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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For instance, the two -analogues for the exponential function denoted by and are given, respectively, by with. Consider the first degree -differential equation: Also, from Definition 1we can easily observe the linearity relation where, andare real numbers. The possibility of deriving more properties for these -transforms is an interesting recommendation. Given the transfor function of the second kind forwe get for a positive integer.
On a -Analogue of the Elzaki Transform Called Mangontarum -Transform
Let be the Heaviside function. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
Comtet, Advanced CombinatoricsD. Theorem 17 duality relation.
Given the -exponential function the -sine and -cosine functions can be defined as where. Let Then, we have the following definition. Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind andrespectively, then 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.
Forlet and. Researchers are encouraged to further investigate other applications of these -transforms, especially the second kind. For and, the -convolution of and is defined in [ 10 ] as where is the -binomial theorem.
For example, for any integers and with andwe have the following -analogues of the integerteansform factorialfactorialand traansform coefficientrespectively: 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. Theorem 7 -derivative of transforms. The -difference calculus or quantum calculus was first studied by Jackson [ 15 ], Carmichael [ 16 ], Mason [ 17 ], Adams [ 18 ], and Trjitzinsky [ 19 ] trasform the early 20th century.
Applying Theorem 3we have Hence, Using the inverse -transform in 61 yields the solution. Also, note that there were no applications shown for the Mangontarum -transform of the second kind in this paper.
Theorem 19 -derivative of transforms. Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution Example Find the solution of when and. View at Google Scholar T. Much is yet to be discovered regarding the Mangontarum -transforms. Table of Contents Alerts. Note that 82 may be expressed as Hence, from 86we have the following results: The Mangontarum -transform of the first kind, denoted byis defined by over the set in 21where and.
View at Google Scholar L.
Application of In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations. Let denote the th -derivative of the function.
Discrete Dynamics in Nature and Society
Then, we have the following theorem. From Definition 1 and 18Since and by 12Further simplifications lead to Ifwe say elzakii is an inverse Mangontarum -transform of the first kindor an inverse -transform of the functionand we write. Then, one has Proof. That is Let so that.
To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below. Elzak 3 transform of -derivatives. The author declares that there is no conflict of interests regarding the publication of this paper.
Now, suppose 43 holds for. 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.
The Mangontarum -transform of the second kind, denoted byis defined by over the setwhere and. Replacing with in 19 yields The next theorem is obtained by multiplying both sides of this equation by.
A -analogue of the gamma function for a complex number, is given by for and where is a positive integer see [ 101314 ]. Theorem 16 transform of -trigonometric functions. Since then the Mangontarum -transform of 72 yields Applying 61 yields the solution.