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

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 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

**Received** 22 May 2022; **Accepted** 02 September 2022; **Issue published** 05 January 2023

## 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

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

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:

Definition 1.2. For

where the

Definition 1.3. [29] Suppose that the

where

Definition 1.4. [29] Let

where

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

Definition 1.5. Let

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

Theorem 1.1. [20] If

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

Theorem 1.2. Let

Definition 1.6. [20] The generalization of the weighted convolution of

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

where

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

I. if we set

II. If we set

III. If we set

IV. If we set

V. If we set

VI. For

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

Definition 2.2. Let

where

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

I. If we choose

II. If we choose

III. If we choose

IV. If we set

V. If we set

VI. (3) reduces to k-Hadamard fractional derivative for

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

Note that

Theorem 2.1. Let

Proof. For

Substituting

By using Minkowski’s inequality, we have

Applying Hölder’s inequality, we get

where

For

Hence the proof is done.

Theorem 2.2. Let

where

Proof. Consider

Substitute

which gives

This proved the inverse property.

Corollary 2.1. Let the function

where

Corollary 2.2. Let the function

where

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

where

Theorem 2.3. Let the function

for all

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

Substitute

This completes the proof.

Theorem 2.4. Let

where

Proof. By Definition 2.1, we get

Substitute

The proof is done.

Example 2.1. Corresponding to the choice of the parameters

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

holds for all values of u.

Proposition 3.1.

Proof. By the Definition 3.1, we have

Substitute

the proof is done.

Theorem 3.1. Let the function

where

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

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

The above equation implies that

Taking

By using the inverse Laplace transform, we obtain

the result is completed.

Remark 4.1. If we set

The following is the cauchy problem based on Theorem 4.1.

Example 4.1. The solution of the cauchy problem

where

subject to the condition

with

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

Consider

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 [36–38]. 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

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

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

Solution 5.1. By choosing

The graph of the function

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**

*Computer Modeling in Engineering & Sciences*,

*136*

*(1)*, 901-919. https://doi.org/10.32604/cmes.2023.024029

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**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 136, no. 1, pp. 901-919. 2023. https://doi.org/10.32604/cmes.2023.024029