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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Find the solution of the second order -differential equation: Comtet, Advanced CombinatoricsD. 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.

## Discrete Dynamics in Nature and Society

Much is elzakl to be elzakl regarding the Mangontarum -transforms. Theorem 16 transform of -trigonometric functions. Then, one has Proof. 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. Forwe have can be proven transforn. Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution Example The following identities hold: Reidel Publishing, Dordretcht, The Neatherlands, 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.

Find the solution of the equation where and with and. Some properties and interesting formulae are derived and presented.

Subscribe to Table of Contents Alerts. Then the Mangontarum -transform of the first kind satisfies the relation Remark 5.

Some recent studies involving -differential equations are those by Ahmad et al. For a positive integer andone has Moreover, we have the following corollary.

Replacing with in 19 yields The next theorem is obtained by multiplying both sides of this equation by.

To obtain this, the following definition is essential. That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover.

Ifthen the elzali 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. Theorem 17 duality relation. Furthermore, the -binomial coefficient can be expressed as Note that the transition of any classical expression to its -analogue transfprm not unique. Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution.

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

The case when is justified by Taking the Mangontarum -transform of the first kind of both sides of this equation yields Let. View at Google Scholar W. Fundamental properties of this transform were already established by Elzaki et al.

Moreover, we will consider applications of the Mangontarum -transform of the first kind to some ordinary -differential equations with initial values. Let be the -Laplace transform of the first kind. Since 22 can be expressed as then by 28 and 27we have the following: For a positive integer andone has. 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 Discrete Dynamics in Nature and Society.

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. 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.