Shehu Transform of Hilfer-Prabhakar Fractional Derivatives and Applications on some Cauchy Type Problems

In this paper, we are interested on the Shehu transform of both Prabhakar and Hilfer Prabhakar fractional derivative and its regularized version. These results are presented in terms of Mittag-Le er type function and also utilized to obtain the solutions of some Cauchy type problems, such as Space-time Fractional Advection-Dispersion equation and Generalized fractional Free Electron Laser (FEL) equation, at which Hilfer-Prabhakar fractional derivative of fractional order and its regularized version are involved.

These denitions, although they do not always lead to identical results, they are equivalent for a wide range of functions. Among them, we have, in particular, Riemann-Liouville integral, Riemann-Liouville fractional derivative, Caputo fractional derivatives, the Prabhakar integral, the Hilfer-Prabhakar fractional derivative, ..., etc. We refer the reader to check e.g. [23,11,10,8,19,14].
Many researchers use the Hilfer-Prabhakar fractional derivative operator for the purpose of modelling some physical aspects due to its specic properties, especially the combination with several integral transforms founded in the literature of fractional dierentiations and integrations like Laplace, Fourier, Sumudu, Elzaki, ..., etc. In [10], the Laplace transformation of Hilfer-Prabhakar and its regularized version is considered, where the authors have also implemented their results in classical equations of mathematical physics such as heat, free electron laser equations and homogeneous Poisson process while in [21] W. Panchal et al. applied its Sumudu transform to some non-homogeneous Cauchy type problems. Indeed, V. Gill et al. in [12] derived the analytical solution of generalized space-time fractional advection-dispersion equation by coupling the Sumudu and Fourier transforms, associated with the Hilfer-Prabhakar fractional derivative. Recently, in [26] the authors use found the Elzaki transform of Hilfer-Prabhakar fractional derivative and its regularized version and gathering these results for solving free electron laser type integro-dierential equation.
On the other hand, the Shehu transform is a new integral transform which is introduced at the rst time in 2019 by Maitama and Zhao [25]. In fact, Shehu transform is a generalization of the Laplace and the Sumudu integral transforms and has some good features. In this context, the main objective of this paper is to nd the Shehu transform of: Prabhakar fractional derivative, HilferPrabhakar derivative and theirs regularized version. These results are presented in terms of Mittag-Leer type function and employed to nd the solutions of some Cauchy type problems such as space-time fractional advection-dispersion equation and generalized fractional free electron laser (FEL) equation, at which Hilfer-Prabhakar fractional derivative of fractional order and its regularized version are involved.

Preliminaries and notations
In this section, we study some important basic denition related to fractional calculus which are used in the sequel. Denition 2.1 (RiemannLiouville integral [8,19,3]). Let a, b ∈ R such that a < b and f ∈ L 1 (a, b) .
Denition 2.6 ([23]). The three parameter M-L function, also called Prabhakar function is given by In applications it is usually used a further generalization of (9) which is given by where ω ∈ C is a parameter and t > 0 the independent real variable.

Fundamental Facts of the Shehu Transform
The Sumudu transform ( [2], [4]) is obtained over the set of functions by Shehu transform of function f (t) is recently introduced by Shehu Maitama and Weidong Zhao [25] and it is a generalization of the Laplace and the Sumudu integral transforms.
The Shehu transform is obtained over the set A is dened as where s and u are the Shehu transform variables and a is a real constant .
Obviously, the Shehu transform is linear as the Laplace and Sumudu transformations. Inversion formula of (20), is given by Theorem 3.1. [25] (Derivative of Shehu transform). Let f (t) ∈ A. If the function f (n) (t) is the n th derivative of the function f (t) with respect to t, then for n ≥ 1 its Shehu transform is dened by Where V (s, u) denotes the Shehu transform of f (t).
Proposition 3.1. [28] Shehu transform of n th order partial derivative is dened as Where V (x, s, u) denotes the shehu transform of the partial derivative of the function u (x, t).
In the next theorem, we nd relation between Sumudu and shehu transform.

Main Result
In this section, we nd the Shehu transforms of: Prabhakar fractional derivative, regularized version of Prabhakar fractional derivative, Hilfer-Prabhakar fractional derivative and its regularized version. in the following, let f (t) ∈ A with Shehu transform V (s, u) .

Lemma 4.2. The Shehu transform of regularized version of Prabhakar fractional derivative
Proof. We follow the same method used in the previous proof, so, taking Shehu transforms of Prabhakar fractional derivative (15) with respect to variable t and using (22), (27) and (25), we get This is the desired result (29).

Applications
In this section, we will provide some applications of HilferPrabhakar derivatives using Shehu transform to nd the solutions of Cauchy problems such as space-time fractional advection-dispersion equation and generalized fractional Free Electron Laser (FEL) equation [12,10].

Generalized Space-time Fractional Advection-Dispersion Equation
Here we nd, the solution of the generalized space-time advection-dispersion equation (32) under the initial condition (33) and the boundary condition(33).
Proof. First, applying the Fourier transform of equation (32) with respect to the space variable x, we can write where u (k, t) represent Fourier transform of u(x, t) and the Fourier transform of λ 2 is given in [7], as Then, taking the Shehu transform on left sided of the above equation (36) with respect to the space variable t and by using (30), we obtain where V (k, s, u) represents Shehu transform of u (k, t) .
Again, apply Shehu transform on right hand side of the equation (36) and using the initial condition (33), we get after some simplications ,we can write so, it gives Now, taking inverse Shehu transform of equation (41) using (27), we have Again, taking inverse Fourier transform of (42), get our required result (35).
Example 5.1. If η = 0, ξ = ih 2m in above theorem 5.1, the solution of the resulting equation called one dimensional space-time Schrödinger equation of fractional order, for a free nature particle of mass m with h Planck constant, is Where λ, x, t, β, ν and λ 2 are the same as we identied previously.
The analytical expression of solute concentration of the resulting Cauchy type problem dened by equation (32) subject to constraints to (33) and (33), is where µ = d v L , L is the packing length, d is the dispersion coecient and v is the Darcy velocity .

Fractional Free Electron Laser (FEL) equation
Here we study the following fractional generalization of the FEL equation, involving HilferPrabhakar derivatives.
Finally, by inverting the Snehu transform to (55) we obtain the required solution (52).

Conclusion
In this work, we present the Shehu transform of Hilfer-Prabhakar fractional derivative and its regularized version. We also present some its application of Cauchy type problems such as Space-time Fractional Convection-dispersion Equation and Generalized fractional Free Electron Laser (FEL) equation using the results of the third and the fourth section. The results shows that Shehu transform is very useful for solving fractional dierential equations.