ)m(a+).

Proof. By the Definition 2.2, Theorem 1.2 and Theorem 3.1, we get

LΥρ{(Υ,ksDa+,ρζΩ)(α)}(u)=LΥρ{ρ1(α)(kΥs(α)Υ(α)ddα)nρ(α)(Υ,ksJa+,ρnkζΩ)(t)}(u)=knunLΥρ{(Υ,ksJa+,ρnkζΩ)(t)}(u)knm=0n1unm1(Υ,ksJa+,ρnkζΩ)m(a+)=(uk)n((s+1)uk)nkζkLΥρ{Ω(α)}(u)knm=0n1unm1(Υ,ksJa+,ρnkζΩ)m(a+)=(s+1)nkζk(ku)ζkLΥρ{Ω(α)}(u)knm=0n1unm1(Υ,ksJa+,ρnkζΩ)m(a+).

The proof is completed.

4  Fractional Free Electron Laser Equation with Solution

In this section, we investigate the fractional generalization FEL by using the introduced fractional integral given in (2) and the fractional derivative presented in (3). The series form solution is obtained by employing the weighted generalized Laplace transform introduced by Jarad et al.  [20].

Theorem 4.1. The solution of the cauchy problem

Υ,ksDa+,ρζΩ(α)=λΥ,ksJa+,ρηΩ(α)+f(α),(9)

Υ,ksJa+,ρnkζΩ(a+)=d,d0,(10)

where α(0,), fL1[a,), a0, ρ0and λR is given by

Ω(α)=dρ1(α)m=0λm(s+1)(ζ+ηk)m+ζkkΓk((ζ+η)m+ζ)(Υs+1(α)Υs+1(a))(ζ+η)m+ζk1+m=0λm(s+1)m+1(Υ,ksJa+,ρ(ζ+η)m+ζf)(α).(11)

Proof. Applying generalized weighted Laplace transform on (9) and using Theorems 3.1 and 3.2, we get

LΥρ{Υ,ksDa+ρζΩ(α)}(u)=λLΥρ{Υ,ksJa+ρηΩ(α)}(u)+LΥρ{f(α)}(u).

The above equation implies that

LΥρ{Ω(α)}(u)=dk[(s+1)kζk(ku)ζk1λ(s+1)kζηk(ku)(ζ+η)k]+[(s+1)kζk(ku)ζk1λ(s+1)kζηk(ku)(ζ+η)k]LΥρ{f(α)}(u).

Taking |λ(s+1)ζ+ηkk(ku)ζ+ηk|1 and using the binomial expansion, we get

LΥρ{Ω(α)}(u)={dk(s+1)kζk(ku)ζk(s+1)kζk(ku)ζkLΥρ{f(α)}(u)}×m=0λm(s+1)(ζ+ηk)mk(ku)(ζ+η)mk=dkm=0λm(s+1)(ζ+ηk)m+ζkk(ku)(ζ+η)m+ζk+m=0λm(s+1)(ζ+ηk)m+ζkk(ku)(ζ+η)m+ζk{LΥρ{f(α)}(u)}.

By using the inverse Laplace transform, we obtain

Ω(α)=dρ1(α)m=0λm(s+1)(ζ+ηk)m+ζkkΓk((ζ+η)m+ζ)(Υs+1(α)Υs+1(a))(ζ+η)m+ζk1+m=0λm(s+1)m+1(Υ,ksJa+,ρ(ζ+η)m+ζf)(α),

the result is completed.

Remark 4.1. If we set s=0, k=1, ζ=η=1, f(α)=0, ρ=ir, λ=iΠp, (r,pR) and Υs+1(α)=α, in 9 and 10, then the original free electron laser equation given in [35] is obtained.

The following is the cauchy problem based on Theorem 4.1.

