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General solution of a fractional Parker diffusion-convection equation describing the superdiffusive transport of energetic particles

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Abstract.

Anomalous diffusion models of energetic particles in space plasmas are developed by introducing the fractional Parker diffusion-convection equation. Analytical solution of the space-time fractional equation is obtained by use of the Caputo and Riesz-Feller fractional derivatives with the Laplace-Fourier transforms. The solution is given in terms of the Fox H-function. Profiles of particle densities are illustrated for different values of the space fractional order and the so-called skewness parameter.

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References

  1. Haitao Qi, Xiaoyun Jiang, Physica A 390, 1876 (2011)

    Article  MathSciNet  Google Scholar 

  2. Xiaoyun Jiang, Haitao Qi, Nonlinear Dyn. 62, 895 (2010)

    Article  Google Scholar 

  3. Bo Yu, Xiaoyun Jiang, Adv. Math. Phys. 2013, 479634 (2013)

    Google Scholar 

  4. Yujie Li, German Farrher, Rainer Kimmich, Phys. Rev. E 74, 066309 (2006)

    Article  ADS  Google Scholar 

  5. L. Zunino, D.G. Pérez, A. Kowalski, M.T. Martín, M. Garavaglia, A. Plastino, O.A. Rosso, Physica A 387, 6057 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  6. A.A. Stanislavsky, Karina Weron, Astrophys. Space Sci. 312, 343 (2007)

    Article  ADS  Google Scholar 

  7. A.H. Bhrawy, Eur. Phys. J. Plus 131, 12 (2016)

    Article  Google Scholar 

  8. Igor Goychuk, Phys. Rev. E 86, 021113 (2012)

    Article  Google Scholar 

  9. Ralf Metzler, Joseph Klafter, Phys. Rep. 339, 1 (2000)

    Article  Google Scholar 

  10. Ralf Metzler, Jae-Hyung Jeon, Andrey G. Cherstvy, Eli Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014)

    Article  Google Scholar 

  11. Diego del Castillo-Negrete, B.A. Carreras, V.E. Lynch, Phys. Rev. Lett. 94, 065003 (2005)

    Article  ADS  Google Scholar 

  12. G. Zimbardo, P. Pommois, P. Veltri, Astrophys. J. Lett. 639, L91 (2006)

    Article  ADS  Google Scholar 

  13. G. Qin, W.H. Matthaeus, J.W. Bieber, Geophys. Res. Lett., https://doi.org/10.1029/2001GL014035 (2002)

  14. S. Perri, G. Zimbardo, Astrophys. Space Sci. Trans. 4, 27 (2008)

    Article  ADS  Google Scholar 

  15. Silvia Perri, Gaetano Zimbardo, J. Geophys. Res.: Space Phys. 113, A03107 (2008)

    ADS  Google Scholar 

  16. Vladimir V. Uchaikin, JETP Lett. 92, 200 (2010)

    Article  ADS  Google Scholar 

  17. Vladimir Vasilyevich Uchaikin, JETP Lett. 91, 105 (2010)

    Article  Google Scholar 

  18. Vladimir V. Uchaikin, Phys. Usp. 56, 1074 (2013)

    Article  ADS  Google Scholar 

  19. Yuri E. Litvinenko, Frederic Effenberger, Astrophys. J. 796, 125 (2014)

    Article  ADS  Google Scholar 

  20. M.C. Rocca, A.R. Plastino, A. Plastino, G.L. Ferri, A. de Paoli, Physica A 447, 402 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  21. Silvia Perri, Gaetano Zimbardo, Frederic Effenberger, Horst Fichtner, Astron. Astrophys. 578, A2 (2015)

    Article  Google Scholar 

  22. Silvia Perri, Elena Amato, Gaetano Zimbardo, Astron. Astrophys. 596, A34 (2016)

    Article  Google Scholar 

  23. Robin Stern, Frederic Effenberger, Horst Fichtner, Tobias Schäfer, Fractional Calculus Appl. Anal. 17, 171 (2014)

    MathSciNet  Google Scholar 

  24. Ashraf M. Tawfik, Horst Fichtner, Reinhard Schlickeiser, A. Elhanbaly, Physica A 491, 810 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  25. A.M. Tawfik, H. Fichtner, R. Schlickeiser, A. Elhanbaly, J. Phys.: Conf. Ser. 869, 012050 (2017)

