Advertisement

Astrophysics and Space Science

, Volume 305, Issue 3, pp 297–303 | Cite as

Reaction-Diffusion Systems and Nonlinear Waves

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

Abstract

The authors investigate the solution of a nonlinear reaction-diffusion equation connected with nonlinear waves. The equation discussed is more general than the one discussed recently by Manne et al. (2000). The results are presented in a compact and elegant form in terms of Mittag-Leffler functions and generalized Mittag-Leffler functions, which are suitable for numerical computation. The importance of the derived results lies in the fact that numerous results on fractional reaction, fractional diffusion, anomalous diffusion problems, and fractional telegraph equations scattered in the literature can be derived, as special cases, of the results investigated in this article.

Keywords

Fractional reaction-diffusion Integral transforms Generalized Mittag-Leffler 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. Doetsch, G.: Anleitung zum praktischen Gebrauch der Laplace-Transformation. Oldenbourg, Munich (1956)Google Scholar
  3. Dzherbashyan, M.M.: Integral transforms and representation of functions in complex domain (in Russian). Nauka, Moscow (1966)Google Scholar
  4. Dzherbashyan, M.M.: Harmonic analysis and boundary value problems in the complex domain. Birkhäuser-Verlag, Basel (1993)Google Scholar
  5. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, vol. 1. McGraw-Hill, New York, Toronto, and London (1953)Google Scholar
  6. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of integral transforms, vol. 1. McGraw-Hill, New York, Toronto, and London (1954a)zbMATHGoogle Scholar
  7. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of integral transforms, vol. 2. McGraw-Hill, New York, Toronto, and London (1954b)zbMATHGoogle Scholar
  8. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, vol. 3. McGraw-Hill, New York, Toronto, and London (1955)zbMATHGoogle Scholar
  9. Gilding, B.H., Kersner, R.: Travelling waves in nonlinear diffusion-convection reaction, Birkhäuser-Verlag, Basel-Boston-Berlin (2004)zbMATHGoogle Scholar
  10. Grindrod, P.: Patterns and waves: The theory and applications of reaction-diffusion equations. Clarendon Press, Oxford (1991)Google Scholar
  11. Haken, H.: Synergetics: Introduction and advanced topics. Springer-Verlag, Berlin-Heidelberg (2004)Google Scholar
  12. Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448–455 (2000)MathSciNetCrossRefADSGoogle Scholar
  13. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 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
  15. Kilbas, A.A., Saigo, M.: H-transforms: Theory and applications. Chapman and Hall/CRC, New York (2004)Google Scholar
  16. Kilbas, A.A., Saigo, M., Saxena, R.K.: Generalized Mittag-Leffler function and generalized fractional calculus. Integr. Transf. Spec. Funct. 15, 31–49 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  17. Kulsrud, R.M.: Plasma physics for astrophysics. Princeton University Press, Princeton and Oxford (2005)Google Scholar
  18. Manne, K.K., Hurd, A.J., Kenkre, V.M.: Nonlinear waves in reaction-diffusion systems: The effect of transport memory. Phys. Rev. E 61, 4177–4184 (2000)CrossRefADSGoogle Scholar
  19. Metzler, R., Klafter, J.: The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Reports 339, 1–77 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  20. 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–R208 (2004)zbMATHMathSciNetCrossRefADSGoogle Scholar
  21. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. John Wiley & Sons, New York (1993)zbMATHGoogle Scholar
  22. Mittag-Leffler, M.G.: Sur la nouvelle fonction Eα(x). Comptes Rendus Acad. Sci. Paris (Ser. II) 137, 554–558 (1903)Google Scholar
  23. Mittag-Leffler, M.G.: Sur la representation analytique d'une branche uniforme d'une fonction monogene. Acta Math. 29, 101–181 (1905)zbMATHCrossRefGoogle Scholar
  24. Nicolis, G., Prigogine, I.: Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. John Wiley & Sons, New York (1977)Google Scholar
  25. Oldham, K.B., Spanier, J.: The fractional calculus: Theory and applications of differentiation and integration to arbitrary order. Academic Press, New York (1974)Google Scholar
  26. Orsingher, E., Beghin, L.: Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Rel. Fields 128, 141–160 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  27. Prabhakar, T.R.: A singular integral equation with generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)zbMATHMathSciNetGoogle Scholar
  28. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon and Breach, New York (1990)Google Scholar
  29. Saxena, R.K., Mathai, A.M., Haubold, H.J.: On fractional kinetic equations. Astrophys. Space Sci. 282, 281–287 (2002)CrossRefADSGoogle Scholar
  30. Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized fractional kinetic equations. Physica A 344, 657–664 (2004)MathSciNetCrossRefADSGoogle Scholar
  31. Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 290, 299–310 (2004a)CrossRefADSGoogle Scholar
  32. Saxena, R.K., Mathai, A.M., Haubold, H.J.: Astrophysical thermonuclear functions for Boltzmann-Gibbs statistics and Tsallis statistics. Physica A 344, 649–656 (2004b)MathSciNetCrossRefADSGoogle Scholar
  33. Saxena, R.K., Mathai, A.M., Haubold, H.J.: Fractional reaction- diffusion equations. Astrophys. Space Sci. 305(1), DOI 10.1007/s10509-006-9189-6 (2006)Google Scholar
  34. Smoller, J.: Shock waves and reaction-diffusion equations. Springer-Verlag, New York-Heidelberg-Berlin (1983)zbMATHGoogle Scholar
  35. Strier, D.E., Zanette, D.H., Wio, H.S.: Wave fronts in a bistable reaction-diffusion system with density-dependent diffusivity. Physica A 226, 310 (1995)CrossRefADSGoogle Scholar
  36. Wilhelmsson, H., Lazzaro, E.: Reaction-diffusion problems in the physics of hot plasmas. Institute of Physics Publishing, Bristol and Philadelphia (2001)CrossRefGoogle Scholar
  37. Wiman, A.: Ueber den Fundamentalsatz in der theorie der Functionen Eα(x). Acta Math. 29, 191–201 (1905)zbMATHCrossRefGoogle 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