Example 4.1. The solution of the cauchy problem

Υ,ksDa+,ρζΩ(α)=λΥ,ksJa+,ρηΩ(α)+f(α),

where

f(α)=ρ1(α)(Ω(s+1)(α)Ω(s+1)(a+))(12)

subject to the condition

Υ,ksJa+,ρnkζΩ(a+)=0(13)

with α(0,), a0, ρ0 and λR is given by

Ω(α)=m=0λm(s+1)(m+1)ζkΓk((ζ+η)m+ζ)(Υs+1(α)Υs+1(a))(ζ+η)m+2ζk1ρ1(α)Γk((ζ+η)m+2ζ)(14)

Solution 4.1. For the function given by (12) subjected to the condition presented in (13) the Eq. (11) becomes

Ω(α)=m=0λm(s+1)m+1(Υ,ksJa+,ρ(ζ+η)m+ζf)(α).(15)

Consider

(Υ,ksJa+,ρ(ζ+η)m+ζf)(α)=Υ,ksJa+,ρ(ζ+η)m+ζ(ρ1(α)(Ω(s+1)(α)Ω(s+1)(a+)))=Γk((ζ+η)m+ζ)(Ωs+1(α)Ωs+1(a))(ζ+η)m+2ζk1ρ1(α)(s+1)ζkΓk((ζ+η)m+2ζ)(16)

Using (16) in (15), we obtain (14).

5  Fractional Kinetic Differ-Integral Equation with Solution

In the last decade, fractional calculus has opened up new vistas of research and brought a revolution in the study of fractional PDE’s and ODE’s [3638]. Fractional kinetic equation has been successfully used to predict physical phenomena such as diffusion in permeable media, reactions and unwinding forms in complicated framework. The fractional form of the kinetic equation has gained attention due to the its relationship with the CTRW-theory [39]. This section is dedicated to investigating a new weighted fractional kinetic equations to explain the continuity of the motion of the material and the fundamental equations of natural sciences. The series solution of this new fractional kinetic equation by applying weighted generalized fractional laplace is also part of this section. The fractional kinetic equation is

b(Υ,ksDa+,ρζN)(t)N0Ω(t)=c(Υ,ksJa+,ρηN)(t),ΩL1[a,)(17)

subject to

(Υ,ksJa+,ρnkζN)(a+)=d,d0,(18)

where a,ζ0, b,cR(b0), k>0, n=[ζk]=1.

Theorem 5.1. The solution of (17) with initial condition (18) is

N(t)=dρ1(t)m=0(cb)m(s+1)(ζk)(m+1)+mηkΓk((ζ+η)m+ζ)(Υs+1(α)Υs+1(a))(ζ+η)m+ζk1+N0bm=0(cb)m(s+1)(m+1)(Υ,ksJa+,ρ(ζ+η)m+ζΩ)(t).

Proof. By applying the modified weighted Laplace transform on both side of (17), we get

bLΥρ{(Υ,ksDa+,ρζN)(t)}(u)LΥρ{N0Ω(t)}(u)=cLΥρ{(Υ,ksJa+,ρηN)(t)}(u).

Using Theorems 3.1 and 3.2, we get

b(s+1)kζk(ku)ζkLΥρ{N(t)}(u)k(Υ,ksJa+,ρkζN)(a)N0LΥρ{Ω(t)}(u)=c(s+1)ζk(uk)ζkLΥρ{N(t)}(u)[bc(s+1)ζk+ηk(ku)ζ+ηk(s+1)ζkk(ku)ζk]LΥρ{N(t)}=bkd+N0LΥρ{Ω(t)}(u)LΥρ{N(t)}=bkd[(s+1)ζkk(ku)ζkbc(s+1)ζk+ηk(ku)ζ+ηk]+[(s+1)ζkk(ku)ζkbc(s+1)ζk+ηk(ku)ζ+ηk]×N0LΥρ{Ω(t)}(u).

