Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FRACTIONAL CALCULUS OPERATORS AND THEIR IMAGE FORMULAS

  • Received : 2015.08.03
  • Published : 2016.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

During the past four decades or so, due mainly to a wide range of applications from natural sciences to social sciences, the so-called fractional calculus has attracted an enormous attention of a large number of researchers. Many fractional calculus operators, especially, involving various special functions, have been extensively investigated and widely applied. Here, in this paper, in a systematic manner, we aim to establish certain image formulas of various fractional integral operators involving diverse types of generalized hypergeometric functions, which are mainly expressed in terms of Hadamard product. Some interesting special cases of our main results are also considered and relevant connections of some results presented here with those earlier ones are also pointed out.

Keywords

Acknowledgement

Supported by : Dongguk University

References

  1. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, 1985.
  2. P. Agarwal, Further results on fractional calculus of Saigo operators, Appl. Appl. Math. 7 (2012), no. 2, 585-594.
  3. P. Agarwal, Generalized fractional integration of the $\overline{H}$-function, Matematiche (Catania) 67 (2012), no. 2, 107-118.
  4. P. Agarwal, Fractional integration of the product of two multivariables H-function and a general class of polynomials, in: Advances in Applied Mathematics 161 and Approximate Theory, 41, 359-374, Springer Proceedings in Mathematics and Statistics 162, 2013.
  5. P. Agarwal, J. Choi, and R. B. Paris, Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl. 8 (2015), no. 5, 451-466. https://doi.org/10.22436/jnsa.008.05.01
  6. P. Agarwal and S. Jain, Further results on fractional calculus of Srivastava polynomials, Bull. Math. Anal. Appl. 3 (2011), no. 2, 167-174.
  7. P. Agarwal, S. Jain, M. Chand, S. K. Dwivedi, and S. Kumar, Bessel functions associated with Saigo-Maeda fractional derivateive operators, J. Fract. Calc. Appl. 5 (2014), no. 2, 9606.
  8. F. AI-Musallam and S. L. Kalla, Asymptotic expansions for generalized gamma and incomplete gamma functions, Appl. Anal. 66 (1997), no. 1-2, 173-187. https://doi.org/10.1080/00036819708840580
  9. F. AI-Musallam and S. L. Kalla, Further results on a generalized gamma function occurring in diffraction theory, Integral Transforms Spec. Funct. 7 (1998), no. 3-4, 175-190. https://doi.org/10.1080/10652469808819198
  10. P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
  11. M. Caputo, Elasticitae dissipazione Zanichelli, Bologna, 1969.
  12. M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math. 78 (1997), no. 1, 19-32. https://doi.org/10.1016/S0377-0427(96)00102-1
  13. M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004), no. 2, 589-602.
  14. M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall, (CRC Press), Boca Raton, London, New York, and Washington, D.C., 2001.
  15. J. Choi and P. Agarwal, Certain class of generating functions for the incomplete hyper-geometric functions, Abstr. Appl. Anal. 2014 (2014), Article ID 714560, 5 papes.
  16. J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, FILOMAT (2015), accepted for publication.
  17. J. Choi and D. Kumar, Certain unified fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials, J. Inequal. Appl. 2014 (2014), Article ID: 499, 15 papes.
  18. R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Singapore , New York, 2000.
  19. S. L. Kalla and R. K. Saxena, Integral operators involving hypergeometric functions, Math. Z. 108 (1969), 231-234. https://doi.org/10.1007/BF01112023
  20. A. A. Kilbas and M. Saigo, Fractional calculus of the H-function, Fukuoka Univ. Sci. Rep. 28 (1998), no. 2, 41-51.
  21. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
  22. V. S. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res Notes Math. 301, Longman Scientific & Technical; Harlow, Co-published with John Wiley, New York, 1994.
  23. V. S. Kiryakova, All the special functions are fractional differintegrals of elementary functions, J. Phys. A 30 (1997), no. 14, 5083-5103
  24. V. S. Kiryakova, Multiple (multi-index) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000), no. 1-2, 241-259. https://doi.org/10.1016/S0377-0427(00)00292-2
  25. V. S. Kiryakova, On two Saigo's fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9 (2006), no. 2, 159-176.
  26. H. Kober, On fractional integrals and derivatives, Quart. J. Math. Oxford Ser. 11 (1940), 193-212.
  27. Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, San Francisco, and London, 1975.
  28. M.-J. Luo, G. V. Milovanovic, and P. Agarwal, Some results on the extended beta and extended hypergeometric functions, Appl. Math. Comput. 248 (2014), 631-651.
  29. O. I. Marichev, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izv. AN BSSR Ser. Fiz.-Mat. Nauk 1 (1974), 128-129.
  30. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010.
  31. A. C. McBride and G. F. Roach (Editors), Fractional Calculus, (University of Strathclyde, Glasgow, Scotland, August 5-11, 1984) Research Notes in Mathematics 138, Pitman Publishing Limited, London, 1985.
  32. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
  33. K. Nishimoto, Fractional calculus, 1 (1984), 2 (1987), 3 (1989), 4 (1991), 5 (1996), Descartes Press, Koriyama, Japan.
  34. K. Nishimoto, An Essence of Nishimoto's Fractional Calculus, (Calculus of the 21st Century): Integration and Differentiation of Arbitrary Order, Descartes Press, Koriyama, 1991.
  35. K. Nishimoto (Editor), Fractional Calculus and Its Applications, (May 29-June 1, 1989), Nihon University (College of Engineering), Koriyama, 1990.
  36. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York, 1974.
  37. E. Ozergin, Some Properties of Hypergeometric Functions, Ph. D. Thesis, Eastern Mediterranean University, North Cyprus, Turkey, 2011.
  38. E. Ozergin, M. A. Ozarslan, and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), no. 16, 4601-4610. https://doi.org/10.1016/j.cam.2010.04.019
  39. R. K. Parmar, A new generalization of Gamma, Beta, hypergeometric and confluent hypergeometric functions, Matematiche (Catania) 69 (2013), no. 2, 33-52.
  40. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  41. T. Pohlen, The Hadamard product and universal power series (Dissertation), Universitat Trier, 2009.
  42. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  43. B. Ross (Editor), Fractional Calculus and Its Applications, (West Haven, Connecticut; June 15-16, 1974), Lecture Notes in Mathematics 457, 1975.
  44. M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1977/78), no. 2, 135-143.
  45. M. Saigo, On generalized fractional calculus operators, In: Recent Advances in Applied Mathematics (Proc. Internat. Workshop held at Kuwait Univ.), Kuwait Univ., Kuwait, (1996), 441-450.
  46. M. Saigo and A. A. Kilbas, Generalized fractional calculus of the H function, Fukuoka Univ. Sci. Rep. 29 (1999), no. 1, 31-45.
  47. M. Saigo and N. Maeda, More generalization of fractional calculus, In: Transform methods & special functions, Varna '96, 386-400, Bulgarian Acad. Sci., Sofia, 1998.
  48. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon et alibi, 1993.
  49. R. K. Saxena and M. Saigo, Generalized fractional calculus of the H-function associated with the Appell function, J. Frac. Calc. 19 (2001), 89-104.
  50. R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russ. J. Math. Phys. 20 (2013), no. 1, 121-128. https://doi.org/10.1134/S1061920813010111
  51. R. Srivastava, Some generalizations of Pochhammer's symbol and their associated families of hypergeometric functions and hypergeometric polynomials, Appl. Math. Inf. Sci. 7 (2013), no. 6, 2195-2206. https://doi.org/10.12785/amis/070609
  52. R. Srivastava and N. E. Cho, Generating functions for a certain class of incomplete hypergeometric polynomials, Appl. Math. Comput. 219 (2012), no. 6, 3219-3225.
  53. R. Srivastava and N. E. Cho, Some extended Pochhammer symbols and their applications involving general- ized hypergeometric polynomials, Appl. Math. Comput. 234 (2014), 277-285.
  54. H. M. Srivastava and P. Agarwal, Certain fractional integral operators and the generalized incomplete hypergeometric functions, Appl. Appl. Math. 8 (2013), no. 2, 333-345.
  55. H. M. Srivastava, A. Cetinkaya, and I. O. Kiymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 (2014), 484-491.
  56. H. M. Srivastava, M. A. Chaudhry, and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), no. 9, 659-683. https://doi.org/10.1080/10652469.2011.623350
  57. 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.
  58. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
  59. H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001), no. 1, 1-52.
  60. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1996.
  61. F. G. Tricomi, Sulla funzione gamma incompleta, Ann. Mat. Pura Appl. (4) 31 (1950), 263-279. https://doi.org/10.1007/BF02428264
  62. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions.Reprint of the fourth (1927) edition.Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996.

