Quarterly of Applied Mathematics

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
		Online ISSN 1552-4485; Print ISSN 0033-569X

Fractional radial diffusion in an infinite medium with a cylindrical cavity

Author(s): Y. Z. Povstenko
Journal: Quart. Appl. Math. 67 (2009), 113-123.
MSC (2000): Primary 26A33
Posted: January 7, 2009
MathSciNet review: 2495074
Retrieve article in: PDF

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:

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)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|