Natural transform of fractional order and some properties

In this work, a new fractional integral transform is proposed, and some of its properties are mentioned. Further, the relation between it and others fractional transforms is given and some of its applications are presented.


Introduction
Natural transform is closely related to Laplace and Sumudu transforms. The Natural transform was first introduced in Khan and Khan (2008) which was called N-transform and its properties were investigated by Al-Omari (2013), Belgacem and Silambarasan (2012b). In Silambarasan (2011, 2012a) the Natural transform was applied to solve Maxwell's equations. More studies regarding the Natural transform can be found from Silambarasan (2011, 2012c).
The Natural transform usually deals with continuous and continuously differentiable functions, or if we assume that the function is fractional derivative and continuous. However, the function is not derivative; therefore, the Natural transform fails to apply similarly as Laplace and Sumudu transforms. Thus, analogously, we need to set a new definition that we name fractional Natural transform.
First of all in the following part, definition of fractional derivative and some basic notations are given.

Fractional derivative
is a continuous function and not necessarily differentiable function, then forward operator FW(h) is defined as follows where h > 0 denotes a constant discretization span. Moreover, fractional difference of g(t) is known as and the -derivative of g(t) is known as For further details, we refer to Almeida, Malinowska, and Torres (2010), Jumarie (2006Jumarie ( , 2009aJumarie ( , 2009b. Jumarie (2009b) proposed the alternative definition of the Riemann-Liouville fractional derivative.

Modified fractional Riemann-Liouville derivative
is a continuous function, but not necessarily differentiable, then (i) Let us presume that g(t) = K, where K is a constant; thus, -derivative of the function g(t) is On the other hand, when g(t) ≠ K, and hence fractional derivative of the function g(t) will be known as at any negative , ( < 0) one has while for positive , we will put When m < < m + 1, we place

Integral with respect to (dt)
The fractional differential equation: h .

Main results
The main results of this work are to define fractional Natural transform and some of its properties.
Definition 2.1 Let f(x) be a function defined for all x ≥ 0; then, fractional Natural transform of order which is denoted by ℕ + can be defined as the next expression Corollary 2.1 From the above definition, we show that (i) when u = 1, we have fractional Laplace transform which is proposed in Jumarie (2009a), (ii) when s = 1, we get fractional Sumudu transform which is proposed in Gupta, Shrama, and Kiliçman (2010).

Some properties of fractional Natural transform
Theorem 2.2 Let a, b be any constants and f(x), g(x) are functions; then, (1) Scaling property (2) Linearity property (3) Shifting property

Proof
(1) The result can be obtained directly using Definition 2.1 as (2) By applying Definition 2.1, we can easily get the result By changing the variable v → x − a and taking into account the formula then we have (4) By substituting x = st (s+au) , then ✷ Remark 2.2 All the results above in Theorem 2.2 satisfy the properties of Natural transform when = 1.

The fractional Natural transform coupled with fractional Laplace transform
First, we mention the next definition that is presented in Jumarie (2009a).

Proof
By making the change of the variable v → ux, it follows that ✷ Remark 2.3 The same result is obtained (see Theorem 2.1 in Belgacem & Silambarasan, 2012b) when = 1 in the above theorem.

The fractional Natural transform coupled with fractional Sumudu transform
We recall the next definition from Gupta et al. (2010).

Theorem 2.7 Let f(x) be a fractional differentiable function; then,
Proof Using the Laplace-Natural duality formula and fractional integration by parts which is presented in Jumarie (2009a)