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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Abstract Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. The case when is justified by View at Google Scholar W. Let be the -Laplace transform of the first kind. Thus, by 42 and 43 we have Hence, the proof is done.

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

Moreover, we will consider applications of the Mangontarum -transform of the first kind to some ordinary -differential equations with initial values. Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution Example For instance, the two -analogues for the exponential function denoted by and are given, respectively, by with. Theorem 6 convolution identity. For a positive integer and.

Also, from Definition 1we can easily observe the linearity relation where, and yransform, are real numbers. Since then from 39we have Thus, from Definition 1.

Note that 82 may be expressed as Hence, from 86we have the following results: Theorem 3 transform of -derivatives. In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations. That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover.

Application of In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations.

One may see [ 4 — 8 ] and the references therein for a brief overview and examples of -analogues and their applications. From [ 10 ], for any positive integerwhere. 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.

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

Discrete Dynamics in Nature and Society. For a positive integer andMuch is yet to be discovered regarding the Mangontarum -transforms. Among these properties are the following: Given the -gamma function of the second kind forwe get for a positive integer.

Then the Mangontarum -transform of the second kind satisfies the relation. Theorem 7 -derivative of transforms.

For a positive transfomr andone has The next corollary follows directly from View at Google Scholar T. The possibility of deriving more properties for these -transforms is an interesting recommendation.

Other -analogues of the Laplace transform were earlier considered by Hahn [ 11 ] and Abdi [ 12 ]. For example, for any integers and with tramsformwe have the following -analogues of the integerfalling factorialfactorialand binomial coefficientrespectively: Hence, the -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution. Then the Mangontarum -transform of the second kind satisfies the relation From [ 10 ], for any positive integerwhere.

## Discrete Dynamics in Nature and Society

Ifwe say that is an inverse Mangontarum -transform of the first kindor an inverse -transform of the functionand we write Observe that tdansform also holds for the inverse -transform of the function. Since 22 can be expressed as then by 28 and 27we have the following: 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.

That is Let so that. The -derivative of the -Laplace transform of the second kind [ 10 ] is Replacing with and applying Theorem 6 yield Thus, the next theorem holds.

Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution.