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 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. https://doi.org/10.32604/cmes.2023.024029
Received 22 May 2022; Accepted 02 September 2022; Issue published 05 January 2023
AbstractIn 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.
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  and Atangana et al.  investigated the fractional derivative with non-local and non-singular kernel. In  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 -Laplacian operators are briefly discussed in [18,19]. Currently, Jarad et al.  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  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  are defined as follows:
Definition 1.1. The k-gamma function is defined by
Note that: and
Definition 1.2. For and , the k-beta function is defined as
where the and functions are related with an identity
Definition 1.3.  Suppose that the be a continuous function on interval [a,b]. Then weighted (k, s)-RLF integral of order is given by
where , and .
Definition 1.4.  Let be a continuous function on and , with , , and . Then for all
where is a weighted (k, s)-RLF integral.
Jarad et al.  defined the generalized weighted Laplace transform as follows:
Definition 1.5. Let , be functions with values in Furthermore, is continuous and on The weighted generalized Laplace transform of is given by
and is true for all values of u for which (1) exists.
Theorem 1.1.  If and of weighted -exponential order. Suppose that the be a piecewise continuous function on every interval [a, T], then the weighted generalized Laplace transform of exists and
The generalized form of Theorem 1.1 is stated in the next result.
Theorem 1.2. Let , such that , , 1, 2, …, n−1 are of weighted -exponential order. If is a continuous function on all intervals [a, T], the weighted generalized Laplace transform of exists and
Definition 1.6.  The generalization of the weighted convolution of and is defined by
In this section, we introduce the generalized weighted (k, s)-RLF operators and describe some of their features.
Definition 2.1. Suppose that the be a continuous function on the finite real interval [a,b] and is strictly increasing function. Then the generalized weighted (k, s)-RLF integral of order is defined by
where , and .
The integral operator defined in 2 cover many fractional integral operators. For instance,
The corresponding weighted generalized fractional derivative is defined by the following definition.
Definition 2.2. Let be continuous function on and , , , and . Then for all the inverse derivative operator of integral operator 2 is defined by
where is a generalized weighted (k, s)-RLF integral.
There are many other fractional derivative operators as special cases of the operator (3).
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 , be the space of all Lebesgue measurable functions for which , where
Note that for and
Theorem 2.1. Let , , and . Then is bounded in and
Proof. For , we have
Substituting and on the right side of (4), we obtain
By using Minkowski’s inequality, we have
Applying Hölder’s inequality, we get
For , we obtain
Hence the proof is done.
Theorem 2.2. Let be continuous on and and , . Then for all .
where and .
Substitute on the right side of (5), we get
This proved the inverse property.
Corollary 2.1. Let the function be continuous on and and , , . Then for all
Corollary 2.2. Let the function be continuous on and , , , and . Then for all
Proof. By using Definition 2.2, we have
By using Theorem 2.2, we have
Hence the semi-group property of new derivative operator is proved.
Corollary 2.3. Suppose that the be a continuous function on and , and . Then for all
where , and .
Theorem 2.3. Let the function be continuous on [a,b] and , and
for all and .
Proof. By utilizing the Definition 2.1 and Dirichlet’s formula, we get
Substitute on the right side of (6), we obtain
This completes the proof.
Theorem 2.4. Let , , , and . Then we have
where represents the k-Gamma function.
Proof. By Definition 2.1, we get
Substitute on the right side of (7), we get
The proof is done.
Example 2.1. Corresponding to the choice of the parameters and we get the following graphs with different choices of the function .
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 be a real valued function defined on and . The weighted generalized Laplace transform of is given by
holds for all values of u.
Proof. By the Definition 3.1, we have
Substitute on the right side of (8), we get
the proof is done.
Theorem 3.1. Let the function be continuous on each interval and of weighted -exponential order. Then
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.
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. .
Theorem 4.1. The solution of the cauchy problem
where , , , and 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 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 , , , and , in 9 and 10, then the original free electron laser equation given in  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 , and is given by
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 . 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
where , , , .
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 , 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 , , . The solution to the growth model (19) is
The graph of the function is presented as follows:
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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