Fractional radial diffusion in an infinite medium with a cylindrical cavity

Povstenko, Y.

doi:10.1090/S0033-569X-09-01114-3

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
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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

Francesco Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals 7 (1996), no. 9, 1461–1477. MR 1409912, DOI https://doi.org/10.1016/0960-0779%2895%2900125-5
F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 291–348. MR 1611587, DOI https://doi.org/10.1007/978-3-7091-2664-6_7
Theo F. Nonnenmacher and Ralf Metzler, Applications of fractional calculus techniques to problems in biophysics, Applications of fractional calculus in physics, World Sci. Publ., River Edge, NJ, 2000, pp. 377–427. MR 1890112, DOI https://doi.org/10.1142/9789812817747_0008
Ralf Metzler and Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77. MR 1809268, DOI https://doi.org/10.1016/S0370-1573%2800%2900070-3

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.

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), no. 6, 461–580. MR 1937584, DOI https://doi.org/10.1016/S0370-1573%2802%2900331-9

R. Metzler and J. Klafter, Accelerated Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett. 51 (2000), 492–498.

R. Kimmich, Strange kinetics, porous media, and NMR, Chem. Phys. 284 (2002), 43–85.

Elliott W. Montroll and Michael F. Shlesinger, On the wonderful world of random walks, Nonequilibrium phenomena, II, Stud. Statist. Mech., XI, North-Holland, Amsterdam, 1984, pp. 1–121. MR 757002

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.

G. Zumofen and J. Klafter, Scale-invariant motion in intermittent chaotic systems, Phys. Rev. E 47 (1993), 851–863.

R. Metzler and A. Compte, Stochastic foundation of normal and anomalous Cattaneo-type transport, Physica A 268 (1999), 454–468.

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett. 9 (1996), no. 6, 23–28. MR 1419811, DOI https://doi.org/10.1016/0893-9659%2896%2900089-4

Walter Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986), no. 11, 2782–2785. MR 861345, DOI https://doi.org/10.1063/1.527251

W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), no. 1, 134–144. MR 974464, DOI https://doi.org/10.1063/1.528578

Ralf Metzler and Joseph Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (2000), no. 1-2, 107–125. MR 1763650, DOI https://doi.org/10.1016/S0378-4371%2899%2900503-8

R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B 104 (2000), 3914–3917.

Andrzej Hanyga, Multidimensional solutions of time-fractional diffusion-wave equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2020, 933–957. MR 1898095, DOI https://doi.org/10.1098/rspa.2001.0904

Y. Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Thermal Stresses 28 (2005), no. 1, 83–102. MR 2119353, DOI https://doi.org/10.1080/014957390523741

Y. Z. Povstenko, Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation, Internat. J. Engrg. Sci. 43 (2005), no. 11-12, 977–991. MR 2163182, DOI https://doi.org/10.1016/j.ijengsci.2005.03.004

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 equation, Internat. J. Solids Structures 44 (2007), no. 7-8, 2324–2348. MR 2295355, DOI https://doi.org/10.1016/j.ijsolstr.2006.07.008

Y. Z. Povstenko, Fundamental solutions to three-dimensional diffusion-wave equation and associated diffusive stresses, Chaos Solitons Fractals 36 (2008), no. 4, 961–972. MR 2379364, DOI https://doi.org/10.1016/j.chaos.2006.07.031

Y. Z. Povstenko, Fractional radial diffusion in a cylinder, J. Mol. Liq. 137 (2008), 46–50.

R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 223–276. MR 1611585

Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073

Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. Based, in part, on notes left by Harry Bateman. MR 0066496

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294

A. S. Galitsyn and A. N. Zhukovsky, Integral Transforms and Special Functions in Heat Conduction Problems, Naukova Dumka, Kiev, 1976. (In Russian).

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integraly i ryady, “Nauka”, Moscow, 1983 (Russian). Spetsial′nye funktsii. [Special functions]. MR 737562