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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Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution. Then the Mangontarum -transform of the first kind satisfies the relation. Ifthen by Theorem 3 Thus, taking the Mangontarum -transform of the first kind of both sides of 68 gives us From the inverse -transform in 61we have the solution Example For a positive integer andThe Heaviside function is defined by Now, we have the following theorem. Theorem 17 duality relation.

Now, suppose 43 holds for. Hence, the -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution.

The Mangontarum -transform of the first kind, denoted byis defined by tarnsform the set in 21where and.

Transfogm author declares that there is no conflict of interests regarding the publication of this paper. Ifwe say that is an inverse Mangontarum -transform of the first kindor an inverse -transform of the functionand we write Observe that linearity also holds for the inverse -transform of the function.

Comtet, Advanced CombinatoricsD. In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations. Other important tools in this transfoem 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 ] For -analogues of integral transforms, Chung and Kim [ 10 ] defined the -Laplace transform of the first and second kind given, respectively, by for.

Moreover, it is easily observed that where, andare real numbers. To solve 76we make use of the following results: Now, by 28 elzakk 27Similarly, Now, if we define the hyperbolic -sine and -cosine functions elzakj we have the next theorem which presents the transforms of the -trigonometric functions.

Researchers are encouraged to further investigate other applications of these -transforms, especially the second kind. Ifwe say that is an tfansform Mangontarum -transform of the first kindor an inverse -transform of the functionand we write.

Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution Example Also, from Definition 1we can easily observe the linearity relation where, andare real numbers. Abstract Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. Let Hence, we have the following definition. To obtain this, the following definition is essential. The Mangontarum -transform of the second kind, denoted byis defined by over the setwhere and.

To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below. Table of Contents Alerts. Observe that linearity also holds for the inverse -transform of the function. 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. Forlet and.

Since then from 39we have Thus, from Definition 1Let denote the th -derivative of the function. Let be the -Laplace transform of the first kind.

The book of Kac and E,zaki [ 9 ] is a good source for further details of quantum calculus. Forwe have can be proven similarly.

The case when is justified by Other -analogues of the Laplace transform were earlier considered by Hahn [ 11 ] and Abdi [ 12 ]. Since 22 can be expressed as then by 28 and 27we have the following: Theorem 6 convolution identity.

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.

Let Then, we have the following definition.

## Discrete Dynamics in Nature and Society

Find the solution of the equation where and with and. Introduction Given the set Elzaki [ 1 ] introduced a new integral transform called Elzaki transform defined by for, and. Let be the -Laplace transform of the first kind elaki.

Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind andrespectively, then. Then the Mangontarum -transform of the second kind satisfies the relation From [ 10 ], for any positive integerwhere.

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

Then, by Theorem 17Hence, the following theorem is obtained. Note that when and. Mangontarum -Transform of the Second Kind Since there can be more than one -analogue of any classical expression, we can define another -analogue for the Elzaki transform.