Composition Formulas of Bessel-Struve Kernel Function

Nisar, K. S.; Mondal, S. R.; Agarwal, P.

doi:https://doi.org/10.1155/2016/9560346

Mathematical Problems in Engineering

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Theory and Applications of Fractional Order Systems 2016

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Research Article | Open Access

Volume 2016 | Article ID 9560346 | https://doi.org/10.1155/2016/9560346

Composition Formulas of Bessel-Struve Kernel Function

K. S. Nisar,¹S. R. Mondal,²and P. Agarwal³

Academic Editor: José A. T. Machado

Received30 Dec 2015

Accepted24 Feb 2016

Published04 May 2016

Abstract

The object of this paper is to study and develop the generalized fractional calculus operators involving Appell’s function due to Marichev-Saigo-Maeda. Here, we establish the generalized fractional calculus formulas involving Bessel-Struve kernel function to obtain the results in terms of generalized Wright functions. The representations of Bessel-Struve kernel function in terms of exponential function and its relation with Bessel and Struve function are also discussed. The pathway integral representations of Bessel-Struve kernel function are also given in this study.

1. Introduction

Fractional calculus has found applications in various and extensive fields of engineering and science such as electromagnetics, fluid mechanics, signals processing, and control theory. It has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Recent researches observed that the solutions of fractional-order differential equations could model real-life situations better, particularly in reaction-diffusion-type problems. Due to the potential applicability to a wide variety of problems, fractional calculus is developed to a large area of mathematics and other engineering applications [1–4]. The fractional integral operator involving several special functions has found great importance and applications in many subfields such as statistical distribution theory, control theory, fluid dynamics, stochastic dynamical system, nonlinear biological systems, astrophysics, and quantum mechanics (see [5–7]).

The influence of fractional integral operators involving various special functions in fractional calculus is very important due to its significance and applications in various subfields of applied mathematical analysis. Many studies related to the fractional calculus are found in the papers of Love [8], McBride [9], Agarwal and Nieto [10], Kalla [11], Kalla and Saxena [12, 13], Saigo [14–16], Saigo and Maeda [17], and Kiryakova [18]. A comprehensive explanation of such operators is given by Miller and Ross [19] and Kiryakova [18].

Recently, researchers investigated and studied about the fractional integration formulas for the Bessel function and generalized Bessel functions (see [20, 21]). The generalization of Bessel function and its applications in fractional transforms are found in [22, 23]. A useful generalization of the hypergeometric fractional integrals, including the Saigo operators ([14–16]), has been introduced by Marichev [24] (see details in Samko et al. [25, page 194]) and later extended and studied by Saigo and Maeda ([17, page 393]) in terms of any complex order with Appell function in the kernel, as follows.

Let and ; then the generalized fractional calculus operators involving the Appell function are defined as follows:with . For more details about the above operators, see [17, 24]. The generalized fractional integral operators of the type (1) and (2) have been introduced by Marichev [24] and later extended and studied by Saigo and Maeda [17] (this operator known as the Marichev-Saigo-Maeda operator). For the definition of the Appell function the interested readers may refer to the monograph by Srivastava and Karlson [26] (see also Erdélyi et al. [27] and Prudnikov et al. [28]). The applications of fractional integral operators are found in many papers ([19, 22, 29–31]). The following results given in [17, 32] are needed in sequel.

Lemma 1. Let such that and Then there exists the relation where

Lemma 2. Let such that and Then there exists the relation

The generalized Wright hypergeometric function is defined by the seriesHere , and (). Asymptotic behavior of this function for large values of argument of was studied in [33] and under the conditionin [34–38]. The Bessel and modified Bessel functions of first kind, the Struve function , and modified Struve function possess power series representation of the form [39]

The Bessel-Struve kernel , [40] which is unique solution of the initial value problem with the initial conditions and , is given by , where and are the normalized Bessel and Struve functions.