    Google Scholar 

  26. L. O’C Drury, Rep. Prog. Phys. 46, 973 (1983)

    Article  ADS  Google Scholar 

  27. Frank C. Jones, Donald C. Ellison, Space Sci. Rev. 58, 259 (1991)

    Article  ADS  Google Scholar 

  28. V. Uchauikin, R. Sibatov, Fractional Kinetics in Space: Anomalous Transport Models (World Scientific, 2018)

  29. G.P. Zank, P. Hunana, P. Mostafavi, J.A. Le Roux, Gang Li, G.M. Webb, O. Khabarova, A. Cummings, E. Stone, R. Decker, Astrophys. J. 814, 137 (2015)

    Article  ADS  Google Scholar 

  30. G. Zimbardo, S. Perri, F. Effenberger, H. Fichtner, Astron. Astrophys. 607, A7 (2017)

    Article  ADS  Google Scholar 

  31. Michele Caputo, Geophys. J. Int. 13, 529 (1967)

    Article  Google Scholar 

  32. K.M. Saad, Eur. Phys. J. Plus 133, 94 (2018)

    Article  Google Scholar 

  33. Igor Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198 (Academic Press, 1998)

  34. William Feller, An Introduction to Probability Theory and its Applications, volume I, Vol. 3 (John Wiley & Sons New York, 1968)

  35. H.M. Srivastava, K.C. Gupta, S.P. Goyal, The H-functions of one and two variables with applications (South Asian Publ., New Delhi-Madras, 1982)

  36. Richard B. Paris, David Kaminski, Asymptotics and Mellin-Barnes Integrals, Vol. 85 (Cambridge University Press, 2001)

  37. R.K. Saxena, A.M. Mathai, H.J. Haubold, Astrophys. Space Sci. 305, 289 (2006)

    Article  ADS  Google Scholar 

  38. Rudolf Gorenflo, Anatoly Aleksandrovich Kilbas, Francesco Mainardi, Sergei V. Rogosin. Mittag-Leffler Functions, Related Topics and Applications (Springer, 2014)

  39. E. Maitland Wright, J. London Math. Soc. 1, 71 (1933)

    Article  Google Scholar 

  40. Farhad Ali, Syed Aftab Alam Jan, Ilyas Khan, Madeha Gohar, Nadeem Ahmad Sheikh, Eur. Phys. J. Plus 131, 310 (2016)

    Article  Google Scholar 

  41. Francesco Mainardi, Antonio Mura, Gianni Pagnini, Int. J. Differ. Equ. 2010, 104505 (2010)

    Google Scholar 

  42. Olivier Vallée, Manuel Soares, Airy Functions and Applications to Physics (World Scientific Publishing Company, 2010)

  43. R. Schlickeiser, Phys. Plasmas 22, 091502 (2015)

    Article  ADS  Google Scholar 

  44. Hans J. Haubold, Arak M. Mathai, Ram K. Saxena, J. Comput. Appl. Math. 235, 1311 (2011)

    Article  MathSciNet  Google Scholar 

  45. R.K. Saxena, A.M. Mathai, H.J. Haubold, Astrophys. Space Sci. 290, 299 (2004)

    Article  ADS  Google Scholar 

  46. E.M. Trotta, G. Zimbardo, Astron. Astrophys. 530, A130 (2011)

    Article  ADS  Google Scholar 

  47. T. Sugiyama, D. Shiota, Astrophys. J. Lett. 731, L34 (2011)

    Article  ADS  Google Scholar 

  48. W.M. Macek, A. Wawrzaszek, V. Carbone, J. Geophys. Res. 117, A12101 (2012)

    Article  ADS  Google Scholar 

  49. A. Shalchi, Ioannis Kourakis, Phys. Plasmas 14, 092903 (2007)

    Article  ADS  Google Scholar 

  50. F Effenberger, Anisotropic diffusion of energetic particles in galactic and heliospheric magnetic fields, PhD thesis, Ruhr-Universitaet Bochum, 2012

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Authors and Affiliations

  1. Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780, Bochum, Germany

    Ashraf M. Tawfik, Horst Fichtner & Reinhard Schlickeiser

  2. Theoretical Physics Research Group, Mansoura University, 35516, Mansoura, Egypt

    Ashraf M. Tawfik & A. Elhanbaly

Authors
  1. Ashraf M. Tawfik
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  2. Horst Fichtner
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  3. A. Elhanbaly
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  4. Reinhard Schlickeiser
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Correspondence to Ashraf M. Tawfik.

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Tawfik, A.M., Fichtner, H., Elhanbaly, A. et al. General solution of a fractional Parker diffusion-convection equation describing the superdiffusive transport of energetic particles. Eur. Phys. J. Plus 133, 209 (2018). https://doi.org/10.1140/epjp/i2018-12049-3

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