A New Integral Transform: ARA Transform and Its Properties and Applications

: In this paper, we introduce a new type of integral transforms, called the ARA integral transform that is defined as: We prove some properties of ARA transform and give some examples. Also, some applications of the ARA transform are given.


Introduction
The integral transforms play a vital role in finding solutions to initial value problems and initial boundary value problems. An integral transform [1] has the form where the input function of the transform is ( ) and the output is [ ( )]( ), and the function ( , ) is a kernel function. Moreover, the inverse transform related to the inverse kernel function is given by: The integral transform was introduced by the French mathematician and physicist P.S. Laplace [2,3] in 1780. In 1822, J. Fourier [4] introduced the Fourier transform. Laplace and Fourier transforms form the foundation of operational analysis, a branch of mathematics that has very powerful applications, not only in applied mathematics but also in other branches of science like physics, engineering, astronomy, etc.
In recent years, mathematicians have been interested in developing and establishing new integral transforms. In 1993, Watugula [5] introduced the Sumudu transform. The natural transform was introduced by Khan and Khan [6] in 2008. In 2011, the Elzaki transform [7] was devised by Elzaki. Atangana and Kiliçman [8] in 2013, introduced the Novel transform. In 2015, Srivastava, Luo and Raina [9] introduced the M-transform. In 2016, many transforms were introduced, like the ZZ transform by Zafar [10], Ramadan Group (RG) transform [11], a polynomial transform by Barnes [12], also, a new integral transform was presented by Yang [13]. In the year 2017, other transforms were introduced, such as the Aboodh transform [14] and Rangaig transform [15], while the Shehu transform [16] was established in 2019, by Shehu and Weidong.
In this paper, we proclaim a new integral transform called the ARA integral transform. This transform is a powerful and versatile generalization that unifies some variants of the classical Laplace transform, namely, the Sumudu transform, the Elzaki transform, the Natural transform, the Yang transform, and the Shehu transform.
In section 2, we state our definition of the ARA transform and some related theorems. In section 3, we provide the properties of the ARA transform, and in the last section, we give some applications.

Definitions and Theorems
Definition 1. The ARA integral transform of order of the continuous function ( ) on the interval (0, ∞) is defined as:

Definition 2. The inverse of the ARA transform is given by
where In fact, from the definition of ARA transform of a function ( ), we have It follows that: .
then ARA transform exists for all > .
Proof of Theorem 1. We have since the function ( ) is piecewise continuous then the first integral on the right side exists. Also, the second integral on the right side converges because:

Dualities between ARA Transform and Some Integral Transform:
Duality between ARA and Laplace Transforms [17]: i: where iii: where Proof: Relations i and ii are obvious. Here, we prove relation iii.
and for relation iv, the Laplace transform for the function does not exist, while: The duality between ARA and Laplace Carson transforms [18] Duality between ARA and Aboodh Transforms [19] [ Duality between ARA and Mohand Transforms [20]:

Properties of ARA Transform
In this section, we establish some properties of the ARA transform, which enable us to calculate further transform of functions in applications.
Proof of Property 2: Using the definition of ARA transform for ( ), we get and a substitution of = in equation (5) Proof of Property 6: Moreover: Using the properties of the convolution of Laplace transform, we get the following property. □        We present a list of ARA transform of some special functions and General properties of the ARA transform in (Appendix Table A1).

Applications of the ARA Transform
In this section, we give some applications of ordinary differential equations, in which the efficiency and high accuracy of ARA transform are illustrated.
Example 5.2: Consider the initial value problem: Solution: Applying ARA transform on both side of equation (19) [ Taking the inverse ARA transform we get: ( ) = . Taking the inverse ARA transform and using the second initial condition (0) = 1, we get: ( ) = cos + sin . Taking the inverse ARA transform and using second initial condition (0) = −3, we get: Example 5.5: Solution: Applying ARA transform on both side of the differential equation (22) Taking the inverse ARA transform and using the second initial condition (0) = 1 we get:  Taking the inverse ARA transform : ( , ) = sin(2 ).

Example 5.7
Consider the initial boundary value problem: Using boundary conditions, we get: Taking the inverse ARA transform we get the solution: ( , ) = sin( ) (1 − cos( )).

Conclusions
In this paper, we introduced a new integral operator transform called the ARA transform. We presented its existence and inverse transform. We presented some properties and their application in the solving of ordinary and partial differential equations that arise in some branches of science like physics, engineering, etc.

Acknowledgments:
The authors express their gratitude to the dear unknown referees and the editor for their helpful suggestions, which improved the final version of this paper.

Conflicts of Interest:
The authors declare no conflict of interest.