×

Similarity solution for fractional diffusion equation. (English) Zbl 1474.35644

Summary: Fractional diffusion equation in fractal media is an integropartial differential equation parametrized by fractal Hausdorff dimension and anomalous diffusion exponent. In this paper, the similarity solution of the fractional diffusion equation was considered. Through the invariants of the group of scaling transformations we derived the integro-ordinary differential equation for the similarity variable. Then by virtue of Mellin transform, the probability density function \(p(r, t)\), which is just the fundamental solution of the fractional diffusion equation, was expressed in terms of Fox functions.

MSC:

35R11 Fractional partial differential equations
35C06 Self-similar solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mandelbrot, B. B., The Fractal Geometry of Nature (1982), New York, NY, USA: W. H. Freeman, New York, NY, USA · Zbl 0504.28001
[2] Havlin, S.; Ben-Avraham, D., Diffusion in disordered media, Advances in Physics, 51, 1, 187-292 (2002) · doi:10.1080/00018730110116353
[3] Liu, D.; Li, H.; Chang, F.; Lin, L., Anomalous diffusion on the percolating networks, Fractals, 6, 2, 139-144 (1998)
[4] Ren, F.-Y.; Liang, J.-R.; Wang, X.-T., The determination of the diffusion kernel on fractals and fractional diffusion equation for transport phenomena in random media, Physics Letters A, 252, 3-4, 141-150 (1999)
[5] Zeng, Q.; Li, H., Diffusion equation for disordered fractal media, Fractals, 8, 1, 117-121 (2000)
[6] Cattani, C., Fractals and hidden symmetries in DNA, Mathematical Problems in Engineering, 2010 (2010) · Zbl 1189.92015 · doi:10.1155/2010/507056
[7] Li, M.; Zhao, W., On bandlimitedness and lag-limitedness of fractional Gaussian noise, Physica A, 392, 9, 1955-1961 (2013) · doi:10.1016/j.physa.2012.12.035
[8] Li, M., A class of negatively fractal dimensional Gaussian random functions, Mathematical Problems in Engineering, 2011 (2011) · Zbl 1209.28015 · doi:10.1155/2011/291028
[9] Li, M.; Cattani, C.; Chen, S.-Y., Viewing sea level by a one-dimensional random function with long memory, Mathematical Problems in Engineering, 2011 (2011) · doi:10.1155/2011/654284
[10] Cattani, C.; Pierro, G., On the fractal geometry of DNA by the binary image analysis, Bulletin of Mathematical Biology, 75, 9, 1544-1570 (2013) · Zbl 1272.92013 · doi:10.1007/s11538-013-9859-9
[11] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[12] Giona, M.; Roman, H. E., Fractional diffusion equation for transport phenomena in random media, Physica A, 185, 1-4, 87-97 (1992)
[13] Metzler, R.; Glöckle, W. G.; Nonnenmacher, T. F., Fractional model equation for anomalous diffusion, Physica A, 211, 1, 13-24 (1994)
[14] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (1993), Amsterdam, The Netherlands: Gordon and Breach, Amsterdam, The Netherlands · Zbl 0818.26003
[15] Podlubny, I., Fractional Differential Equations (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[16] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0428.26004
[17] Băleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus: Models and Numerical Methods. Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos (2012), Boston, Mass, USA: World Scientific, Boston, Mass, USA · Zbl 1248.26011 · doi:10.1142/9789814355216
[18] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[19] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Amsterdam, The Netherlands: Elsevier, Amsterdam, The Netherlands · Zbl 1092.45003
[20] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity (2010), London, UK: Imperial College Press, London, UK · Zbl 1210.26004 · doi:10.1142/9781848163300
[21] Xu, M. Y.; Tan, W. C., Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion, Science in China A, 44, 11, 1387-1399 (2001) · Zbl 1138.76320
[22] Li, C. P.; Deng, W. H.; Xu, D., Chaos synchronization of the Chua system with a fractional order, Physica A, 360, 2, 171-185 (2006) · doi:10.1016/j.physa.2005.06.078
[23] Duan, J.-S., Time- and space-fractional partial differential equations, Journal of Mathematical Physics, 46, 1, 13504-13511 (2005) · Zbl 1076.26006 · doi:10.1063/1.1819524
[24] Duan, J.-S., The periodic solution of fractional oscillation equation with periodic input, Advances in Mathematical Physics, 2013 (2013) · Zbl 1291.34008 · doi:10.1155/2013/869484
[25] Duan, J. S.; Rach, R.; Baleanu, D.; Wazwaz, A. M., A review of the Adomian decomposition method and its applications to fractional differential equations, Communications in Fractional Calculus, 3, 2, 73-99 (2012)
[26] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Applied Mathematics and Computation, 191, 1, 12-20 (2007) · Zbl 1193.76093 · doi:10.1016/j.amc.2006.08.162
[27] Wang, Z. H.; Wang, X., General solution of the Bagley-Torvik equation with fractional-order derivative, Communications in Nonlinear Science and Numerical Simulation, 15, 5, 1279-1285 (2010) · Zbl 1221.34020 · doi:10.1016/j.cnsns.2009.05.069
[28] Yang, A. M.; Cattani, C.; Jafari, H.; Yang, X. J., Analytical solutions of the onedimensional heat equations arising in fractal transient conduction with local fractional derivative, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.35016 · doi:10.1155/2013/462535
[29] Bluman, G. W.; Anco, S. C., Symmetry and Integration Methods for Differential Equations (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 1013.34004
[30] Gorenflo, R.; Luchko, Y.; Mainardi, F., Wright functions as scale-invariant solutions of the diffusion-wave equation, Journal of Computational and Applied Mathematics, 118, 1-2, 175-191 (2000) · Zbl 0973.35012 · doi:10.1016/S0377-0427(00)00288-0
[31] Wyss, W., The fractional diffusion equation, Journal of Mathematical Physics, 27, 11, 2782-2785 (1986) · Zbl 0632.35031 · doi:10.1063/1.527251
[32] Buckwar, E.; Luchko, Y., Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, Journal of Mathematical Analysis and Applications, 227, 1, 81-97 (1998) · Zbl 0932.58038 · doi:10.1006/jmaa.1998.6078
[33] Davies, B., Integral Transforms and Their Applications (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 0996.44001 · doi:10.1007/978-1-4684-9283-5
[34] Mathai, A. M.; Saxena, R. K., The H-Function with Applications in Statistics and Other Disciplines (1978), New Delhi, India: John Wiley & Sons, New Delhi, India · Zbl 0382.33001
[35] Srivastava, H. M.; Gupta, K. C.; Goyal, S. P., The H-Functions of One and Two Variables with Applications (1982), New Delhi, India: South Asian, New Delhi, India · Zbl 0506.33007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.