Fractional Tarig Transform and Mittag-Leffler Function

In the present paper the Tarig transform of fractional order is studied by employing Mittag - Leffler function. Properties of Tarig transform are proved using the same fractional Tarig transform.


Introduction
One may, among several , can justify efficiencies of the applications of integral transforms, viz.Fourier, Laplace and Hankel (among other) to solve ordinary and partial differential equations besides their applications in other areas.Consider a set A of function f (t) of exponential order, expressed as where M is a constant of finite number and k 1 and k 2 may be finite or infinite.

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Deshna Loonker and P. K. Banerji Properties of the said transform are given as follows : 1.The Tarig transform of derivative and nth order derivative of f (t), respectively, are defined 2. When f (t) = δ(t) (the Dirac delta function), the Tarig transform becomes more such for other values are tabulated in [2,3].
When f (t) = 1, the Tarig transform becomes and when f (t) = t n ,the same yields = Γ(n + 1)u 2n+1 . (1.8) 3. If F (u) and G(u) are Tarig transforms of the functions f (t) and g(t), then the convolution is given by (1.9) 4. If α, β are any constants and f (t) and g(t) are real functions, then the linear property is defined by (1.10) 5.The relation between the Laplace transform F (s) and Tarig transform G(u) is defined by .11)In [12], the Tarig transform is extended to the distribution spaces and some other properties have been formulated.In [11] the Parseval equation of the Tarig Tarig transform for fractional integrals and derivatives for distribution spaces are employed in [13].The Mittag -Leffler function and its applications with the integral transforms viz., Fourier, Laplace, Sumudu and Natural transform are given in [6,7,8,9,10].Different techniques are employed to solve fractional differential equations [1,9,15,16,17,18].Using the Mittag -Leffler function and its properties, some functions are defined and studied by the researchers [9,15,16,17,18].
The Mittag -Leffler function [15] is a direct generalization of the exponential function, and has an affinity for fractional calculus.
One parameter representation of the Mittag -Leffler function is given by [14] , α > 0 (1.12) Whereas two parameter Mittag -Leffler function [4] is represented as with C being the set of complex numbers.
Special cases of the Mittag -Leffler function are Following relations, related to the Mittag -Leffler function, may be useful.
Having accommodated possible sources and terminologies on Tarig transform and the Mittag -Leffler function, in Section 2 we write concepts of fractional derivative, define modified Riemann -Liouville fractional derivative, and Taylor series of fractional order.In Section 3, we establish main results.
Deshna Loonker and P. K. Banerji 2. Preliminaries on Fractional Derivatives 2.1.Fractional derivative via fractional difference Definition 2.1.Let there be a continuous function f : R → R, t → f (t) (but not necessarily differentiable).Let h > 0 be a constant discretization span.The forward operator F W (h) is given by .
With regard to (2.11), the fractional difference of order α, 0 < α < 1, of the function f (t) is given by and the fractional derivative of order α is defined by the limit (2.12)

Modified Riemann-Liouville fractional derivative
. To overcome some drawbacks with the Riemann -Liouville fractional derivative, the modified version is devised [6,7].
Definition 2.2.Let f : R → R, t → f (t) is a continuous function.
(i) When f (t) is constant K, its fractional derivative of order α , is given by For f (t) being not a constant, we have and its fractional derivative is defined by which, when α < 0 , is given by

Integration with respect to (dt) α
. The integral with respect to (dt) α , is defined as the solution of the fractional differential equation Lemma 2.4.[6,7] Let f (t)be a continuous function.Then the solution y(t), y(0) = 0 , is given by (2.42)

Tarig Transform of Fractional Order and the Mittag-Leffler Function
In this section Tarig transform of fractional order is defined by using the Mittag -Leffler function, which is the generalization of the exponential function.Properties and convolution theorem are proved using the Tarig transform of fractional order.
By virtue of terminologies used in the preceding sections and recalling those described for Fourier and the Lapalce transforms, respectively , through the Mittag -Leffler function [6,7], following definition results.Definition 3.1.Let f (t) be a function that vanishes for negative values of t .Then the Tarig transform of order α, for finite f (t), is defined by where E α is the Mittag -Leffler function, given by (1.12).
Theorem 3.2.(Tarig Laplace Duality of Fractional order) If the Laplace transform of fractional order of a function Proof: Invoking Eqn.(2.42) in the definition of Tarig transform of fractional order (3.1), we write By using the change of variable ut → w , i.e., dt = dw u , we get the right hand side Using the definition of Laplace transform, we have which, proves the theorem.✷ Theorem 3.3.(Change of Scale Property) Let f (at) be a function in the set A , where a is non -zero constant.Then Conditions as mentioned above, are applicable.
Fractional Tarig Transform and Mittag-Leffler Function

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Proof: Using (3.3), we have By using the change of variable at → t ′ , dt = dt ′ a , we get Theorem is proved.✷ Theorem 3.
Considering t − b = x, we have the right hand side .
, then the Tarig transform is given by (3.13) Proof: Using (3.3) again of Definition 3.1, we have.
Setting (1 − au 2 )t = w, we have the right hand side, reduced to .
Hence, the theorem is proved.✷ Theorem 3.6.Let the convolution of two functions f (t) and g(t) of order α is given by i.e.

Fractional
Tarig Transform and Mittag-Leffler Function 85 transform for distribution spaces is established and solution of Abel integral equation is obtained related to the distribution spaces.