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.

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 [

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 [

Motivated by the recent studies presented in [

In the beginning, we recall some related definitions and notions. The integral form of the

Note that:

Jarad et al. [

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

In this section, we introduce the generalized weighted

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

if we set

If we set

If we set

If we set

If we set

For

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

There are many other fractional derivative operators as special cases of the operator

If we choose

If we choose

If we choose

If we set

If we set

In the following definition, we define the space where the generalized weighted

Note that

Substituting

By using Minkowski’s inequality, we have

Applying Hölder’s inequality, we get

For

Hence the proof is done.

Substitute

This proved the inverse property.

By using Theorem 2.2, we have

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

Substitute

This completes the proof.

Substitute

The proof is done.

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.

Substitute

the proof is done.

This completes the proof.

The proof is completed.

In this section, we investigate the fractional generalization FEL by using the introduced fractional integral given in

The above equation implies that

Taking

By using the inverse Laplace transform, we obtain

The following is the cauchy problem based on Theorem 4.1.

Consider

Using

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 [

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.

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.

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