In this paper a new integral transform namely Elzaki transform was applied to Elzaki transform was introduced by Tarig ELzaki to facilitate the process of. The ELzaki transform, whose fundamental properties are presented in this paper, is little known and not widely The ELzaki transform used to. Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. Properties such as the transforms of.

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Discrete Dynamics in Nature and Society.

The author would like to thank the academic editor for his invaluable work during the editorial workflow and the referees for their corrections and suggestions which helped improve the clarity flzaki this paper. Then, by Theorem 17Hence, the following theorem is obtained.

## Discrete Dynamics in Nature and Society

Taking the Mangontarum -transform of eozaki first kind of both sides of this equation yields Let. Note that when and. Given the -gamma function of the second kind forwe get for a positive integer. A -analogue of the gamma function for a complex number, is given by for and where is a positive integer see [ 101314 ]. That is Let so that. View at Google Scholar T.

### On a -Analogue of the Elzaki Transform Called Mangontarum -Transform

Properties such as the transforms of -trigonometric functions, transform of -derivatives, duality relation, convolution identity, -derivative of transforms, and transform of transfogm Heaviside function are derived and presented.

Find the solution of the second order -differential equation: Mangontarum -Transform of the Second Kind Since there can be more than flzaki -analogue of any classical expression, we can define another -analogue for the Elzaki transform.

Given the -exponential function the -sine and -cosine functions can be defined as where. The following identities hold: For a positive integer andThe Heaviside function is defined by Now, we have the following theorem. Theorem 18 transform of -derivatives. In the following examples, fransform effectiveness of the Mangontarum -transform of the first kind in solving certain initial value problem involving ordinary -differential equations ttansform illustrated.

Ifwe say that is an inverse Mangontarum -transform of the first kindor an inverse -transform of the functionand we write. Since there can be more than one -analogue elzaik any classical expression, we can define another -analogue for the Elzaki transform. In this paper, we will define two kinds of -analogues of the Elzaki transform, and to differentiate them from other possible -analogues, we will refer to these transforms as Mangontarum -transforms.

One may read [ 1 — 3 ] and the references therein for guidance regarding this matter.

One may see [ 4 — 8 ] and the references therein for a brief overview and examples of -analogues and their applications. Reidel Publishing, Dordretcht, The Neatherlands, Moreover, we will consider applications of the Mangontarum -transform of the first kind to some ordinary -differential equations with initial values.

Abstract Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. Consider the first degree -differential equation: From Definition 1 and 18Since and by 12Further simplifications lead to The -difference calculus or quantum calculus was first studied by Jackson [ 15 ], Carmichael [ 16 ], Mason [ 17 ], Adams [ 18 ], and Trjitzinsky [ 19 ] in the early 20th century. The Heaviside function is defined by Now, we have the following theorem.

Acknowledgments The author would like to thank the academic editor for his invaluable work during the editorial workflow and the referees for their corrections and suggestions which helped improve the clarity of this paper. The Mangontarum -transform of the first kind, denoted byis defined by over the set in 21where and.

Let be the -Laplace transform of the first kind of. Then, we have the following theorem.

Then the Mangontarum -transform of the first kind satisfies the relation. Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. Moreover, it is easily observed that where, andare real numbers. Fundamental properties of this transform were already established by Elzaki et al.

Ifthen by Theorem 3 Thus, taking the Mangontarum -transform of the first transfork of both sides of 68 gives us From the inverse -transform in 61we have the solution. Among these properties are the following: Theory and Practicalvol.

The following statements are true: