×

zbMATH — the first resource for mathematics

The fundamental solution of the space-time fractional advection-dispersion equation. (English) Zbl 1086.35003
Summary: A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order \(\alpha\in(0,1]\), and the second-order space derivative is replaced with a Riesz-Feller derivative of order \(\beta\in(0,2]\). We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

MSC:
35A08 Fundamental solutions to PDEs
35K57 Reaction-diffusion equations
26A33 Fractional derivatives and integrals
49K20 Optimality conditions for problems involving partial differential equations
44A10 Laplace transform
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V.V. Anh and N.N. Leonenka,Spectral analysis of fractional kinetic equations with random data, J. Stat. Physics,104, N5/6 (2001), 1349–1387. · Zbl 1034.82044
[2] V.V. Anh and N.N. Leonenko,Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Rel. Fields,124 (2002), 381–408. · Zbl 1031.60043
[3] V.V. Anh and N.N. Leonenko,Harmmonic analysis of fractional diffusion-wave equations, Applied Math. Comput.,48(3) (2003), 239–252. · Zbl 0970.35174
[4] M. BAsu and D.P. Acharya,On quadratic fractional generalized solid bi-criterion, J. Appl. Math. and Computing(old:KJCAM)2(2002), 131–144. · Zbl 1007.90038
[5] D.A. Benson,The fractional advection dispersion equation: Development and application, Ph.D. thesis, Univ. of Nev. Reno, 1998.
[6] D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert,Application of a fractional advection-dispersion equation, Water Resources Research,36(6) (2000), 1403–1412.
[7] M. Caputo,Linear model of dissipation whose Q is almost frequency indepent-H, Geophys. J. R. Astr. Soc.,13 (1967), 529–539.
[8] M.M. Djrbashian,Integral transforms and representations of functions in the complex plane, Nauka, 1966 (in russian).
[9] A.M.A. El-Sayed and M.A.E. Aly,Continuation theorem of fractionalorder evolutionary integral equations, J. Appl. Math. and Computing (old:KJCAM)2(2002), 525–534. · Zbl 1011.34047
[10] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi,Higer transcendental functions,3, New York, McGraw-Hill, (1953-1954).
[11] Y. Fujita,Integro differential equation which interpolates the heat equation and the wave equation, Osaka, J. Math.27 (2004), 309–321. · Zbl 0790.45009
[12] A.A. Kilbas, T. Pierantozzi, J. Trujillo,On the solution of fractional evolution equations, J. Phys. A: Math. Gen.37 (2004), 3271–3283. · Zbl 1059.35030
[13] F. Liu, I. Turner and V. Anh,An unstructured mesh finite volume method for modelling saltwater intrusion into coatal aquifer, J. Appl. Math. and Computing (old:KJCAM)9 (2002), 391–407. · Zbl 1002.76074
[14] F. Liu, L.W. Turner, V. Anh and N. Su,A two-dimensional finite volume method for transient simulation of time-, scale-and density-dependent transport in heterogeneous aquifer systems, J. Appl. Math. and Computing11 (2003a), 215–241. · Zbl 1145.76407
[15] F. Liu, I.W. Turner, V. Anh and P. Zhuang,Time fractional advection-dispersion equation, J. Appl. Math. and Computing13(2003b), 233–245. · Zbl 1068.26006
[16] F. Liu, V.V. Anh and I. Turner:Numerical solution of the space fractional Fokker-Plank Equation, J. Comp. Appl. Math.166 2004, 209–319. · Zbl 1036.82019
[17] W. Feller,On a generalization of Marcel Riesz’s potentials and the semigroups generated by them, Meddekanden lunds Universitets Matematiska Seminarium (Comm. Sém. M.athém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, (1952) 73–81.
[18] R. Gorenflo and F. Mainardi,Approximation of Lévy-Feller diffusion by random walk, ZAA,18 (1999), 231–246. · Zbl 0948.60006
[19] R. Gorenflo, Yu. Luchko and F. Mainardi,Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math.118 (2000), 175–191. · Zbl 0973.35012
[20] R. Gorenflo and F. Mainardi,Fractional calculus: integral and differential equations of fractional order, in A. Carpinteri and Mainardi (Editors) Fractals and Fractional Calculus in Continuum Mechanics, Wien and New York, Springer Verlag, (1997), 223–273.
[21] F. Huang and F. Liu,The time fractional diffusion equation and advection-dispersion equation, Submitted to the Australian and New Zealand Industrial and Applied Mathematic Journal (ANZIAM), 2004, in press.
[22] F. Mainardi,Fraction calculus: some basic problems in continuum, and statistical mechanics (A. Carpinteri, F. Mainardi, Eds.),Fractal and Fractional Colin Continuum Mechanics, Springer, Wien (1997), 291–348. · Zbl 0917.73004
[23] F. Mainardi, Y. Luchko, G. Pagnini,The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis,4 (2001), 153–1925. · Zbl 1054.35156
[24] M.M. Meerschaert, D.A. Benson and B. Bäumer,Multidimensional advection and fractional dispersion, Phys. Rev. E.59(5), (1999), 5026–5028.
[25] M.M. Meerschaert and C. Tadjeran,Finite difference approximations for fractional advection-dispersion equations. · Zbl 1126.76346
[26] I. Podlubny,Fractional differential equations, Academic press, San Diego, 1999. · Zbl 0924.34008
[27] R. Schunner, D.A. Benson, M.M. Meerschaert, S.W. Wheatcraft,Eulerian derivation of the factional adverction-dispersion equation, Journal of Contaninant Hydrology48 (2001), 69–88.
[28] A. Saichev and G. Zaslavsky,Fractional kinetic, equations: solutions and applications, Chaos7 (1997), 753–764. · Zbl 0933.37029
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.