Triple Shehu transform and its properties with applications

In the current paper, the concept of one-dimensional Shehu Transform have been generalized into three-dimensional Shehu Transform namely, Triple Shehu Transform (TRHT). Further, some main properties, several theorems and properties related to the TRHT have been established. Triple Shehu transform was used in solving fractional partial differential equations, with the fractional derivative described in Caputo sense. The proposed scheme finds the solution without any discretization, transformation or restrictive assumptions. Several examples are given to check the reliability and efficiency of the proposed technique.


INTRODUCTION
integral transforms is one of the most easy and effective methods for solving problems arising in Mathematical Physics, Applied Mathematics and Engineering Science which are defined by differential equations, difference equations and integral equations. The main idea in the application of the method is to transform the unknown function of some variable t to a different function of a complex variable. With this, the associated differential equation can be directly reduced to either a differential equation of lower dimension or an algebraic equation in the new variable. There are several forms of integral transforms such as Laplace transform (Papoulis, 1957Debnath and Bhatta, 2014, Rehman et al., 2014and Dhunde et al., 2013, Sumudu transform (Kılıcman and Gadain, 2010;Mahdy et al., 2015 andMahdy et al., 2015a), Eltayeb and Kilicman, 2010, and Mechee and Naeemah, 2020. Aboodh transform (Aboodh, 2013, Elzaki transform (Elzaki, 2011), Variational homotopy perturbation method (Mahdy et al., 2015b), Alternative variational iteration method  and one form may be obtained from the other by a transformation of the coordinates and the functions. Recently, in 2019 Maitama and Zhao introduced a new type of integral transform as a generalization of both Laplace transform and Sumudu transform for solving differential equations in the time domain, and provided some theorems on this transform. The study was further reinforced by Issa and Mensah (2020), Alfaqeih and Misirli (2020), Aggarwal et al. (2019), Mahdy and Mtawal (2016), Mtawal and Alkaleeli (2020). In this paper, we extend and generalized some results of Thakur et al. (2018), ‫‬ Abdon (2013) and Alfaqeih (2019); in particular, we extend one-dimensional Shehu transform into three-dimensional Shehu transform, and provide some examples to show the effectiveness of our results.

PRELIMINARIES
Here, we recall the definitions of Shehu and Double Shehu transform.

Definition 1
The Shehu Transform ( ) (Maitama and Zhao, 2019) is defined over the set of the functions And the inverse Shehu transform is defined by (Podlubny, 1999;He, 2014).

Definition 3
The left-sided Riemann-Liouville fractional integral of order of a function , 1, Podlubny, 1999;He, 2014) is defined as: Here ( ) is the gamma function.
n m f C n N  The left Caputo fractional derivative of f in the Caputo sense (Podlubny, 1999;He, 2014) is defined as follows:

Definition 5
The Mittag-Leffler function , (Kilbas et al., 2004) are defined as These functions are generalization of the exponential function. Some special cases of the Mittag-Leffler function are as follows:

Definition 6
The single Shehu Transform ( ) of a function ( ) with respect to the variables x, y and z respectively (Maitama and Zhao, 2019) are defined by:

Definition 8
The double Shehu Transform ( 2 H ) of a function ( ) with respect to xy, xz and yz respectively (Alfaqeih and Misirli, 2020), are defined by:

RESULTS
Here, we introduce the definition of Triple Shehu transform and Triple Shehu transform of partial and fractional derivatives which are used further in this paper; moreover, we apply Triple Shehu transform for some basic functions.

Definition 9
Let be a continuous function of three variables; then, the Triple Shehu transform (TRST) of ( , , z) is defined by Where are Shehu variables, provided the integral exists.
Also, the inverse Triple Shehu transform is defined by

Existence and uniqueness of TRST
Here, we debate the existence and uniqueness of the Triple Shehu transform and prove it.

Definition 10
A function   ,, f x y z is said to be of exponential order x y z ).
Or, equivalently, This proves the uniqueness of the TRHT.

TRHT of some elementary functions
( x .. , n xyz n uvk n sqr

TRST of derivative of a function of three variables
1) The TRST of mixed derivative of a function of three variables is given by: 2) The TRST of nth partial derivative of a function of three variables is given by 3) The TRST of the partial fractional Caputo derivatives of a function of three variables is given by: Shehu transform method is an appropriate method for solving the fractional partial differential equations. As a new work, it will be interesting to extend known results on a Triple Laplace transform, Triple Aboodh transform, etc, to our results on a Triple Shehu transform. Finally, based on the mathematical formulations, simplicity and the findings of the proposed Triple Shehu transform, we conclude that it is highly efficient.