New Transform Fundamental Properties and Its Applications

In this paper, new transform with fundamental properties are presented. The new transform has many interesting properties and applications which make it rival to other transforms. Furthermore, we generalize all existing differentiation, integration, and convolution theorems in the existing literature. New results and new shifting theorems are introduced. Finally, comprehensive list of this transforms of functions will be providing.


Introduction
Recently, integral transformations played an important role in many fields of science and engineering [1][2][3][4], especially mathematical physics [5], optics [6], engineering mathematics [7,8], Cryptography [9], image processing [10] and, few others, because they have been successfully used in solving many problems in those fields. The possibility of solving a problem is required transforming the problem from a space to another space where the solving is possible or easy. As well as the possibility of decreasing the independent variables in some problems. Many of these transforms have been introduced which were extensively used and applied on theory and applications, such as Laplace [11,12], Fourier [2], Sumudu [13,14], Elzaki [15][16][17], Aboodh [18], Natural, and ZZ Transforms [19]. Among these the most widely used is Laplace transform.Here, new integral transform is proposed to avoid the complexity of previous transforms.

Definition of New Transform
The transform of a function f(t)is defined by Where u is a real number,for those values of u which the improper integral converges.

Lemma 1
The improper integral ∫ converges only for a > 0. Proof In the case that a = 0, the improper integral is diverging, since we are computing the area of a rectangle with sides equal to one and infinity. In the case a < 0, holds (t-a) e -au /u u>0 ln(at) , a>0 ln(a/u)-γ u>0 In the next section, we discuss the existence of the new transform Existence of the New Transform Does the new transform always exist?It can be shown that For every real number u. Hence, the function does not have a transform.
In this section, we will establish the conditions that ensure the existence of the new transform of a function. We first review some relevant definitions from calculus,and for simplicity, we take the following coding and equivalent to ( ) ( ) ( )

Definition 3 (Jump discontinuity)
If f(t 0 +)  ( )then we say that f has a jump discontinuity at t 0 , and the value f(t 0 +) ( ) is called the jump in f at t 0 . Definition 4 (Removable discontinuity) If the limit of a function f at a point t 0 exists, but either f is not defined at t 0 or it's defined but ( ) ( ) Then, we say that f has a removable discontinuity at t 0 .

Theorem 1(Sufficient conditions).
If a function f is piecewise continuous on[0,)and of exponential order s 0 , then the transform of f exists for all u> s 0 .

Proof
To proof the existences of the transform of f, we must show ̅ ( )is well-defined, that is, finite, by bounding its absolute value by a finite number.
Since fof exponential order s 0 , then there exist M> 0 such that | ( )| , ) , then (4) becomes If the transforms * +and * + of thefunctions f and g are well-defined and a, b are constants, then the following equation holds:

Corollary 1
If the functions * + and * + are well-defined then: The proof is clear by using the Mathematical induction.

The Convolution of Two Functions
Definition7 Assume that f and g are piecewise continuous functions, or one of them is a Dirac's delta generalized function. The convolution of f and g is a function denotedby f *g and given by the following expression.
In [20] Gabriel Nagy summarized the main properties of the convolution illustrated in the following lemma.

Lemma 2 (Properties)
For every piecewise continuous functions f, g, and h, the following properties hold: Then we can change the order of integration, i.e.,equation (8) becomes: Now,ifγ= λ-τ then dγ=dλ, since the variable τ is constant, hence when we integrate with respect to τ, we get Ifx=uτ and y=uγthen we have dx=udτ anddy=udγ, i.e.,

Proof
We will use the Mathematical induction If n=1 then the corollary (2) holds by theorem (6). Now suppose it holds for n=m, i.e., and suppose n=m+1: Then the corollary is performing for any nonnegative integer n.

Theorem 7 (Derivatives of other variables)
If the function * ( )+is well-defined then:

Corollary3
Let f satisfy the conditions in theorem (1)and has the transform ̅ ( ). Then the Laplace transform ̂( )of f(t) is given by

The Advantages of the New Transform
The new transform has many interesting properties which make it rival to the Laplace transform. Some of these properties are: 1. The domain of the new transform is wider than or equal to the domain of Laplace transform as illustrated in the

Conclusion
The new transform has some strength points for feature among others transform such that:The new transform easier than the Laplace transform for beginners to understand and use. Itcan be used to solve problems without resorting to the frequency domain. Especially, with respect to applications in problems with physical dimensions.We showed it to be the theoretical dual to the Laplace transform, and hence ought to rival it in solving intricate problems in engineering mathematics and applied science.Also, the new transform is a convenient tool for solving differential equations in the time domain without the need for performing its inverse.The connection of the new transform with the Laplace transform goes much deeper.We also present many of the new transform properties that make it uniquely qualified to address and solve some applied problems, especially ones in which the units of the problem must be preserved.