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Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

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Central European Journal of Physics

Abstract

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ 1,…,δ m ) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

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References

  1. R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)

    Book  MATH  Google Scholar 

  2. V.V. Uchaikin, Method of fractional derivatives (Artishok, Ul’janovsk, 2008), in Russian

    Google Scholar 

  3. R.R. Nigmatullin, D. Baleanu, Fract. Calc. Appl. Anal. 15, 718 (2012)

    MathSciNet  Google Scholar 

  4. R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, Singapore, 2011)

    Book  Google Scholar 

  5. J.L.A. Dubbeldam, A. Milchev, V.G. Rostiashvili, T.A. Vilgis, Phys. Rev. E 76, 010801 (2007)

    Article  ADS  Google Scholar 

  6. A. Freed, K. Diethelm, and Yu. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus (NASA’s Glenn Research Center, Ohio, 2002)

    Google Scholar 

  7. R. Klages, G. Radons, and I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications (Wiley-VCH, Weinheim, 2008)

    Book  Google Scholar 

  8. Yu. Luchko, Forum der Berliner mathematischen Gesellschaft 19, 53 (2011)

    Google Scholar 

  9. Yu. Luchko, A. Punzi, Int. J. Geom. 1, 257 (2011)

    Article  MathSciNet  Google Scholar 

  10. R.L. Magin, Crit. Rev. Biomed. Eng. 32, 195 (2004)

    Article  Google Scholar 

  11. F. Mainardi, Chaos. Soliton. Fract. 7, 1461 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press & World Sci., London — Singapore, 2010)

    Book  Google Scholar 

  13. R. Metzler and J. Klafter, J. Phys. A. Math. Gen. 37, R161 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordon & Breach. Sci. Publ., London-N. York, 1993)

    MATH  Google Scholar 

  15. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering 198 (Academic Press, N. York-Boston, 1999)

    MATH  Google Scholar 

  16. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  17. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods (World Scientific, Singapore, 2012)

    MATH  Google Scholar 

  18. M. Caputo, Geophys. J. R. Astr. Soc. 13, 529 (1967)

    Article  Google Scholar 

  19. Ch. Li, D. Qian, YQ Chen, Discrete Dynamics in Nature and Society 2011, 562494 (2011)

    MathSciNet  Google Scholar 

  20. I. Dimovski, Compt. Rend. Acad. Bulg. Sci. 19, 1111 (1966)

    MathSciNet  Google Scholar 

  21. V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes in Math. 301(Longman Sci. Tech. & J. Wiley, Harlow-N. York, 1994)

    MATH  Google Scholar 

  22. S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Ser.: Mathematics and Its Applications 287 (Kluwer Acad. Publ., Dordrecht-Boston-London, 1994)

    Book  MATH  Google Scholar 

  23. A.P. Prudnikov, Yu. Brychkov, O.I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Gordon and Breach Sci. Publ., N. York-London-Paris-Tokyo, etc., 1990)

    Google Scholar 

  24. H.M. Srivastava, R.G. Buschman, Theory and Applications of Convolution Integral Equations (Kluwer Acad. Publ., Dordrecht-Boston-London, 1992)

    Book  MATH  Google Scholar 

  25. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, 3 (McGraw-Hill, New York, 1953)

    Google Scholar 

  26. V. Kiryakova, Recent Advances in Appl. Mathematics’ 96 (Proc. Intern. Workshop, Kuwait University, 1996) 281

    Google Scholar 

  27. V. Kiryakova, Fract. Calc. Appl. Anal. 2, 445 (1999)

    MATH  MathSciNet  Google Scholar 

  28. Yu. Luchko, Fract. Calc. Appl. Anal. 2, 463 (1999)

    MATH  MathSciNet  Google Scholar 

  29. V. Kiryakova, J. Comput. Appl. Math. 118, 214 (2000)

    MathSciNet  Google Scholar 

  30. Yu. Luchko, S.B. Yakubovich, Math. Balkanica 7, 119 (1993)

    MATH  MathSciNet  Google Scholar 

  31. E. Buckwar, Yu. Luchko, J. Math. Anal. Appl. 227, 81 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. R. Gorenflo, Yu. Luchko, F. Mainardi, Fract. Calc. Appl. Anal. 2, 383 (1999)

    MATH  MathSciNet  Google Scholar 

  33. R. Gorenflo, Yu. Luchko, F. Mainardi, J. Comput. Appl. Math. 11, 175 (2000)

    Article  MathSciNet  Google Scholar 

  34. Yu. Luchko, R. Gorenflo, Fract. Calc. Appl. Anal. 1, 63 (1998)

    MATH  MathSciNet  Google Scholar 

  35. V. Kiryakova, Comput. Math. Appl. 59, 1885 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. V. Kiryakova, Yu. Luchko, American Institute of Physics-Conf. Proc. 1301, 597 (2010)

