Abstract
In this paper, we introduce and investigate a fractional integral operator which contains Fox’s H-function in its kernel. We find solutions to some fractional differential equations by using this operator. The results derived in this paper generalize the results obtained in earlier works by Kilbas et al. [7] and Srivastava and Tomovski [23]. A number of corollaries and consequences of the main results are also considered. Using some of these corollaries, graphical illustrations are presented and it is found that the graphs given here are quite comparable to the physical phenomena of decay processes.
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References
M. Abramowitz, and I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables(Applied Mathematics Series 55, National Bureau of Standards, Washington, D.C., 1964; Reprinted by Dover Publications, New York, 1965 (see also [10])).
R. G. Buschman, and H. M. Srivastava, “The H Function Associated with a Certain Class of Feynman Integrals,” J. Phys. A: Math. Gen. 23, 4707–4710 (1990).
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953).
R. Gorenflo, F. Mainardi, and H. M. Srivastava, “Special Functions in Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena,” Proceedings of the Eighth International Colloquium on Differential Equations (Plovdiv, Bulgaria; August 18–23 (1997) (D. Bainov, Editor), VSP Publishers, Utrecht and Tokyo, 195–202 (1998).
R. Hilfer (Editor), Applications of Fractional Calculus in Physics (World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000).
A. A. Inayat-Hussain, “New Properties of Hypergeometric Series Derivable from Feynman Integrals. II: A Generalisations of the H-Function,” J. Phys. A: Math. Gen. 20, 119–128 (1987).
A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler Function and Generalized Fractional Calculus Operators,” Integral Transforms Spec. Funct. 15, 31–49 (2014).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006).
K. S. Miller, and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore, 1993).
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Editors), NIST Handbook of Mathematical Functions ([With 1 CD-ROM (Windows, Macintosh and UNIX)], U. S. Department of Commerce, National Institute of Standards and Technology, Washington, D. C., 2010; Cambridge University Press, Cambridge, London and New York, 2010 (see also [1])).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Reading, Tokyo, Paris, Berlin and Langhorne (Pennsylvania), 1993).
H. M. Srivastava, “Some Formulas for the Bernoulli and Euler Polynomials at Rational Arguments,” Math. Proc. Cambridge Philos. Soc. 129, 77–84 (2000).
H. M. Srivastava, “Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler and Genocchi Polynomials,” Appl. Math. Inform. Sci. 5, 390–444 (2011).
H. M. Srivastava, “Generating Relations and Other Results Associated with Some Families of the Extended Hurwitz-Lerch Zeta Functions,” SpringerPlus 2, Article ID 2:67, 1–14 (2013).
H. M. Srivastava, “A New Family of the λ-Generalized Hurwitz-Lerch Zeta Functions with Applications,” Appl. Math. Inform. Sci. 8, 1485–1500 (2014).
H. M. Srivastava, and J. Choi, Series Associated with the Zeta and Related Functions (Kluwer Academic Publishers, Dordrecht, Boston and London, 2001).
H. M. Srivastava, and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals (Elsevier Science Publishers, Amsterdam, London and New York, 2012).
H. M. Srivastava, D. Jankov, T. K. Pogány, and R. K. Saxena, “Two-Sided Inequalities for the Extended Hurwitz-Lerch Zeta Function,” Comput. Math. Appl. 62, 516–522 (2011).
H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions of One and Two Variables with Applications (South Asian Publishers, New Delhi and Madras, 1982).
H. M. Srivastava, S.-D. Lin, and P.-Y. Wang, “Some Fractional-Calculus Results for the H-Function Associated with a Class of Feynman Integrals,” Russian J. Math. Phys. 13, 94–100 (2006).
H. M. Srivastava, and R. K. Saxena, “Operators of Fractional Integration and Their Applications,” Appl. Math. Comput. 118, 1–52 (2011).
H. M. Srivastava, R. K. Saxena, T. K. Pogány, and R. Saxena, “Integral and Computational Representations of the Extended Hurwitz-Lerch Zeta Function,” Integral Transforms Spec. Funct. 22, 487–506 (2011).
H. M. Srivastava, and Ž. Tomovski, “Fractional Calculus with an Integral Operator Containing a Generalized Mittag-Leffler Function in the Kernel,” Appl. Math. Comput. 211, 198–210 (2009).
Ž. Tomovski, R. Hilfer, and H. M. Srivastava, “Fractional and Operational Calculus with Generalized Fractional Derivative Operators and Mittag-Leffler Type Functions,” Integral Transforms Spec. Funct. 21, 797–814 (2010).
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Srivastava, H.M., Harjule, P. & Jain, R. A general fractional differential equation associated with an integral operator with the H-function in the kernel. Russ. J. Math. Phys. 22, 112–126 (2015). https://doi.org/10.1134/S1061920815010124
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DOI: https://doi.org/10.1134/S1061920815010124