Cited by

  1. On homogeneous second order linear general quantum difference equations vol.2017, pp.1, 2017, https://doi.org/10.1186/s13660-017-1471-3
  2. Non-Standard Finite Difference Schemes for Solving Variable-Order Fractional Differential Equations 2017, https://doi.org/10.1007/s12591-017-0378-2
  3. Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations vol.73, pp.2, 2017, https://doi.org/10.1016/j.camwa.2016.11.012
  4. Certain fractional kinetic equations involving the product of generalized k-Bessel function vol.55, pp.4, 2016, https://doi.org/10.1016/j.aej.2016.07.025
  5. (p, q)-Extended Bessel and Modified Bessel Functions of the First Kind vol.72, pp.1-2, 2017, https://doi.org/10.1007/s00025-016-0649-1
  6. On matrix fractional differential equations vol.9, pp.1, 2017, https://doi.org/10.1177/1687814016683359
  7. Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons vol.9, pp.3, 2017, https://doi.org/10.1177/1687814017692946
  8. On new applications of fractional calculus vol.37, pp.3, 2017, https://doi.org/10.5269/bspm.v37i3.18626
  9. The Nehari manifold for a boundary value problem involving Riemann–Liouville fractional derivative vol.2018, pp.1, 2018, https://doi.org/10.1186/s13662-018-1722-8
  10. On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function vol.2018, pp.1, 2018, https://doi.org/10.1186/s13662-018-1694-8
  11. Incomplete Caputo fractional derivative operators vol.2018, pp.1, 2018, https://doi.org/10.1186/s13662-018-1656-1
  12. Ulam–Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions vol.2019, pp.1, 2019, https://doi.org/10.1186/s13662-018-1943-x