Moreover, the Bessel-Struve kernel is a holomorphic function on and it can be expanded in a power series in the form

The present paper is organized as follows. The composition of integral transforms (1) and (2) with Bessel-Struve kernel function and the relation between Bessel-Struve function and other functions are given in Section 2. Section 3 investigates the pathway fractional integration of the Bessel-Struve kernel function and finally the concluding remark is drawn in Section 4.

2. Fractional Integral Formulas

In this section we will investigate the composition of integral transforms (1) and (2) with the Bessel-Struve kernel function defined in (11).

Theorem 3. Let . Suppose that and Then

Proof. Applying (11) and using the definition (1) to (12), we get By changing the order of integration and summation, Hence replacing by in Lemma 1, after some simplification, we obtain the following expression: whose last summation, in view of (8), is easily seen to arrive at the expression (12).

Theorem 4. Let . Suppose that and Then

Proof. Using (2) and (11) and then changing the order of integration and summation, Using Lemma 2, after a little simplification, we obtain the following expression: In view of (8), we obtained the desired result (16).

2.1. Representation of Bessel-Struve Kernel Function in terms of Exponential Function

In this subsection we represent the Bessel-Struve function in terms of exponential function. Also, we derive the Marichev-Saigo-Maeda operator representation of these special cases. The representation of Bessel-Struve kernel function in terms of exponential function isNow, we have the following theorems.

Theorem 5. Let . Suppose that and Then

Proof. From (1), (19), and the definition of Bessel-Struve kernel function (11), we have This together with Lemma 1 yields which is the desired result.

Theorem 6. Let . Suppose that and Then

Proof. The proof of the fractional integration formula (24) would run parallel to the proof of (21) by considering (20). Therefore, we omit the details.

2.2. Relation between Bessel-Struve Kernel Function and Bessel and Struve Function of First Kind

In this subsection we show the relation between and modified Bessel function and modified Struve function by choosing particular values of :

In the coming two theorems, we give the fractional integral representations of (25) and (26).

Theorem 7. Let . Suppose that and Then

Proof. Applying (26) and using the definition (1) to (27), we get Using Lemma 1, we obtain The use of (8) will give the desired result (27).

Theorem 8. Let . Suppose that and Then

Proof. The details of the proof are omitted because it runs as parallel as the proof of (27).

3. Pathway Fractional Integration of Bessel-Struve Kernel Function

The pathway fractional integral operator was introduced and studied by Mathai [41] and Nair [42] and developed further by Mathai and Haubold ([43, 44]) as follows.

Let and the pathway parameter ; thenFor a real scalar , the pathway model for scalar random variables is represented by the following probability density function (p.d.f.):provided that , and , where is the normalizing constant and is called the pathway parameter (for details, see [42]).

Here, we investigate the composition formula of integral transform operator due to Nair, which is expressed in terms of the generalized Wright hypergeometric function, by inserting the generalized Bessel-Struve kernel which is defined in (11). The results given in this section are based on the preliminary assertions given by composition formula of pathway fractional integral (31) with a power function.

Lemma 9 (see [42]). Let , , and . If and , then

Now, we have the following theorems.

Theorem 10. Let and such that , and ; then the following formula holds:

Proof. Applying (11) and (31) and changing the order of integration and summation, we get Using the conditions mentioned in the statement of the theorem and Applying Lemma 9 and using (33) with replaced by , we get which gives the desired result.

By considering the relations given in (19) and (20), we obtain various new integral formulas for Bessel-Struve functions involved in the pathway fractional integration operators.

Theorem 11. Let and such that , and ; then the following formula holds:

Proof. Applying (19) and using (31) with the help of Lemma 9 we can easily prove (37), so the details are omitted.

Theorem 12. Let and such that and ; then the following formula holds:

Proof. The details of proof are omitted because the result (38) can easily derive as similar as the procedure of the proof of Theorem 10 using (20), (11), and (31).

Remark 13. As similar as the method discussed in Theorem 10, one can easily derive the pathway integral representation of (25) and (26).

