Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Fractional radial diffusion in an infinite medium with a cylindrical cavity

Author: Y. Z. Povstenko
Journal: Quart. Appl. Math. 67 (2009), 113-123
MSC (2000): Primary 26A33
DOI: https://doi.org/10.1090/S0033-569X-09-01114-3
Published electronically: January 7, 2009
MathSciNet review: 2495074
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The time-fractional diffusion equation is employed to study the radial diffusion in an unbounded body containing a cylindrical cavity. The Caputo fractional derivative is used. The solution is obtained by application of Laplace and Weber integral transforms. Several examples of problems with Dirichlet and Neumann boundary conditions are presented. Numerical results are illustrated graphically.

References [Enhancements On Off] (What's this?)

  • 1. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals 7 (1996), 1461-1477. MR 1409912 (97i:26011)
  • 2. F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics. In: A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997, 291-348. MR 1611587 (99f:26010)
  • 3. T. F. Nonnenmacher and R. Metzler, Applications of fractional calculus techniques to problems in biophysics. In: R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000, 377-427. MR 1890112 (2003b:26006)
  • 4. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), 1-77. MR 1809268 (2001k:82082)
  • 5. R. Metzler and T. F. Nonnenmacher, Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chem. Phys. 284 (2002), 67-90.
  • 6. G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), 461-580. MR 1937584 (2003i:70030)
  • 7. R. Metzler and J. Klafter, Accelerated Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett. 51 (2000), 492-498.
  • 8. R. Kimmich, Strange kinetics, porous media, and NMR, Chem. Phys. 284 (2002), 43-85.
  • 9. E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks. In: J. L. Lebowitz and E. W. Montroll (Eds.), Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics, North-Holland, Amsterdam, The Netherlands, 1984, 1-121. MR 757002 (86g:82002)
  • 10. R. Metzler, J. Klafter, and I. M. Sokolov, Anomalous transport in external fields: Continuous time random walks and fractional diffusion equation extended, Phys. Rev. E 58 (1998), 1621-1633.
  • 11. G. Zumofen and J. Klafter, Scale-invariant motion in intermittent chaotic systems, Phys. Rev. E 47 (1993), 851-863.
  • 12. R. Metzler and A. Compte, Stochastic foundation of normal and anomalous Cattaneo-type transport, Physica A 268 (1999), 454-468.
  • 13. F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett. 9 (1996), 23-28. MR 1419811 (97h:35132)
  • 14. W. Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986), 2782-2785. MR 861345 (87m:44008)
  • 15. W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), 134-144. MR 974464 (89m:45017)
  • 16. R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A 278 (2000), 107-125. MR 1763650 (2001b:35138)
  • 17. R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B 104 (2000), 3914-3917.
  • 18. A. Hanyga, Multidimensional solutions of time-fractional diffusion-wave equations, Proc. Roy. Soc. London A 458 (2002), 933-957. MR 1898095 (2003e:35039)
  • 19. Y. Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Thermal Stresses 28 (2005), 83-102. MR 2119353 (2005i:74024)
  • 20. Y. Z. Povstenko, Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation, Int. J. Engng. Sci. 43, (2005), 977-991. MR 2163182 (2006c:35158)
  • 21. Y. Z. Povstenko, Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equations, Int. J. Solids Struct. 44 (2007), 2324-2348. MR 2295355 (2007j:74024)
  • 22. Y. Z. Povstenko, Fundamental solutions to three-dimensional diffusion-wave equations and associated diffusive stresses, Chaos Solitons Fractals 36 (2008), 961-972. MR 2379364
  • 23. Y. Z. Povstenko, Fractional radial diffusion in a cylinder, J. Mol. Liq. 137 (2008), 46-50.
  • 24. R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, In: A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997, 223-276. MR 1611585 (99g:26015)
  • 25. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. MR 2218073 (2007a:34002)
  • 26. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, vol. 3, McGraw-Hill, New York, 1955. MR 0066496 (16:586c)
  • 27. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, 1959. MR 0022294 (9:188a)
  • 28. A. S. Galitsyn and A. N. Zhukovsky, Integral Transforms and Special Functions in Heat Conduction Problems, Naukova Dumka, Kiev, 1976. (In Russian).
  • 29. E. C. Titchmarsh, Eigenfunction Expansion Associated with Second-Order Differential Equations, Clarendon Press, Oxford, UK, 1946. MR 0019765 (8:458d)
  • 30. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions. Nauka, Mocsow, 1983. (In Russian). MR 737562 (85b:33002)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 26A33

Retrieve articles in all journals with MSC (2000): 26A33

Additional Information

Y. Z. Povstenko
Affiliation: Institute of Mathematics and Computer Science, Jan Długosz University of Czȩstochowa, al.Armii Krajowej 13/15, 42–200 Czȩstochowa, Poland
Email: j.povstenko@ajd.czest.pl

DOI: https://doi.org/10.1090/S0033-569X-09-01114-3
Received by editor(s): July 14, 2007
Published electronically: January 7, 2009
Article copyright: © Copyright 2009 Brown University

American Mathematical Society