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Generalized Fractional Integral Operators Involving Mittag-Leffler Function.

1. Introduction and Preliminaries

M-L Function. In 1903, Mittag-Leffler [1] introduced the function [E.sub.[lambda]] (z), defined by

[mathematical expression not reproducible] (1)

A further, two-index generalization of this function was given by Wiman [2] as

[mathematical expression not reproducible] (2)

where R([lambda]) > 0 and R([beta]) > 0.

By means of the series representation a generalization of M-L function (2) is introduced by Prabhakar [3] as

[mathematical expression not reproducible], (3)

where [lambda], [beta], [gamma] [member of] C (R([lambda]) > 0). Further, it is an entire function of order [[R([lambda])].sup.-1].

Generalized Fractional Integral Operator. Now, we recall the definition of generalized fractional integral operators involving Fox's H-function as kernel, defined by Saxena and Kumbhat [4] means of the following equations:

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible], (5)

where U and V represent the expressions

[mathematical expression not reproducible], (6)

respectively, with [tau], v > 0. The sufficient conditions of operators are given below:

(i) 1 [less than or equal to] p, q < [infinity], [p.sup.-1] +[q.sup.-1] - 1;

(ii) [mathematical expression not reproducible];

(iii) f(x) [member of] [L.sub.p](0, [infinity]);

(iv) [mathematical expression not reproducible].

An interest in the study of the fractional calculus associated with the Mittag-Leffler function and H-function, its application in the form of differential, and integral equations of, in particular, fractional orders (see [5-10]).

H-Function. Symbol [H.sup.m,n.sub.p,q](x) stands for well known Fox H-function [11], in operator (4) and (5) defined in terms of Mellin-Barnes type contour integral as follows:

[mathematical expression not reproducible], (7)

where

[mathematical expression not reproducible], (8)

[mathematical expression not reproducible].

For the conditions of analytically continuations together with the convergence conditions of H-function, one can see [12,13]. Throughout the present paper, we assume that these conditions are satisfied by the function.

2. Images of M-L Function Involving the Generalized Fractional Integral Operators

In this section, we consider two generalized fractional integral operators involving the Fox's H-function as the kernels and derived the following theorems.

Theorem 1. Let [mathematical expression not reproducible]; then the fractional integration [R.sup.[mu],[alpha].sub.x,r] of the product of M-L function exists, under the condition

[mathematical expression not reproducible]; (9)

then there holds the following formula:

[mathematical expression not reproducible]. (10)

Proof. Let l be the left-hand side of (10); using (3) and (4), we have

[mathematical expression not reproducible]. (11)

Changing the order of the integration valid under the condition given with the theorem, we obtain

[mathematical expression not reproducible]. (12)

Let the substitution [t.sup.r]/[x.sup.r] = w; then t = x[w.sup.(1/r)] in the above term; we get

[mathematical expression not reproducible]. (13)

Using beta function for (13), the inner integral reduces to

[mathematical expression not reproducible]. (14)

Interpreting the right-hand side of (14), in view of the definition (7), we arrive at the result (10).

Theorem 2. Let [mathematical expression not reproducible]; then the fractional integration [K.sup.[epsilon],[alpha].sub.x,r] of the product of M-L function exists, under the condition

[mathematical expression not reproducible] (15)

and then the following formula holds:

[mathematical expression not reproducible]. (16)

Proof. Let p be the left-hand side of (16); using (3) and (5), we have

[mathematical expression not reproducible]. (17)

Changing the order of the integration valid under the condition given with the theorem statement, we obtain

[mathematical expression not reproducible]. (18)

Letting the substitution [x.sup.r]/[t.sup.r] = u, then t = x/[u.sup.(1/r)] in the above term and, using beta function, we get

[mathematical expression not reproducible]. (19)

Interpreting the right-hand side of (19), in view of definition (7), we arrive at the result (16).

3. Integral Transforms of Fractional Integral Involving M-L Function

In this section, Mellin, Laplace, Euler, Whittaker, and K-transforms of the results established in Theorems 1 and 2 have been obtained.

Euler Transform (Sneddon [14]). The Euler transform of a function f(t) is defined as

[mathematical expression not reproducible]. (20)

Theorem 3. Let [mathematical expression not reproducible]; then

[mathematical expression not reproducible]. (21)

Proof. Using (10) and (20) gives

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible]. (23)

Now, we obtain the result (23). This completes the proof of the theorem.

Theorem 4. Let [mathematical expression not reproducible]; then

[mathematical expression not reproducible]. (24)

Proof. In similar manner, in proof of Theorem 3, we obtain the result (24).

Mellin Transform (Debnath and Bhatta [15]). The Mellin transform of a function f(t) is defined as

[mathematical expression not reproducible]. (25)

Theorem 5. All conditions follow from that stated in Theorem 1 with R(s) > R(v); the following result holds:

[mathematical expression not reproducible]. (26)

Proof. From (10) and (25), it gives

[mathematical expression not reproducible]. (27)

Now, evaluating the Mellin transform of [t.sup.[??]+vn-1] using formula given by Mathai et al. [16]. we arrive at (26).

Theorem 6. All conditions follow from what is stated in Theorem 2 with R(1 - [??]) < 1, R(s) > R(v); the following result holds:

[mathematical expression not reproducible]. (28)

Proof. In similar manner, in proof of Theorem 5, we obtain the result (28).

