Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 825618, 9 pages
http://dx.doi.org/10.1155/2014/825618

Research Article

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

Mahid M. Mangontarum

Department of Mathematics, Mindanao State University, Main Campus, 9700 Marawi City, Philippines

Received 21 June 2014; Revised 19 September 2014; Accepted 19 September 2014; Published 13 October 2014

Academic Editor: Seenith Sivasundaram

Copyright © 2014 Mahid M. Mangontarum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

T. M. Elzaki, “The new integral transform ‘Elzaki transform’,” Global Journal of Pure and Applied Mathematics, vol. 7, no. 1, pp. 57–64, 2011.

T. Elzaki, S. Elzaki, and E. Elnour, “On the new integral transform “Elzaki Transform”, fundamental propeties, investigations and applications,” Global Journal of Mathematical Sciences: Theory and Practical, vol. 4, no. 1, pp. 1–13, 2012.

T. Elzaki, S. Elzaki, and E. Elnour, “On some applications of new integral transform ‘elzaki transform’,” The Global Journal of Mathematical Sciences: Theory and Practical, vol. 4, no. 1, pp. 15–23, 2012.

L. Comtet, Advanced Combinatorics, D. Reidel Publishing, Dordretcht, The Neatherlands, 1974.

L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000, 1948.

H. W. Gould, “The q-Stirling numbers of first and second kinds,” Duke Mathematical Journal, vol. 28, pp. 281–289, 1961.

R. Corcino and C. Montero, “A q-analogue of rucinski-voight numbers,” ISRN Discrete Mathematics, vol. 2012, Article ID 592818, 18 pages, 2012.

C. A. Charalambides and N. Papadatos, “The q-factorial moments of discrete q-distributions and a characterization of the Euler distribution,” in Advances on Models, Characterizations and Applications, N. Balakrishnan, G. Bairamov, and O. L. Gebizlioglu, Eds., pp. 57–71, Chapman and Hall/CRC Press, Boca Raton, Fla, USA, 2005.

V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.

W. Chung and T. Kim, “On the q-analogue of Laplace transform,” http://arxiv.org/abs/1307.6752

W. Hahn, “Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der HYPergeometrischen -DIFferenzengleichung. DAS -Analogon der Laplace-TRAnsformation,” Mathematische Nachrichten, vol. 2, pp. 340–379, 1949.

W. H. Abdi, “On q-Laplace transforms,” Proceedings of the National Academy of Sciences, India, vol. 29, pp. 389–408, 1960.

A. de Sole and V. Kac, “On integral representations of q-gamma and q-beta functions,” http://arxiv.org/abs/math/0302032v1

R. Díaz and C. Teruel, “-generalized gamma and beta functions,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 1, pp. 118–134, 2005.

F. H. Jackson, “q-difference equations,” American Journal of Mathematics, vol. 32, no. 4, pp. 305–314, 1910.

R. D. Carmichael, “The general theory of Linear -difference equations,” The American Journal of Mathematics, vol. 34, no. 2, pp. 147–168, 1912.

T. Mason, “Properties of the Solutions of Linear q-difference Equations with Entire Function Coefficient,” The American Journal of Mathematics, vol. 37, no. 4, pp. 439–444, 1915.

C. Adams, “On the linear ordinary q-difference equation,” American Mathematics II, vol. 30, no. 1, pp. 439–449, 1915.

W. J. Trjitzinsky, “Analytic theory of linear q-difference equations,” Acta Mathematica, vol. 61, no. 1, pp. 1–38, 1933.

B. Ahmad, A. Alsaedi, and S. K. Ntouyas, “A study of second-order -difference equations with boundary conditions,” Advances in Difference Equations, vol. 2012, article 35, 2012.

C. Yu and J. Wang, “Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives,” Advances in Difference Equations, vol. 2013, article 124, 2013.

T. Sitthiwirattham, J. Tariboon, and S. K. Ntouyas, “Three-point boundary value problems of nonlinear second-order -difference equations involving different numbers of ,” Journal of Applied Mathematics, vol. 2013, Article ID 763786, 12 pages, 2013.