    ADS  MathSciNet  Google Scholar 

  37. M. Dzrbashjan, Izv. Akad. Nauk Armen. SSR 13, 21 (1960), In Russian

    Google Scholar 

  38. I.N. Sneddon, Lecture Notes in Math. 457, Fractional Calculus and Its Applications (Proc. Internat. Conf. Held in New Haven, Springer, N. York, 1975) 37

    Book  Google Scholar 

  39. O.P. Agrawal, Fract. Calc. Anal. Appl. 15, 700 (2012)

    Google Scholar 

  40. S.L. Kalla, V.S. Kiryakova, Math. Japonica 35, 1 (1990)

    MathSciNet  Google Scholar 

  41. V. Kiryakova, American Institute of Physics-Conf. Proc. 1410, 247 (2011)

    ADS  MathSciNet  Google Scholar 

  42. Yu. Luchko, Fract. Calc. Appl. Anal. 7, 339 (2004)

    MATH  MathSciNet  Google Scholar 

  43. Yu. Luchko, J.J. Trujillo, Fract. Calc. Appl. Anal. 10, 249 (2007)

    MATH  MathSciNet  Google Scholar 

  44. G. Pagnini, Fract. Calc. Appl. Anal. 15, 117 (2012)

    MATH  MathSciNet  Google Scholar 

  45. G. Pagnini, A. Mura, F. Mainardi, Intern. J. of Stochastic Analysis 2012, 427383 (2012)

    MathSciNet  Google Scholar 

  46. L. Rodriguez-Germa, J.J. Trujillo, L. Vazquez, M. Pilar-Velasco, Fract. Calc. Appl. Anal. 11, 431 (2008)

    MATH  MathSciNet  Google Scholar 

  47. Yu. Luchko, Integr. Transforms Spec. Funct. 19, 809 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  48. Yu. Luchko, V. Kiryakova, Fract. Calc. Appl. Anal. 16, 405 (2013)

    MathSciNet  Google Scholar 

  49. H. Kober, Quart. J. Math. Oxford ll, 193 (1940)

  50. V. Kiryakova, Fract. Calc. Appl. Anal. 11, 201 (2008)

    Google Scholar 

  51. J. Mikusinski, Cz. Ryll-Nardzewski, Studia Math. 12, 51 (1951)

    MATH  MathSciNet  Google Scholar 

  52. I. Dimovski, V. Kiryakova, Numer. Funct. Anal. Opt. 21, 121 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  53. Yu. Luchko, V. Kiryakova, C.R. Acad. Bulg. Sci. 53, 17 (2000)

    MATH  MathSciNet  Google Scholar 

  54. Yu. Luchko, V. Kiryakova, Algebraic Analysis and Related Topics, Banach Center Publ. 53 (Proc. Conf., Warsaw, 2000) 155

    Google Scholar 

  55. I. Ali, V. Kiryakova, S. Kalla, J. Math. Anal. Appl. 269, 172 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  56. K. Furati, Fract. Calc. Appl. Anal. 16, 171 (2013)

    MathSciNet  Google Scholar 

  57. M.A. Al-Bassam, Yu. Luchko, Journal of Fractional Calculus 7, 69 (1995)

    MATH  MathSciNet  Google Scholar 

  58. S.B. Hadid, Yu. Luchko, Panamerican Mathematical Journal 6, 57 (1996)

    MATH  MathSciNet  Google Scholar 

  59. Yu. Luchko, H.M. Srivastava, Comput. Math. Appl. 29, 73 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  60. I. Dimovski, V. Kiryakova, Proc. Conf. Complex Anal. and Appl., Varna’ 81, 148 (1984)

    MathSciNet  Google Scholar 

  61. F. Al-Musallam, V. Kiryakova, Vu Kim Tuan, Rocky Mountain J. Math. 32, 409 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  62. M. Saigo, Mathematical Reports Kyushu Univ. 11, 135 (1978)

    MathSciNet  Google Scholar 

  63. A. Rao, M. Garg, S.L. Kalla, Kuwait J. Sci. Engg. 37, 15 (2010)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bontchev” Str., Block 8, 1113, Sofia, Bulgaria

    Virginia Kiryakova

  2. Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, D - 13353, Berlin, Germany

    Yuri Luchko

Authors
  1. Virginia Kiryakova
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  2. Yuri Luchko
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Correspondence to Virginia Kiryakova.

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Kiryakova, V., Luchko, Y. Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. centr.eur.j.phys. 11, 1314–1336 (2013). https://doi.org/10.2478/s11534-013-0217-1

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  • DOI: https://doi.org/10.2478/s11534-013-0217-1

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