Advertisement

Astrophysics and Space Science

, Volume 305, Issue 3, pp 289–296 | Cite as

Fractional Reaction-Diffusion Equations

  • R. K. SaxenaEmail author
  • A. M. Mathai
  • H. J. Haubold
Original Article

Abstract

In a series of papers, Saxena et al. (2002, 2004a, 2004b) derived solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which provide the extension of the work of Haubold and Mathai (1995, 2000). The subject of the present paper is to investigate the solution of a fractional reaction-diffusion equation. The results derived are of general nature and include the results reported earlier by many authors, notably by Jespersen et al. (1999) for anomalous diffusion and del-Castillo-Negrete et al. (2003) for reaction-diffusion systems with Lévy flights. The solution has been developed in terms of the H-function in a compact form with the help of Laplace and Fourier transforms. Most of the results obtained are in a form suitable for numerical computation.

Keywords

Reaction-diffusion Fractional calculus Mittag-Leffler function Laplace transform Mellin transform Fox H-function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Caputo, M.: Elasticita e dissipazione. Zanichelli, Bologna (1969)Google Scholar
  2. Compte, A.: Stochastic foundations of fractional dynamics. Phys. Rev. E 53, 4191–4193 (1996)CrossRefADSGoogle Scholar
  3. Del-Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Front dynamics in reaction-diffusion systems with Lévy flights: A fractional diffusion approach. Phys. Rev. Lett. 91, 018302 (2003)CrossRefADSGoogle Scholar
  4. Del-Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Front propogation and segregation in a reaction-diffusion model with cross-diffusion. Physica D 168–169, 45–60 (2002)MathSciNetCrossRefGoogle Scholar
  5. Debnath, L.: Fractional integral and fractional differential equations in fluid mechanics. Fractional Calculus Appl. Anal. 6, 119–155 (2003)MathSciNetGoogle Scholar
  6. Dzherbashyan, M.M.: Integral transforms and representation of functions in complex domain (in Russian), Nauka, Moscow (1966)Google Scholar
  7. Dzherbashyan, M.M.: Harmonic analysis and boundary value problems in the complex domain. Birkhaeuser-Verlag, Basel (1993)Google Scholar
  8. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, Vol. 1. McGraw-Hill, New York, Toronto, and London (1953)Google Scholar
  9. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, Vol. 2. McGraw-Hill, New York, Toronto, and London (1953)Google Scholar
  10. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of integral transforms, Vol. 1. McGraw-Hill, New York, Toronto, and London (1954)Google Scholar
  11. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G: , Tables of integral transforms, Vol. 2. McGraw-Hill, New York, Toronto, and London (1954b)Google Scholar
  12. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G: , Higher transcendental functions, Vol. 3. McGraw-Hill, New York, Toronto, and London (1955)Google Scholar
  13. Haken, H.: Synergetics: Introduction and advanced topics. Springer-Verlag, Berlin-Heidelberg (2004)Google Scholar
  14. Haubold, H.J., Mathai, A.M.: A heuristic remark on the periodic variation in the number of solar neutrinos detected on Earth. Astrophys. Space Sci. 228, 113–134 (1995)CrossRefADSGoogle Scholar
  15. Haubold, H.J., Mathai, A.M.: The fractional kinetic equation and thermonuclear functions. Astrophys. Space Sci. 273, 53–63 (2000)CrossRefADSzbMATHGoogle Scholar
  16. Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448–455 (2000)MathSciNetCrossRefADSGoogle Scholar
  17. Henry, B.I., Wearne, S.L.: Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math. 62, 870–887 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Turing pattern formation in fractional activator-inhibitor systems. Phys. Rev. E 72, 026101 (2005)MathSciNetCrossRefADSGoogle Scholar
  19. Hundsdorfer, W., Verwer, J.G.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer-Verlag, Berlin-Heidelberg-New York (2003)zbMATHGoogle Scholar
