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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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 T.
One may read [ 1 — 3 ] and the references therein for guidance regarding this matter. Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution Example Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution. Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution Example 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.
Replacing with in 19 yields The next theorem is obtained by multiplying both sides of this equation by. For a positive integer andone has.
On a -Analogue of the Elzaki Transform Called Mangontarum -Transform
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. View at Google Scholar L. Theorem 7 -derivative of transforms.
From [ 10 ], for any positive integerwhere. Other important tools in this sequel are the Jackson -derivative and the definite Jackson -integral Note that elza,i 14 and 15we have Furthermore, given the improper -integral of we get see [ 10 ]. 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.
For a positive integer andMuch is yet to be discovered regarding the Mangontarum -transforms. 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. View at Google Scholar W. View at Google Scholar V. Given the -exponential function the -sine and -cosine functions can be defined as where.
The case when is justified by Let Then, we have the following definition. Clearly, The expressions in 9 are called -integer-falling factorial of of order-factorial ofand -binomial coefficient or Gaussian polynomialrespectively.
In this paper, we will define two kinds of -analogues of the Elzaki transform, and to differentiate them from other possible -analogues, we will refer to these transforms as Mangontarum -transforms. Note that 82 may be expressed as Hence, from 86we have the following results: Consider the first degree -differential equation: To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below.
Since there can be more than one -analogue of any classical expression, we can define another -analogue for the Elzaki transform. Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. Note that when and. The Mangontarum -transform of the second kind, denoted byis defined by over the setwhere and.
Let be the -Laplace transform of the second kind of. To obtain this, the following definition is essential. Clearly, This makes 22 a -analogue of the Elzaki transform in 2.
Discrete Dynamics in Nature and Society
Then the Mangontarum -transform of the first kind satisfies the relation Remark 5. 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 That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends towe recover. 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.
Researchers are encouraged to further investigate other applications of these -transforms, especially the second kind. Given the set Elzaki [ 1 ] introduced a new integral transform called Elzaki transform defined by for, and. In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations.
Indexed in Science Citation Index Expanded. Find the solution of the equation where and with and.
Find the solution of elzakii and. Hence, the transtorm of 76 is Further simplifications yield The inverse -transform in 61 gives the solution 4. Theorem 3 transform of -derivatives. Theorem 19 -derivative of transforms. Abstract Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper.
The known – product rule of differentiation is given by Thus, we have Let so that. Subscribe to Table of Contents Alerts.
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.