4. Conclusion

Fractional integral formulas involving Bessel-Struve kernel function have been developed and studied in this paper. The pathway integral representations Bessel-Struve kernel function and its relation between many other functions are also derived in this study. The results thus given in this paper are general in character and likely to find some applications in the theory of special functions.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors have equal contributions. All authors read and approved the final paper.

Acknowledgments

This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project #2015/01/3883.

References

D. Valerio and J. S. Costa, “Introduction to single-input, single-output fractional control,” IET Control Theory & Applications, vol. 5, no. 8, pp. 1033–1057, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Jin, Y.-Q. Chen, and D. Xue, “Time-constant robust analysis of a fractional order [proportional derivative] controller,” IET Control Theory and Applications, vol. 5, no. 1, pp. 164–172, 2011.
View at: Publisher Site | Google Scholar
I. Podlubny, Fractional-Order Systems and Fractional-Order Controllers, UEF-03-94, Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovakia, 1994.
R. Caponetto, G. Dongola, and I. Petras, Fractional Order Systems, Modeling and Control Applications, vol. 72 of World Scientific A, 2010.
D. Baleanu, “About fractional quantization and fractional variational principles,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp. 2520–2523, 2009.
View at: Publisher Site | Google Scholar
V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301, Longman Scientific & Technical, Essex, UK, 1994.
D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010.
View at: Publisher Site | Google Scholar
E. R. Love, “Some integral equations involving hypergeometric functions,” Proceedings of the Edinburgh Mathematical Society, vol. 15, no. 3, pp. 169–198, 1967.
View at: Publisher Site | Google Scholar | MathSciNet
A. C. McBride, “Fractional powers of a class of ordinary differentilal operators,” Proceedings of the London Mathematical Society, vol. 45, no. 3, pp. 519–546, 1982.
View at: Publisher Site | Google Scholar
P. Agarwal and J. J. Nieto, “Some fractional integral formulas for the Mittag-Leffler type function with four parameters,” Open Mathematics, vol. 13, no. 1, pp. 537–546, 2015.
View at: Publisher Site | Google Scholar
S. L. Kalla, “Integral operators involving Fox's H-function II,” Acta Mexicana de Ciencia y Tecnología, vol. 7, pp. 72–79, 1969.
View at: Google Scholar
S. L. Kalla and R. K. Saxena, “Integral operators involving hypergeometric functions,” Mathematische Zeitschrift, vol. 108, pp. 231–234, 1969.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. L. Kalla and R. K. Saxena, “Integral operators involving hypergeometric functions II,” Universidad Nacional de Tucumán Revista Serie A: Matemática y Física Teórica, vol. 24, pp. 31–36, 1974.
View at: Google Scholar
M. Saigo, “A remark on integral operators involving the Gauss hypergeometric functions,” Mathematical Reports, College of General Education, Kyushu University, vol. 11, pp. 135–143, 1978.
View at: Google Scholar
M. Saigo, “A certain boundary value problem for the Euler-Darboux equation,” Mathematica Japonica, vol. 24, no. 4, pp. 377–385, 1979.
View at: Google Scholar | MathSciNet
M. Saigo, “A certain boundary value problem for the Euler-Darboux equation II,” Mathematica Japonica, vol. 25, no. 2, pp. 211–220, 1980.
View at: Google Scholar | MathSciNet
M. Saigo and N. Maeda, “More generalization of fractional calculus,” in Proceedings of the 2nd International Workshop on Transform Methods & Special Functions, pp. 386–400, Varna, Bulgaria, August 1996.
View at: Google Scholar
V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Scientific & Technical, Essex, UK, 1994.
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
A. A. Kilbas and N. Sebastian, “Generalized fractional integration of Bessel function of the first kind,” Integral Transforms and Special Functions, vol. 19, no. 11-12, pp. 869–883, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
H. M. Srivastava and P. Agarwal, “Certain fractional integral operators and the generalized incomplete hypergeometric functions,” Applications and Applied Mathematics, vol. 8, no. 2, pp. 333–345, 2013.
View at: Google Scholar
