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

Abstract

Two -analogues of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper. 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. Moreover, we will consider applications of the Mangontarum -transform of the first kind to some ordinary -differential equations with initial values.

1. Introduction

Given the set Elzaki [1] introduced a new integral transform called Elzaki transform defined by for , , and . This new transform appears to be a convenient tool in solving various problems related to differential equations as seen in [1–3] and the references cited therein. Fundamental properties of this transform were already established by Elzaki et al. in [1–3] and the references therein. Among these properties are the following: (i)the Elzaki transform of special functions such as given by (ii)the Elzaki transform of , given by (iii)the Elzaki transform of derivatives given by (iv)the duality relation given by where is the Laplace transform of ; and(v)the convolution identity given by where and are the Elzaki transforms of and , respectively, and is the convolution of and .

The study of the -analogues of classical identities is a popular topic among mathematicians and physicists. One may see [4–8] and the references therein for a brief overview and examples of -analogues and their applications. The parameter is known to stand for “quantum” which is widely seen in quantum calculus (or -calculus). The book of Kac and Cheung [9] is a good source for further details of quantum calculus. Basically, the term “-analogue” refers to a mathematical expression involving a parameter which generalizes a known identity and reduces back to the known identity when . That is, a polynomial is said to be a -analogue of an integer if by taking its limit as tends to , we recover . For example, for any integers and with and , we have the following -analogues of the integer , falling factorial , factorial , and binomial coefficient , respectively: Clearly, The expressions in (9) are called -integer, -falling factorial of of order , -factorial of , and -binomial coefficient (or Gaussian polynomial), respectively. Furthermore, the -binomial coefficient can be expressed as Note that the transition of any classical expression to its -analogue is not unique. For instance, the two -analogues for the exponential function denoted by and are given, respectively, by with . Other important tools in this sequel are the Jackson -derivative and the definite Jackson -integral Note that from (14) and (15), we have Furthermore, given the improper -integral of we get (see [10])

For -analogues of integral transforms, Chung and Kim [10] defined the -Laplace transform of the first and second kind given, respectively, by for . Other -analogues of the Laplace transform were earlier considered by Hahn [11] and Abdi [12]. This motivates us to define a -analogue for the Elzaki transform in [1].

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. Some properties and interesting formulae are derived and presented.

2. Mangontarum -Transform of the First Kind

Let Hence, we have the following definition.

Definition 1. The Mangontarum -transform of the first kind, denoted by , is defined by over the set in (21), where and .

Clearly, This makes (22) a -analogue of the Elzaki transform in (2). Also, from Definition 1, we can easily observe the linearity relation where , , and , are real numbers.

A -analogue of the gamma function for a complex number, is given by for and where is a positive integer (see [10, 13, 14]).

Since (22) can be expressed as then by (28) and (27), we have the following: Moreover, by (24) and (29), we have where .

Given the -exponential function the -sine and -cosine functions can be defined as where . Now, by (28) and (27), Similarly, Now, if we define the hyperbolic -sine and -cosine functions as we have the next theorem which presents the transforms of the -trigonometric functions.

Theorem 2 (transform of -trigonometric functions). The following identities hold:

Proof. and can be proved parallel to (33). For , we have can be proven similarly.

The known -product rule of differentiation is given by Thus, we have Let so that . Since then from (39), we have Thus, from Definition 1,

Let denote the th -derivative of the function . Then, we have the following theorem.

Theorem 3 (transform of -derivatives). If  , , then

Proof (by induction). The case when is justified by (42). Now, suppose (43) holds for . That is Let so that . Thus, by (42) and (43) we have Hence, the proof is done.

Let be the -Laplace transform of the first kind. Replacing with in (19) yields The next theorem is obtained by multiplying both sides of this equation by .

Theorem 4 (duality relation). Let be the -Laplace transform of the first kind of . Then the Mangontarum -transform of the first kind satisfies the relation

Remark 5. Note that when and , .

