ON DOUBLE SHEHU TRANSFORM AND ITS PROPERTIES WITH APPLICATIONS

In the current paper, we have generalized the concept of one dimensional Shehu transform into two dimensional Shehu transform namely, double Shehu transform (DHT). Further, we have established some main properties and theorems related to the (DHT). To show the efficiency, high accuracy and applicability of the proposed transform, we have implemented the new transform to solve integral equations and partial differential equations.


Introduction
Many problems in the fields of most applied science and engineering encounter double integral equations or partial differential equations describing the physical phenomena [20][21][22]. Solving such equations using single transforms is more difficult than using the double transforms. In the past few decades, integral transforms have been the focus of many authors due to their huge appearance in various applications in modern sciences and engineering, in the current years, a very extensive literature on integral transforms of a function of one but there is a little work available on double integral transform. Consequently, great attention has been given to deal with the double integral transform.
Among the solution methods, integral transform methods are rather and popular, Hence in the literature, there are many different types of integral transforms such as Fourier transform [1] Laplace [2,3] transform, Sumudu transform [4][5][6]19], Natural transform [7], Ezaki transform [8], and so on. These kinds of transforms have a wide variety of applications in various areas in applied physics, applied mathematics, statistics, engineering and in most of other sciences [9,10].
Shehu transform (HT) of single variable [11], is a new transform which was recently introduced by Shehu Maitama and Weidong Zhao in 2019. (HT) is a generalization of Laplace and Sumudu transforms. This transform is used to solve both ordinary and partial differential equations. After the appearance of the (HT), many authors applied this transform to solve partial differential equations including ordinary and fractional, for instance, see [12][13][14][15]20].
The main objective of this paper is to extend the one dimensional Shehu integral transform to two dimensional Shehu integral transform namely double Shehu transform and to get the solution of initial and boundary value problems in different areas of real life science and engineering.
This work is organized as follows. In section 2, we present some notations about Laplace, sumudu, and single Shehu transforms. We then introduce the definition of double Shehu transform and its inverse with examples, and we prove the existence and uniqueness theorems of the new transform in section 3. In section 4, we discuss some properties and theorems related to the double Shehu transform. in section 5, we implement the double Shehu transform method to some examples of double integral equations with convolution and partial differential equations. Section 6 is for the conclusions of this paper.

Preliminaries
In this section, we present some basic notations about the Laplace, Sumudu and the Shehu transforms.
Definition 2.1.Let g : (0, ∞) → be a real valued function. The single Laplace transform of g is defined by: (2.1) Definition 2.2. [16] Let g(x, t), x, t > 0 be a real valued function. The double Laplace transform of g is defined by: Definition 2.5. [11] The single Shehu transforms (ST) of a real valued function g(x, t) with respect to the variables x and t respectively, are defined by:

Double Shehu transforms (DHT)
In this section, we introduce the definitions of (DHT) and the inverse of (DHT).
Definition 3.1. The (DHT) of the function g(x, t) is defined by the double integral as : on the set of functions provided that the integral exist.
3.1. Existence and uniqueness of (DHT). Proof. Using definition of DHT, we have .
Let h(x, t)and l(x, t) be continuous functions and having the double Shehu Proof. Assume α, and γ to be sufficiently large, then since and the theorem is established.
As a consequence of property (3) Similarly, .

Properties of the Double Shehu Transform
(1) The double Shehu transform H 2 xt (.) is a linear operator, that is Proof.
(2) Changing of scale property Proof. Using the definition of (DHT), we deduce Substituting y = ax, and z = bt in equation (4.1), we get (3) Shifting property Proof. Using the definition of (DHT), we deduce Theorem 4.1. If H 2 xt (g (x, t)) = G ((p, q), (u, v)) , Then where H (x, t) is the Heaviside unit step function defined as follows : x > a, t > b 0 otherwise Proof : By using the definition of (DHT), we have (4) The double Shehu transform of the first and the second order partial derivatives with respect to x Similarly, with respect to t Moreover, the double Shehu transform for the mixed double order partial derivative of function of two variables is given by Theorem 4.2. The double Shehu transforms of the, n, m ∈ N times partial derivatives ∂ n g ∂x n , ∂ m g ∂t m , of the function g(x, t) are given by : Proof. the proof can be done by mathematical induction. Theorem 4.3. If the double Shehu transform of ∂ n g ∂x n , ∂ m g ∂t m is given by equations (4.2) then the double Shehu transforms of ∂x n , and t m ∂ m g (x, t) ∂t m are given by , where m, n = 1, 2, 3, ... Proof. By using the definition of (DST), we get taking the n th partial derivative w.r.t p for both sides of equation (4.5), we have simplifying, we obtain (−1) n u n ∂ n ∂p n H 2 xt (g (x, t)) = H 2 xt (x n g (x, t)) .
where L 2 xt denotes the (DLT) of the function g(x, t).
Proof. The proof can be done directly from the definition of (DHT) that is where S 2 xt denotes the (DST) of the function g(x, t). Proof. By letting φ = px u , ϕ = qt v in the (DHT) formula, we get Theorem 4.6. Let g (x, t) , and h (x, t) be of exponential order, having double Shehu transforms , t)) , and H 2 xt (h (x, t)), respectively. The the double Shehu transform of the convolution of g (x, t) and h (x, t)

5.
Application of (DHT) method to integral and partial differential equations In this section, we illustrate the applicability of the (DHT) method, by constructing some examples.

The single Shehu transform of the initial conditions
. (5.4) Substituting equation (5.4) into equation (5.3), we get Simplifying, we get , (5.6) taking the inverse double Shehu transform of equation (5.6) Example 5.2. Consider the following partial double integro-differential equation.
with respect to the initial conditions g(x, 0) = e x , ∂ ∂x g(x, 0) = e x , g(0, t) = e t , ∂ ∂t g(0, t) = e t . (5.9) Taking the (DHT) to both side of equation (5.8), we get substituting the single Shehu transform of initial and boundary conditions, in equation (5.10) we get simplifying, we get , (5.12) taking the inverse double Shehu transform of equation (5.12) Example 5.3. Consider the following partial integro-differential equation.
with respect to the initial conditions Taking the (DHT) to both side of equation (5.15), we get pq uv G ((p, q), (u, v)) − q v H t (g (0, t)) − p u H x (g (x, 0)) + g (0, 0) + (5.16) substituting the single Shehu transform of initial and boundary conditions, Example 5.4. Consider the following Partial differential Telegraph equation with respect to the initial and boundary conditions g (x, 0) = x 2 , ∂ ∂t g (x, 0) = 1, g (0, t ) = t, ∂ ∂x g (0, t ) = 0. Example 5.6. Consider the partial differential Euler -Bernoulli equation other methods to solve non-linear (PDEs) arising in applied mathematics, applied physics and engineering, which will be discussed in subsequent articles.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.