S. R. Mondal and K. S. Nisar, “Marichev-Saigo-Maeda fractional integration operators involving generalized Bessel functions,” Mathematical Problems in Engineering, vol. 2014, Article ID 274093, 11 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
D. Baleanu and P. Agarwal, “On generalized fractional integral operators and the generalized Gauss hypergeometric functions,” Abstract and Applied Analysis, vol. 2014, Article ID 630840, 5 pages, 2014.
View at: Publisher Site | Google Scholar
O. I. Marichev, “Volterra equation of Mellin convolution type with a Horn function in the kernel,” Izvestiya Akademii Nauk BSSR, Seriya Fiziko-Matematicheskikh Nauk, vol. 1, pp. 128–129, 1974 (Russian).
View at: Google Scholar
S. Samko, A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon & Breach Science Publishers, New York, NY, USA, 1993.
H. M. Srivastava and P. W. Karlson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Limited, New York, NY, USA, 1985.
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, NY, USA, 1953.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions, Vol. 1–5, Gordon & Breach, New York, NY, USA, 1986.
Y. C. Kim, K. S. Lee, and H. M. Srivastava, “Some applications of fractional integral operators and Ruscheweyh derivatives,” Journal of Mathematical Analysis and Applications, vol. 197, no. 2, pp. 505–517, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. Kiryakova, “All the special functions are fractional differintegrals of elementary functions,” Journal of Physics A: Mathematical and General, vol. 30, no. 14, pp. 5085–5103, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
H. M. Srivastava, S.-D. Lin, and P.-Y. Wang, “Some fractional-calculus results for the $\bar{H}$ -function associated with a class of Feynman integrals,” Russian Journal of Mathematical Physics, vol. 13, no. 1, pp. 94–100, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
R. K. Saxena and M. Saigo, “Generalized fractional calculus of the H-function associated with the Appell function F3,” Journal of Fractional Calculus, vol. 19, pp. 89–104, 2001.
View at: Google Scholar
C. Fox, “The asymptotic expansion of generalized hypergeometric functions,” Proceedings London Mathematical Society, vol. 27, no. 4, pp. 389–400, 1928.
View at: Google Scholar
E. M. Wright, “The asymptotic expansion of integral functions defined by Taylor series,” Philosophical Transactions of the Royal Society of London. Series A, vol. 238, pp. 423–451, 1940.
View at: Publisher Site | Google Scholar | MathSciNet
E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proceedings of the London Mathematical Society, vol. 46, no. 2, pp. 389–408, 1940.
View at: Google Scholar | MathSciNet
A. A. Kilbas, M. Saigo, and J. J. Trujillo, “On the generalized Wright function,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 437–460, 2002.
View at: Google Scholar | MathSciNet
A. A. Kilbas and N. Sebastian, “Fractional integration of the product of Bessel functions of the first kind,” Fractional Calculus & Applied Analysis, vol. 13, no. 2, pp. 159–175, 2010.
View at: Google Scholar | MathSciNet
H. M. Srivastava, “Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 1, pp. 56–71, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1992.
A. Gasmi and M. Sifi, “The Bessel-Struve intertwining operator on $C$ and mean-periodic functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 59, pp. 3171–3185, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
A. M. Mathai, “A pathway to matrix-variate gamma and normal densities,” Linear Algebra and its Applications, vol. 396, pp. 317–328, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. S. Nair, “Pathway fractional integration operator,” Fractional Calculus & Applied Analysis, vol. 12, no. 3, pp. 237–252, 2009.
View at: Google Scholar | MathSciNet
A. M. Mathai and H. J. Haubold, “On generalized distributions and pathways,” Physics Letters A, vol. 372, no. 12, pp. 2109–2113, 2008.
View at: Publisher Site | Google Scholar
A. M. Mathai and H. J. Haubold, “Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy,” Physica A: Statistical Mechanics and Its Applications, vol. 375, no. 1, pp. 110–122, 2007.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2016 K. S. Nisar et al. 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.

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