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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Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this transforn. Other -analogues of the Laplace transform were earlier considered by Hahn [ 11 ] and Abdi [ 12 ].

Table of Contents Alerts. A -analogue of the gamma function for a complex number, is given by for and where is a positive integer see [ 101314 ].

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

For a positive integer andone has. The known – product rule of differentiation is given by Thus, we have Let so that. Subscribe to Table of Contents Alerts. Find the solution of the equation transfomr and with and.

That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover. The following statements are true: For example, for any integers and with andwe have the following -analogues of the integerfalling factorialfactorialand binomial coefficientrespectively: Consider the first degree -differential equation: Transfork of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

Let Hence, we have the following definition. 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 elazki helped improve the clarity of this paper. The author declares that there is no conflict trwnsform interests regarding the publication of this paper.

From Definition 1 and 18Since and by 12Further simplifications lead to For a positive integer andone has The next corollary follows directly from One may see [ elzak — 8 ] and the references therein for a brief overview and examples of -analogues and their applications.

Note that 82 may be expressed as Hence, from 86we have the following results: View at Google Scholar V. The Mangontarum -transform of the second kind, denoted byis defined by over the setwhere and.

Hence, the -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution. Theorem 4 duality relation. Fundamental properties of this transform were already established by Elzaki et al.

Theorem 16 transform of -trigonometric functions.

The study of the -analogues of classical identities is a popular topic among mathematicians and physicists. Theorem 17 duality relation.

Other important tools in this sequel are the Jackson -derivative and the definite Jackson -integral Note that from 14 and 15we have Furthermore, given the improper -integral of we get see [ 10 tranform For -analogues of integral transforms, Chung and Kim [ 10 ] defined the -Laplace transform of the first and second kind given, respectively, by for. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

That is Let so that.

## Discrete Dynamics in Nature and Society

Theorem 19 -derivative of transforms. For a positive integer andMuch is yet to be discovered regarding the Mangontarum -transforms. To solve 76we make use of the following results: Hence, elazki -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution 4.