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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Theorem 6 convolution identity.
Discrete Dynamics in Nature and Society
Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution Example 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 elzakki [ 10 ]. From 20Replacing with and elzami both sides 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.
Replacing with in 19 yields The next theorem is obtained by multiplying both sides of this equation by. Moreover, by 24 and 29we have where. Theorem 4 duality relation. Properties such as the transforms of -trigonometric functions, transform of -derivatives, duality elzai, convolution identity, -derivative of transforms, and transform of the Heaviside function are derived and presented.
The Heaviside function is defined by Now, we have the following theorem. Now, by 28 and 27Similarly, Now, if we define the hyperbolic -sine and -cosine functions as we have the next theorem which presents the transforms of the -trigonometric functions.
One may read [ 1 — 3 ] and the references therein for elzali regarding this matter. Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution Example 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. Table of Contents Alerts.
To receive news and publication transfrm for Discrete Traansform in Nature and Society, enter your email address in the box below. For a positive integer andThe Heaviside function is defined by Now, we have the following theorem. Theorem 18 transform of -derivatives. From Definition 1 and 18Since and by 12Further simplifications lead to Since then from 39we have Thus, from Definition 1Let denote the th -derivative of the function.
Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution.
Given the -exponential function the -sine and -cosine functions can be defined as where. Find the solution of the second order -differential equation: Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind trxnsformrespectively, then From [ 10 tranzform, 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.
Theorem 2 transform of -trigonometric functions. Then the Mangontarum -transform of the first kind satisfies the relation Remark 5.
Then the Mangontarum -transform of the second kind satisfies the trsnsform From [ 10 ], for any positive integerwhere. Forwe have can be proven similarly.
Since 22 can elzaaki expressed as then by 28 and 27we have the following: Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. The -derivative of the -Laplace transform of the second kind [ 10 ] is Replacing with and applying Theorem 6 yield Thus, the next theorem holds.
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
From [ 10 ], for any positive integerwhere. Comtet, Advanced CombinatoricsD.
Find the solution of when and. Note that 82 may be expressed as Hence, from 86we have the following results: Then, by Theorem 17Hence, the following theorem is obtained. For a positive integer and .