  20. Jespersen, S., Metzler, R., Fogedby, H.C.: Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions. Phys. Rev. E 59, 2736–2745 (1999)CrossRefADSGoogle Scholar
  21. Kilbas, A.A., Saigo, M.: H-transforms: theory and applications. Chapman and Hall/CRC, New York (2004)zbMATHGoogle Scholar
  22. Kulsrud, R.M.: Plasma physics for astrophysics. Princeton University Press, Princeton and Oxford (2005)Google Scholar
  23. Kuramoto, Y.: Chemical oscillations, waves, and turbulence. Dover Publications, Inc., Mineola, New York (2003)Google Scholar
  24. Mathai, A.M.: A handbook of generalized special functions for statistical and physical sciences. Clarendon Press, Oxford (1993)zbMATHGoogle Scholar
  25. Mathai, A.M., Saxena, R.K.: The H-function with applications in statistics and other disciplines. Halsted Press [John Wiley and Sons], New York, London, and Sydney (1978)zbMATHGoogle Scholar
  26. Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: A fractional dynamics approach. Phys. Reports 339 1–77 (2000)MathSciNetCrossRefADSzbMATHGoogle Scholar
  27. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–208 (2004)MathSciNetCrossRefADSzbMATHGoogle Scholar
  28. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. John Wiley and Sons, New York (1993)zbMATHGoogle Scholar
  29. Mittag-Leffler, M.G.: Sur la nouvelle fonction Eα(x). Comptes Rendus Acad. Sci. Paris (Ser.II) 137, 554–558 (1903)Google Scholar
  30. Mittag-Leffler, M.G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogéne. Acta Math. 29, 101–181 (1905)CrossRefzbMATHMathSciNetGoogle Scholar
  31. Murray, J.D.: Mathematical biology. Springer-Verlag, New York (2003)zbMATHGoogle Scholar
  32. Nicolis, G., Prigogine, I.: Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. John Wiley and Sons, New York (1977)zbMATHGoogle Scholar
  33. Oldham, K.B., Spanier, J.: The fractional calculus: Theory and applications of differentiation and integration to arbitrary order. Academic Press, New York; and Dover Publications, New York 2006 (1974)zbMATHGoogle Scholar
  34. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and series, Vol. 3. More Special Functions, Gordon and Breach, New York (1989)Google Scholar
  35. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: Theory and applications. Gordon and Breach, New York (1990)Google Scholar
  36. Saxena, R.K., Mathai, A.M., Haubold, H.J.: On fractional kinetic equations. Astrophys. Space Sci. 282, 281–287 (2002)CrossRefADSGoogle Scholar
  37. Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized fractional kinetic equations. Physica A 344, 657–664 (2004a)MathSciNetCrossRefADSGoogle Scholar
  38. Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 290, 299–310 (2004b)CrossRefADSGoogle Scholar
  39. Tsallis, C.: What should a statistical mechanics satisfy to reflect nature? Physica D 193, 3–34 (2004)MathSciNetCrossRefADSzbMATHGoogle Scholar
  40. Tsallis, C., Bukman, D.J.: Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. Phys. Rev. E 54, R2197–R2200 (1996)CrossRefADSGoogle Scholar
  41. Wilhelmsson, H., Lazzaro, E.: Reaction-diffusion problems in the physics of hot plasmas. Institute of Physics Publishing, Bristol and Philadelphia (2001)CrossRefGoogle Scholar
  42. Wiman, A.: Ueber den Fundamentalsatz in der Theorie der Functionen Eα(x). Acta Mathematica 29, 191–201 (1905)CrossRefzbMATHMathSciNetGoogle Scholar
  43. Wright, E.M.: On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 8, 71–79 (1933)zbMATHGoogle Scholar
  44. Wright, E.M.: The asymptotic expansion of the generalized hypergeo- metric functions. J. Lond. Math. Soc. 10, 387–293 (1935)CrossRefGoogle Scholar
  45. Wright, E.M.: The asymptotic expansion of the generalized hypergeo- metric functions. Proc. Lond. Math. Soc. 46, 389–408 (1940)zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJai Narain Vyas UniversityJodhpurIndia
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Office for Outer Space AffairsUnited NationsViennaAustria

Personalised recommendations