×

zbMATH — the first resource for mathematics

Approximate solution of the fractional advection-dispersion equation. (English) Zbl 1210.65168
The solution of the advection-dispersion equation of fractional order
\[ \frac{\partial^{\alpha}u(x,t)}{\partial t^{\alpha}}+ v(x,t)D_{x}^{\nu}u(x,t)-k(x,t)D_{x}^{\beta}u(x,t)=F(x,t) \]
with homogeneous initial and boundary conditions is presented in the form of Fourier series. The approximate solution is obtained by truncating this series.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
35L50 Initial-boundary value problems for first-order hyperbolic systems
35R11 Fractional partial differential equations
35C10 Series solutions to PDEs
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Meerschaert, M.M.; Scheffler, H.P., Semistable Lévy motion, Frac. calc. appl. anal., 5, 27-54, (2002) · Zbl 1032.60043
[2] Meerschaert, M.; Benson, D.; Scheffler, H.P.; Baeumer, B., Stochastic solution of space – time fractional diffusion equations, Phys. rev. E, 65, 1103-1106, (2002)
[3] West, B.; Bologna, M.; Grigolini, P., Physics of fractal operators, (2003), Springer New York
[4] Zaslavsky, G., Hamiltonian chaos and fractional dynamics, (2005), Oxford University Press Oxford · Zbl 1083.37002
[5] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M., Fractional calculus and continuous-time finance. III. the diffusion limit, (), 171-180 · Zbl 1138.91444
[6] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314, 749-755, (2002) · Zbl 1001.91033
[7] Sabatelli, L.; Keating, S.; Dudley, J.; Richmond, P., Waiting time distributions in financial markets, Eur. phys. J. B, 27, 273-275, (2002)
[8] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[9] Machado, J.T., Discrete time fractional-order controllers, Frac. calc. appl. anal., 4, 47-66, (2001) · Zbl 1111.93307
[10] Baeumer, B.; Meerschaert, M.M.; Benson, D.A.; Wheatcraft, S.W., Subordinated advection – dispersion equation for contaminant transport, Water resource res., 37, 1543-1550, (2001)
[11] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Wheatcraft, S.W., Eulerian derivation of the fractional advection – dispersion equation, J. contaminant hydrol., 48, 69-88, (2001)
[12] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Baeumer, B., Multiscaling fractional advection – dispersion equations and their solutions, Water resource res., 39, 1022-1032, (2003)
[13] Reimus, P.; Pohll, G.; Mihevc, T.; Chapman, J.; Haga, M.; Lyles, B.; Kosinski, S.; Niswonger, R.; Sanders, P., Testing and parameterizing a conceptual model for solute transport in a fractured granite using multiple tracers in a forced-gradient test, Water resour. res., 39, 1356-1370, (2003)
[14] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific Singapore · Zbl 0998.26002
[15] Zaslavsky, G., Chaos, fractional kinetics, and anomalous transport, Phys. rep., 371, 461-580, (2002) · Zbl 0999.82053
[16] 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. physics A, 37, R161-R208, (2004) · Zbl 1075.82018
[17] Metzler, R.; Klafter, J., The random Walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep., 339, 1-77, (2000) · Zbl 0984.82032
[18] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour. res., 36, 6, 1403-1412, (2000)
[19] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Time fractional advection dispersion equation, J. appl. math. comput., 13, 233-245, (2003) · Zbl 1068.26006
[20] Liu, F.; Anh, V.; Turner, I., Numerical solution of space fractional fokker – planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019
[21] Huang, F.; Liu, F., The fundamental solution of the space – time fractional advection – dispersion equation, J. appl. math. comput., 19, 233-245, (2005)
[22] Momani, S.; Odibat, Z., Numerical solutions of the space – time fractional advection – dispersion equation, Numer. methods partial differential equations, 24, 6, 1416-1429, (2008) · Zbl 1148.76044
[23] 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, Appl. math. comput., 191, 12-20, (2007) · Zbl 1193.76093
[24] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[25] Ervin, J.S.; Roop, J.P., Variational solution of fractional advection dispersion equations on bounded domains in \(R^d\), Numer. methods partial differential equations, 23, 2, 256-281, (2007) · Zbl 1117.65169
[26] Hackbusch, Wolfgang, Integral equations: theory and numerical treatment, International series of numerical mathematics, (1995), Birkhäuser Basel, Boston, Berlin · Zbl 0823.65139
[27] Cui, Minggen; Lin, Yingzhen, Nonlinear numerical analysis in the reproducing kernel space, (2009), Nova Science Publisher New York · Zbl 1165.65300
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.