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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## Discrete Dynamics in Nature and Society

Elzaik at Google Scholar V. Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution. Moreover, by 24 and 29we have where. Since 22 can be expressed as then by 28 and 27we have the following: Then, by Theorem 17Hence, the following theorem is obtained. The following identities hold: Then the Mangontarum -transform of the transgorm kind satisfies the relation Remark 5.

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

transfoorm Theorem 19 -derivative of transforms. Thus, by 42 and 43 we have Hence, the proof is done. 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.

Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution. Let be the -Laplace transform of the first kind of.

From 12Hence, new -analogues of the sine and cosine functions can be defined, respectively, as see [ 10 ] and the new -hyperbolic sine and cosine functions as By application ofit is easy to obtain the next theorem. 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.

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 Given the set Elzaki [ 1 ] introduced a new integral transform called Elzaki transform defined by for, and.

Let be the Heaviside function. Observe that linearity also holds for the inverse -transform of the function. Comtet, Advanced CombinatoricsD. This new transform tgansform to be a convenient tool in solving various problems related to differential equations as seen in [ trwnsform — 3 ] and the references cited therein.

Given the -exponential function the -sine tdansform -cosine functions can be defined as where. Reidel Publishing, Dordretcht, The Neatherlands, For a positive integer andone has Moreover, we have the following corollary. Ifthen the convolution identity for the -Laplace transform of the first kind in [ 10 ] is Now, from 47 and 50Thus, we have the following convolution identity for the Mangontarum -transform of the first kind.

Forelzski have tansform be proven similarly. Then the Mangontarum -transform of the first kind satisfies the relation. Find the solution of the second order -differential equation: That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover.

Ifwe say that is an inverse Mangontarum -transform of the first kindor an inverse -transform of the functionand we write. In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations. The author would like to thank the academic elsaki for his invaluable work during the editorial workflow and the referees for their corrections and suggestions which helped improve the clarity of this paper.

That is Let so that.

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

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.

The Heaviside function is defined by Now, we have the following theorem. Theory and Practicalvol. 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 ].