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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Clearly, The expressions in 9 are called -integer-falling factorial of transfform order-factorial ofand -binomial coefficient or Gaussian polynomialrespectively. Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution.

The -derivative of the -Laplace transform of the second kind [ 10 ] is Replacing with and applying Theorem 6 yield Thus, the next theorem holds. Fundamental properties of this transform were already established by Elzaki et al. Taking the Mangontarum -transform of the first kind of both sides of this equation yields Let. That is Let so that. To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below.

From 20Replacing with and multiplying both sides with yield Thus, we have the duality relation between the -Laplace transform of the second kind and the Mangontarum -transform of the second kind as stated in the next theorem. The case trasform is justified by Find the solution of when and.

Some recent studies involving -differential equations are those by Ahmad et al. From Definition 1 and 18Since and by 12Further simplifications lead to Properties such as the transforms of -trigonometric functions, transform of -derivatives, duality relation, convolution identity, -derivative of transforms, and transform of the Heaviside function are derived and presented.

In this section, we will consider tranzform of the Mangontarum dlzaki of the first kind to some -differential equations.

The following statements are true: This new transform appears to be a convenient tool in solving various problems related to differential equations as seen in [ 1 — 3 ] and the references cited therein. For a positive integer tfansformMuch is yet to be discovered regarding the Mangontarum -transforms.

Theorem 17 duality relation.

Theorem 7 -derivative of transforms. The study of the -analogues of classical identities is a popular topic among mathematicians and physicists. Theorem 4 duality relation. For example, for any integers and with andwe have the following -analogues of the integerfalling factorialfactorialand binomial coefficientrespectively: 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 ].

## Discrete Dynamics in Nature and Society

In the following examples, the effectiveness of the Mangontarum -transform of the first kind in solving certain initial value problem involving ordinary -differential equations is illustrated.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. Then the Mangontarum -transform of the first kind satisfies the relation Remark 5. Find the solution of the second order -differential equation: Other -analogues of the Laplace transform were earlier considered by Hahn [ 11 ] and Abdi [ 12 ]. This motivates us to define a -analogue for the Elzaki transform in [ 1 ]. Comtet, Advanced CombinatoricsD.

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

Hence, the -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution. Abstract Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind andrespectively, then From [ 10 ], the -derivative of the -Laplace transform of the first kind is Replacing with and applying Theorem 6 yield Hence, the following theorem is easily observed.

Theorem 16 transfform of -trigonometric functions.

View at Google Scholar T. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, ttansform reproduction in any medium, provided the original work is properly cited. Since 22 can be expressed as then by 28 and 27we have the following: Note that when and.

Given the -gamma function of the second kind forwe get for a positive integer. For a positive integer and. Theorem 6 convolution identity. Also, from Definition 1we elzakii easily observe the linearity relation where, andare real numbers.

The Mangontarum -transform of the first kind, denoted byis defined by over the set in 21where and. Reidel Publishing, Dordretcht, The Neatherlands, Theorem 3 transform of -derivatives.

For a positive integer andone has Moreover, we have the following corollary.

Let be the -Laplace transform of the first kind. That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover. Subscribe to Table of Contents Alerts.