Taking |cb(s+1)ζk+ηk(ku)ζ+ηk|1, we get

LΥρ{N(t)}=[kd[(s+1)ζkk(ku)ζk]+a1N0[(s+1)ζkk(ku)ζk]]×m=0(cb)m(s+1)(ζk+η)mk(ku)(ζ+η)mkLΥρ{Ω(t)}(u)=kd[(s+1)ζkk(ku)ζk]m=0(cb)m(s+1)(ζk+η)mk(ku)(ζ+η)mk+b1N0[(s+1)ζkk(ku)ζk]×m=0(cb)m(s+1)(ζk+η)mk(ku)(ζ+η)mkLΥρ{Ω(t)}(u)=kdm=0(cb)m(s+1)(ζk)(m+1)+mηk(ku)(ζ+η)m+ζk+N0bm=0(cb)m(s+1)(ζk)(m+1)+mηk(ku)(ζ+η)m+ζkLΥρ{Ω(t)}(u)=kdm=0(cb)m(s+1)(ζk)(m+1)+mηk(ku)(ζ+η)m+ζk+N0bm=0(cb)m(s+1)(ζ+η)m+ζk(s+1)(m+1)(ku)(ζ+η)m+ζkLΥρ{Ω(t)}(u).

By applying the inverse Laplace transform, we get

N(t)=dρ1(t)m=0(cb)m(s+1)(ζk)(m+1)+mηkΓk((ζ+η)m+ζ)(Υs+1(α)Υs+1(a))(ζ+η)m+ζk1+N0bm=0(cb)m(s+1)(m+1)(Υ,ksJa+,ρ(ζ+η)m+ζΩ)(t).

Next, we include an example in the field of engineering using our defined operators.

Example 5.1. Consider a famous growth model given by

Υ,ksDa+,ρζN(t)N(t)=0(19)

subject to the condition

Υ,ksJa+,ρnkζN(0)=d0,(20)

where ζ0, k>0, n=[ζk]=1. The solution to the growth model (19) is

N(t)=d0ρ1(t)m=0(s+1)(ζk)(m+1)+mkΓk((ζ)m+ζ)(Υs+1(α)Υs+1(a))(m+1)ζk1.(21)

Solution 5.1. By choosing b=c=1N0=0,η=0 in (17) and a=0,d=d0 in (18), we obtain the growth model with solution (21). Further with the choice of parameters k=1,s=0,ζ=1.5,d0=1,ρ(t)=1 and Υ(α)=α, we get

N(α)=m=0(α)1.5m+0.5Γ((1+m)1.5).

The graph of the function N(α) is presented as follows:

images

Figure 2: For 0 < α < 1, the graph in Fig. 2 indicates the increasing and convergent behaviour of the infinite series

6  Conclusion

In this paper, the weighted generalized fractional integral and derivative operators of Riemann-type are investigated. We discuss some properties of the fractional operators in certain spaces. Specifically, the semi-group and inverse properties are proved for the introduced operators. The modified weighted Laplace transform of novel operators is also examined which is compatible with the introduced operators. It is worth mentioning that many established operators unify some operators that exist in literature. Finally, the solutions of the weighted generalized fractional free electron laser and kinetic equations are obtained by utilizing the skillful technique of the weighted Laplace transform, which has been applied in many mathematical and physical problems. Furthermore, a Cauchy problem and a growth model for a specific choice of parameters involved are designed and sketched in their graphs to check the validity.

Acknowledgement: The authors T. Abdeljawad and K. Shah would like to thank Prince Sultan University for supporting through TAS research lab.

Funding Statement: The authors are thankful to Prince Sultan University for paying the article processing charges.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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Cite This Article

APA Style
Samraiz, M., Umer, M., Abdeljawad, T., Naheed, S., Rahman, G. et al. (2023). On riemann-type weighted fractional operators and solutions to cauchy problems. Computer Modeling in Engineering & Sciences, 136(1), 901-919. https://doi.org/10.32604/cmes.2023.024029
Vancouver Style
Samraiz M, Umer M, Abdeljawad T, Naheed S, Rahman G, Shah K. On riemann-type weighted fractional operators and solutions to cauchy problems. Comput Model Eng Sci. 2023;136(1):901-919 https://doi.org/10.32604/cmes.2023.024029
IEEE Style
M. Samraiz, M. Umer, T. Abdeljawad, S. Naheed, G. Rahman, and K. Shah "On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems," Comput. Model. Eng. Sci., vol. 136, no. 1, pp. 901-919. 2023. https://doi.org/10.32604/cmes.2023.024029


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