Laplace Transform (Sneddon [14]). The Laplace transform of a function /(i), denoted by F(s), is defined by the equation

[mathematical expression not reproducible]. (29)

Provided the integral (29) is convergent and that the function, f(t), is continuous for t > 0 and of exponential order as t [right arrow] [infinity], (29) maybe symbolicallywritten as

F(s) = L{f(t); s} or f(t) - [L.sup.-1] {F(s);i}. (30)

The following result is well known:

[mathematical expression not reproducible]. (31)

Theorem 7. All conditions follow from what is stated in Theorem 1 with R(s) > 0 and R([??] + vn) > 0; the following result holds:

[mathematical expression not reproducible]. (32)

Proof. we can develop similar line by using result of Laplace integral (31).

Theorem 8. All conditions follow from what is stated in Theorem 2 with R(s) > 0 and R(1 - [??] - vn) > 0; the following result holds:

[mathematical expression not reproducible]. (33)

Proof. In a similar manner, in proof of Theorem 7, we obtain the result (33).

Whittaker Transform (Whittaker and Watson [17]). Due to Whittaker transform, the following result holds:

[mathematical expression not reproducible], (34)

where R([omega][+ or -][zeta]) > -1/2 and [W.sub.[chi],[omega]](t) is the Whittaker confluent hypergeometric function:

[mathematical expression not reproducible], (35)

where [M.sub.[chi],[omega]] is defined by

[mathematical expression not reproducible]. (36)

Theorem 9. Following what is stated in Theorem 1for conditions on parameters, with R[[omega][+ or -]([??] + [zeta] + vn - 1)] > 1/2, then the following result holds:

[mathematical expression not reproducible]. (37)

Proof. Using (10) and (34), it gives

[mathematical expression not reproducible]. (38)

Assume that t = k, [??] dt = dk/[phi]; we get

[mathematical expression not reproducible]. (39)

Interpreting the right-hand side of (39), using (34), we arrive at the result (37).

Theorem 10. Following what is stated in Theorem 2 for conditions on parameters, with R[[omega][+ or -](-[??] + [zeta]- vn - 1)] > 1/2,then the following result holds:

[mathematical expression not reproducible]. (40)

Proof. In a similar manner, in proof of Theorem 9, we obtain the result (40).

K-Transform (Erdelyi et al. [18]). This transform is defined by the following integral equation:

[mathematical expression not reproducible], (41)

where R(p) > 0; [K.sub.v](x) is the Bessel function of the second kind defined by ([18], p. 332)

[mathematical expression not reproducible], (42)

where [W.sub.0,v](x) is the Whittaker function defined in Erdelyi et al. [18].

The following result given in Mathai et al. ([16], p. 54, eq. 2.37) will be used in evaluating the integrals:

[mathematical expression not reproducible]. (43)

Theorem 11. Following what is stated in Theorem 1for conditions on parameters, with R([omega]) > 0; R(([rho] + [??] + vn - 1)[+ or -]l) > 0, then the following result holds:

[mathematical expression not reproducible]. (44)

Proof. Using (10) and (44), it gives

[mathematical expression not reproducible], (45)

and we get

[mathematical expression not reproducible]. (46)

Interpreting the right-hand side of (46), we arrive at the result (44).

Theorem 12. Following what is stated in Theorem 2 for conditions on parameters, with R([omega]) > 0; R(([rho] - [??] - vn) [+ or -]l) > 0, then the following result holds:

[mathematical expression not reproducible]. (47)

Proof. In a similar manner, in proof of Theorem 11, we obtain the result (47).

4. Properties of Integral Operators

Here, we established some properties of the operators as consequences of Theorems 1 and 2. These properties show compositions of power function.

Theorem 13. Following all the conditions on parameters as stated in Theorem 1 with R([psi] + [??]) > 0, then the following result holds true:

[mathematical expression not reproducible]. (48)

Proof. From (10), the left-hand side of (48), we have

[mathematical expression not reproducible], (49)

and again, by (10), the right-hand side of (48) follows:

[mathematical expression not reproducible]. (50)

It seems that Theorem 13 readily follows due to (49) and (50).

Theorem 14. Following all the conditions on parameters as stated in Theorem 2 with R([beta] + [??]) > 0, then the following result holds true:

[mathematical expression not reproducible]. (51)

Proof. From (12), the left-hand side of (51), we have

[mathematical expression not reproducible]. (52)

Again by (12), the right-hand side of (51) follows:

[mathematical expression not reproducible]. (53)

It seems that Theorem 14 readily follows due to (52) and (53).

5. Conclusions

In this article, we have investigated and studied two classes of generalized fractional integral operators involving Fox's H-function as kernel due to Saxena and Kumbhat which are applied on M-L function. We discussed the actions of fractional integral operators under Euler, Mellin, Laplace, Whittaker, and K-transforms and results are given in better pragmatic series solutions. The majority of the results derived here are general in nature and compact forms are fairly helpful in deriving a variety of integral formulas in the theory of integral operators which arises in a range of problems of applied sciences like kinematics, diffusion equation, kinetic equation, fractal geometry, anomalous diffusion, propagation of seismic waves, turbulence, etc. We may obtain other special functions such as M-L function and Bessel-Maitland function (see, e.g., ([19-21]) as its special cases and, therefore, various unified fractional integral presentations can be obtained as special cases of our results.

https://doi.org/10.1155/2018/7034124

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

References

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Hafte Amsalu (iD) and D. L. Suthar (iD)

Department of Mathematics, Wollo University, P.O. Box 1145, Dessie, Ethiopia

Correspondence should be addressed to Hafte Amsalu; yohanahafte@gmail.com

Received 2 March 2018; Accepted 11 April 2018; Published 3 June 2018

Academic Editor: Khalil Ezzinbi
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Title Annotation:Research Article
Author:Amsalu, Hafte; Suthar, D.L.
Publication:Abstract and Applied Analysis
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Date:Jan 1, 2018
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