On The Generalized Natural Transform

Integral transform method has wide range of applications in the various fields of science and engineering. In most of the cases the physical phenomenon is converted into an ordinary differential equations and partial differential equations which can be solved by integral transform method. This is the basic thing by which the researchers are being motivated to define new integral transforms and used to solve many problems in the field of applied mathematics. Recently, the new integral transform Natural transform (N-transform) was introduced by (Khan and Khan, 2008) and studied its properties and some applications. Later on (Silambarasan et. al., 2011 and Belgacem et. al., 2012) defined the inverse Natural transform and studied some properties and applications of Natural transforms. The distribution theory provides powerful analytical technique to solve many problems that arises in the applied field. This gives rise to define the various integral transforms to the distribution space (Lookner, 2010, 2012 & 2013, Omari, 2014, Shah, 2015, Pathak, 1997, Schwartz, 1950, 51 and Zemanian, 1987).The aim of this paper is to extend the Natural transform in the distributional space of compact support and to investigate some properties and theorems of the generalized integral transform. 1.1. The Natural Transformation


Introduction
Integral transform method has wide range of applications in the various fields of science and engineering. In most of the cases the physical phenomenon is converted into an ordinary differential equations and partial differential equations which can be solved by integral transform method. This is the basic thing by which the researchers are being motivated to define new integral transforms and used to solve many problems in the field of applied mathematics. Recently, the new integral transform Natural transform (N-transform) was introduced by (Khan and Khan, 2008) and studied its properties and some applications. Later on (Silambarasan et. al., 2011 and defined the inverse Natural transform and studied some properties and applications of Natural transforms.
The distribution theory provides powerful analytical technique to solve many problems that arises in the applied field. This gives rise to define the various integral transforms to the distribution space (Lookner, 2010, Omari, 2014, Shah, 2015, Pathak, 1997, Schwartz, 1950, 51 and Zemanian, 1987.The aim of this paper is to extend the Natural transform in the distributional space of compact support and to investigate some properties and theorems of the generalized integral transform.

The Natural Transformation
The Natural transform of the function f t ( )∈ ℜ 2 is sectionwise continuous, exponential order and defined in the set where s and u are the transform variables and is defined by an integral equation   If R(s,u) is the Natural transform, F(s) is the Laplace transform and G(u)is Sumudo transform of function f(t) ∈ A then we can have Natural-Laplace and Natural-Sumudo duality as and We can extract the Laplace, Sumudu, Fourier and Mellin transform from Natural transform and which shows that Natural transform convergence to Laplace and Sumudu transform (Shah et. al., 2015). Moreover Natural transform plays as a source for other transform and it is the theoretical dual of Laplace transform. Further study and applications of Natural transform can be seen in (Silambarasan et. al

Basic Properties of Natural Transform
6. If F(s, u) and G(s,u) are the Natural transforms of respective functions f(t) and g(t) both defined in set A then  f g u F s u G s u * . , ,

Generalized Natural Transform
The author Deshna Loonkar (Lookner et. al., 2013) have studied distributional Natural transform and motivated from that study, here we construct the testing function space to define the generalized integral transform and prove some theorems of generalized Natural transform.

Testing function space
Where This  a b , is linear space under the pointwise addition of function and their multiplication by complex numbers. Each γ k is clearly a seminorm on  a b , and γ 0 is a norm. We assign the topology generated by the sequence of seminorm γ k k ( ) = ∞ 0 there by making it a countably multinormed space. Note that for each fixed s and u the kernel We call Ω f the region (or strip) of definition for  f t ( )     and w 1 and w 2 the abscissas of definition. Note that the properties like linearity and continuity of generalized Natural transform will follows from

( ) is analytic on Ω f
Proof: Let (s, u) be arbitrary but fixed point in Choose the real positive number a,b and r such that Re .
Let ∆S be the complex increment such that ∆S r < and as ∈  , so that equation (8) is meaningful.
We shall now show that as ∆ ∆ , . Since f a b ∈ ′ ℜ , this will imply that where 0 . We may interchange differentiation on s with differentiation on t and using Cauchy integral formula so that equation (8) becomes Now for all ξ ∈ C and −∞ < <∞ ( ) where M is constant independent of ξ and t. Moreover The RHS is independent of t and converges to zero as ∆S → 0 . This shows that ψ ∆S converges to zero in  a b , as ∆S → 0 which completes the proof of theorem. Similar proof can be made for the another variable u.

Theorem 2.2 [Characteriztion Theroem]
The necessary condition for the function Rf(s,u) to be the Natural transform of generalized function f are that Rf(s,u) is analytic on Ω f and for each closed strip u s a s u b , : Re . The polynomial P will depend in general on a and b.

Proof:
The analyticity of R f (s,u) has been already proved in the previous theorem. By the definition of the Natural transform, f is a member of ′  a b . where ω ω 1 2 < < < a b so that there exists a constant M and non-negative integer r such that for a

Conclusion
In this paper we extended the Natural transform in the distributional space of compact support and so defined generalized Natural transform. The analyticity theorem and inversion theorem are proved. This paper might be a new window for the researcher to study of generalized integral transforms.