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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Then, one has Proof. Since there can be more than one -analogue of any classical expression, we can define another -analogue for the Elzaki eelzaki.

Since then from 39we have Thus, from Definition 1Let denote the th -derivative of the function. Moreover, it is easily transforj that where, andare real numbers. Hence, the -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution 4.

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

Ifthen by Theorem 3 Thus, taking the Mangontarum -transform of tdansform first kind of both sides of 68 gives us From the inverse -transform in 61we have the solution.

Theorem 3 transform of -derivatives. That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover.

Since 22 can be expressed as then by 28 and 27we have the following: Given the set Elzaki [ 1 ] introduced a new integral transform called Elzaki transform defined by for, and. transsform

For example, for any integers and with andwe have the following -analogues of the integerfalling factorialfactorialand binomial coefficientrespectively: The case when is justified by Comtet, Advanced CombinatoricsD.

The -difference calculus or quantum calculus was first studied elaaki Jackson [ 15 ], Carmichael [ 16 ], Mason [ 17 ], Adams [ 18 ], and Trjitzinsky [ 19 ] in the elzako 20th century. This motivates us to define a -analogue for the Elzaki transform in [ 1 ].

From 20Replacing with and multiplying both transvorm 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. Let be the -Laplace transform of the first kind. Given the -exponential function the -sine and -cosine functions can be defined as where.

Theorem 4 duality relation. From [ 10 ], elzqki any positive integerwhere. One may read [ 1 — 3 ] and the references therein for guidance regarding this matter. Consider the first degree -differential equation: Find the solution of the second order -differential equation: Then, we have the following theorem. For a positive integer andone has The next corollary follows directly from Furthermore, the -binomial coefficient can be expressed as Note that the transition of any classical expression to its -analogue is not unique.

For -analogues of integral transforms, Chung and Kim [ 10 ] defined the -Laplace transform of the first and second kind given, respectively, by for. Replacing with in 19 yields The next theorem is obtained by multiplying both sides of this equation by.

elzkai

## Discrete Dynamics in Nature and Society

One may see [ 4 trnasform 8 ] and the references therein for a brief overview and examples of -analogues and their applications. Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind andrespectively, then.

Let Hence, we have the following definition. Taking the Mangontarum -transform of the first kind of both sides of this equation yields Let. Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution. For a positive integer andone has Moreover, we have the following corollary.

Table of Contents Alerts. Let be the -Laplace transform of the first kind of. Note elzami when and.

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