For and (, ), the -convolution of and is defined in [10] as where is the -binomial theorem. If , then the convolution identity for the -Laplace transform of the first kind in [10] is Now, from (47) and (50), Thus, we have the following convolution identity for the Mangontarum -transform of the first kind.

Theorem 6 (convolution identity). If , with -Laplace transforms of the first kind and , respectively, and Mangontarum -transforms of the first kind and , respectively, 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.

Theorem 7 (-derivative of transforms). For a positive integer and , one has

The next corollary follows directly from (55).

Corollary 8. For a positive integer and ,

The Heaviside function is defined by Now, we have the following theorem.

Theorem 9. Let be the Heaviside function. Then, one has

Proof. From Definition 1 and (18), Since and by (12), Further simplifications lead to (58).

3. Application of

In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations. To obtain this, the following definition is essential.

Definition 10. Let . If , we say that is an inverse Mangontarum -transform (of the first kind), or an inverse -transform of the function , and we write

Observe that linearity also holds for the inverse -transform of the function . That is, where .

The -difference calculus (or quantum calculus) was first studied by Jackson [15], Carmichael [16], Mason [17], Adams [18], and Trjitzinsky [19] in the early 20th century. Some recent studies involving -differential equations are those by Ahmad et al. [20], Yu and Wang [21], and Sitthiwirattham et al. [22]. 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.

Example 11. Consider the first degree -differential equation: where with initial condition . Taking the Mangontarum -transform (of the first kind) of both sides of this equation yields Let . Applying Theorem 3, we have Hence, Using the inverse -transform in (61) yields the solution

Example 12. Find the solution of the second order -differential equation: with initial condition and .
Solution. If , then by Theorem 3 Thus, taking the Mangontarum -transform (of the first kind) of both sides of (68) gives us From the inverse -transform in (61), we have the solution

Example 13. Find the solution of the equation where and with and .
Solution. Let . Since then the Mangontarum -transform of (72) yields Applying (61) yields the solution

Example 14. Find the solution of when and .
Solution. Let . To solve (76), we make use of the following results: Hence, the -transform of (76) is Further simplifications yield The inverse -transform in (61) gives the solution

4. 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. Let Then, we have the following definition.

Definition 15. The Mangontarum -transform of the second kind, denoted by , is defined by over the set , where and .

Clearly, . Moreover, it is easily observed that where , , and , are real numbers.

Given the -gamma function of the second kind for , we get for a positive integer. Note that (82) may be expressed as Hence, from (86), we have the following results:

From (12), Hence, 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 of , it is easy to obtain the next theorem.

Theorem 16 (transform of -trigonometric functions). The following statements are true:

From (20), Replacing with and multiplying 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.

Theorem 17 (duality relation). Let be the -Laplace transform of the second kind of . Then the Mangontarum -transform of the second kind satisfies the relation

From [10], for any positive integer , where , . For , let and . Then, by Theorem 17, Hence, the following theorem is obtained.

Theorem 18 (transform of -derivatives). If  , , then

The -derivative of the -Laplace transform of the second kind [10] is Replacing with and applying Theorem 6 yield Thus, the next theorem holds.

Theorem 19 (-derivative of transforms). For a positive integer and , one has

Moreover, we have the following corollary.

Corollary 20. For a positive integer and ,

Much is yet to be discovered regarding the Mangontarum -transforms. The possibility of deriving more properties for these -transforms is an interesting recommendation. One may read [1–3] and the references therein for guidance regarding this matter. Also, note that there were no applications shown for the Mangontarum -transform of the second kind in this paper. Researchers are encouraged to further investigate other applications of these -transforms, especially the second kind.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the academic editor for his invaluable work during the editorial workflow and the referees for their corrections and suggestions which helped improve the clarity of this paper. This research paper is supported by the Mindanao State University-Main Campus, Marawi City, Philippines.