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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:

(This article belongs to this 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.


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.


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


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


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


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<


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


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


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


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


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


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


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




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


Proof. For 1p<, we have


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


By using Minkowski’s inequality, we have


Applying Hölder’s inequality, we get


where 1p+1q=1.


For p=, we obtain


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.


where ζ,k>0 and nkζ>0.

Proof. Consider


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


which gives


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


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


where ζ,η,k>0.

Proof. By using Definition 2.2, we have


By using Theorem 2.2, we have


which implies


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


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}


for all ζ,η>0 and α[a,b].

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



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


This completes the proof.

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


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

Proof. By Definition 2.1, we get


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


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).


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


holds for all values of u.

Proposition 3.1.


Proof. By the Definition 3.1, we have


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


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


where k>0, ρ(α)0, sR{1}.

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


This completes the proof.

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


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


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



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


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


The above equation implies that


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


By using the inverse Laplace transform, we obtain


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




subject to the condition


with α(0,), a0, ρ0 and λR is given by


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




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


subject to


where a,ζ0, b,cR(b0), k>0, n=[ζk]=1.

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


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


Using Theorems 3.1 and 3.2, we get


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


By applying the inverse Laplace transform, we get


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

Example 5.1. Consider a famous growth model given by


subject to the condition


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


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


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


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

Samraiz, M., Umer, M., Abdeljawad, T., Naheed, S., Rahman, G. et al. (2023). On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems. CMES-Computer Modeling in Engineering & Sciences, 136(1), 901–919.

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