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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That is, a polynomial is said to be a -analogue of an integer if by taking its limit transfom tends towe recover. Fundamental properties of this transform were already established by Elzaki et al.

The possibility of deriving more properties for these -transforms is an interesting recommendation. Replacing with in 19 yields The next theorem is obtained by multiplying both sides of this equation by. 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.

To solve 76we make use of the following results: View at Google Scholar W. The known – product rule of differentiation is given by Thus, we have Let so that. Also, from Definition 1we can easily observe the linearity relation where, andare real numbers. Theorem 6 convolution identity. For a positive integer andone has The next corollary follows directly from Let be the Heaviside function.

Since there can be more than one -analogue of any classical expression, we can define another -analogue for the Elzaku transform.

## Discrete Dynamics in Nature and Society

Theorem 7 -derivative of transforms. 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. The study of the -analogues of classical elzaji is a popular topic among mathematicians and physicists.

Furthermore, the -binomial coefficient can be expressed as Note that the transition of any classical expression to its -analogue is not unique.

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

To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below. Let be the -Laplace transform of the first kind of. Moreover, it is easily observed that where, andare real numbers. Let Then, we have the following definition. Note that 82 may be expressed as Hence, from 86we have the following results: Comtet, Advanced CombinatoricsD. Ifwe say that is an inverse Mangontarum -transform of the first kindor an inverse -transform of the function ezaki, and we write Observe that linearity also holds for the inverse -transform of the function.

Reidel Publishing, Dordretcht, The Neatherlands, Introduction Given the set Elzaki [ 1 ] introduced a new integral transform called Elzaki transform defined by for, and. Note that when and. Other -analogues of the Laplace transform were earlier considered by Hahn [ 11 ] and Abdi [ 12 ].

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 eelzaki. Table of Contents Alerts.

Theorem 18 transform of -derivatives. Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind andrespectively, then. The Heaviside function is defined by Now, we have the following theorem. Much is yet to be discovered regarding the Mangontarum elazki.

Find the solution of the second order -differential equation: This motivates us to define a -analogue for the Elzaki transform elzakk [ 1 ].

The following statements are true: The book of Kac and Cheung [ 9 ] is a good source for further details of quantum calculus. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

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. From 20Replacing with and multiplying both sides with yield Thus, we have the duality relation between the -Laplace transflrm of the second kind and the Mangontarum -transform of the second kind as stated in the next theorem. Discrete Dynamics in Nature and Society.

For and, the -convolution of and is defined in [ 10 ] as where is the -binomial theorem.

For a positive integer andMuch is yet to be discovered regarding the Mangontarum -transforms. Theorem 17 duality relation.

Then the Mangontarum -transform of the first kind satisfies the relation Remark 5. Let Hence, we have the following definition.

Abstract Two -analogues of the Elzaki transform, called Mangontarum trznsform, are introduced in this paper. To obtain this, the following definition is essential. 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. View at Google Scholar V. 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 The following identities hold: Indexed in Science Citation Index Expanded.

Theorem 2 transform of -trigonometric functions. Tranwform then from 39we have Thus, from Definition 1 .