×

Variational solution of fractional advection dispersion equations on bounded domains in \(\mathbb R^{d}\). (English) Zbl 1117.65169

The authors introduce fractional directional derivatives and integrals and prove their fundamental properties (semigroup property for integrals, fundamental theorem, adjoint operator, Fourier transform). Similar investigations for two-dimensional fractional differential and integral operators with respect to a probability measure are included as well.
The features of these operators are then used to motivate their use in the derivation of a steady-state fractional advection-dispersion equation which, in particular, is based on a space-fractional version of Fick’s diffusion law. This is followed by an analysis of the space of functions where the solutions of the equations should be sought. A variational form of the fractional advection-dispersion equation is derived next, followed by a finite element approximation method for its solution and a discussion of its convergence properties.

MSC:

65R20 Numerical methods for integral equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carreras, Phys Plasmas 8 pp 5096– (2001)
[2] Shlesinger, Phys Rev Lett 58 pp 1100– (1987)
[3] Zaslavsky, Phys Rev E 48 pp 1683– (1993)
[4] Benson, Water Resour Res 36 pp 1413– (2000)
[5] Meerschaert, Phys Rev E 59 pp 5026– (1999)
[6] and , Stable non-Gaussian random processes: Stochastic models with infinite variance, Chapman and Hall, New York, 1994. · Zbl 0925.60027
[7] Meerschaert, J Comp Appl Math 172 pp 65– (2004)
[8] Meerschaert, Fract Calc Appl Anal 7 pp 61– (2004)
[9] Ervin, Numer Methods Partial Differential Eq
[10] Fix, Computers Math Appl 48 pp 1017– (2004)
[11] Roop, J Comp Appl Math
[12] , and , Fractional integrals and derivatives: Theory and applications, Gordon and Breach, New York, 1993.
[13] and , Generalized functions, Academic Press, New York, 1964.
[14] Lu, Water Resour Res 38 pp 1165– (2002)
[15] Singularities in boundary value problems, Springer-Verlag, New York, 1992.
[16] and , The mathematical theory of finite element methods, Springer-Verlag, New York, 1994. · doi:10.1007/978-1-4757-4338-8
[17] Fractional differential equations, Academic Press, New York, 1999.
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.