Time fractional advection-dispersion equation. (English) Zbl 1068.26006

Applying first a simplifying substitution of the dependent variable (which in the end must be inverted) and then using the transforms of Laplace and Mellin the authors construct in terms of H-functions a representation of the fundamental solution to the linear time-fractional diffusion-convection equation (called by them “advection dispersion-equation”) with constant coefficients. In this way they generalize the solution known for the special case of pure diffusion (dispersion) without convection (advection). By “time-fractional” they mean, as has become common language, replacement of the first time derivative by a fractional time derivative of order \(\alpha \in (0, 1]\). Actually, they use the fractional derivative suitably regularized at zero, usually now called the “Caputo fractional derivative” which is appropriate in handling Cauchy problems.


26A33 Fractional derivatives and integrals
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
44A10 Laplace transform
44A15 Special integral transforms (Legendre, Hilbert, etc.)
45K05 Integro-partial differential equations
35K57 Reaction-diffusion equations
Full Text: DOI


[1] Augulo, J. M.; Ruiz-Medina, M. D.; And, V. V.; Grecksch, W., Fractional diffusion and fractional heat equation, Adv. Appl. Prob., 32, 1077-1099 (2000) · Zbl 0986.60077 · doi:10.1239/aap/1013540349
[2] Anh, V. V.; Leonenko, N. N., Scaling laws for fractional diffusion-wave equations with singular data, Statistics and Probability Letters, 48, 239-252 (2000) · Zbl 0970.35174 · doi:10.1016/S0167-7152(00)00003-1
[3] Anh, V. V.; Leonenko, N. N., Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys., 104, 1349-1387 (2001) · Zbl 1034.82044 · doi:10.1023/A:1010474332598
[4] Basu, M.; Acharya, D. P., On quadratic fractional generalized solid bi-criterion, J. Appl. Math. and Computing, 10, 131-144 (2002) · Zbl 1007.90038 · doi:10.1007/BF02936212
[5] Benson., D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-despersion equation, Water Resour. Res., 36, 6, 1403-1412 (2000) · doi:10.1029/2000WR900031
[6] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., The fractional-order governing equation of Levy motion, Water Resour. Res., 36, 6, 1413-1423 (2000) · doi:10.1029/2000WR900032
[7] Biler, P.; Funaki, T.; Woyczynski, W. A., Fractal Burgers equation, I, Differential Equations, 147, 1-38 (1998) · Zbl 0921.35074 · doi:10.1006/jdeq.1998.3435
[8] Caputo, M., The Green function of the diffusion of fluids in porous media with memory, Rend. Fis. Acc. Lincei (ser. 9), 7, 243-250 (1996) · Zbl 0879.76098
[9] El-Sayed, A. M.A.; Aly, M. A.E., Continuation theorem of fractionalorder evolutionary integral equations, Korean.J. Comput. Appl. Math., 9, 525-534 (2002) · Zbl 1011.34047
[10] Erdelyi, A., Tables of Integral Transforms (1954), New York: McGraw-Hill, New York · Zbl 0055.36401
[11] Giona, R.; Roman, H. E., A theory of transport phenomena in disordered systems, Chem. Eng. J. bf, 49, 1-10 (1992) · doi:10.1016/0300-9467(92)85018-5
[12] Gorenflo, R.; Luchko, Yu.; Mainardi, F., Analytical properties and applications of the Wright function, Fractional Calculus Appl. Anal., 2, 383-414 (1999) · Zbl 1027.33006
[13] Gorenflo, R.; Luchko, Yu.; Mainardi, F., Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math., 118, 175-191 (2000) · Zbl 0973.35012 · doi:10.1016/S0377-0427(00)00288-0
[14] Hilfer, R., Exact solutions for a class of fractal time random walks, Fractals, 3, 211-216 (1995) · Zbl 0881.60066 · doi:10.1142/S0218348X95000163
[15] F. Liu, V. Anh and I. Turner,Numerical solution of the fractional-order advection-dispersion equation, Proceedings of the International Conference on Boundary and Interior Layers, Perth, Australia, (2002), 159-164.
[16] Mainardi, F.; Rionero, S.; Ruggeri, T., On the initial value problem for the fractional diffusion-wave equation, Waves and Stability in Continuous Media, 246-251 (1994), Singapore: World Scientific, Singapore
[17] Mainardi, F.; Wagner, J. L.; Norwood, F. R., Fractional diffusive waves in viscoelastic solids, IUTAM Symposium—Nonlinear Waves in Solids, 93-97 (1995), Fairfield NJ: ASME/AMR, Fairfield NJ
[18] F. Mainardi, Yu. Luchko and G. Pagnini,The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus Appl. Anal.,4, (2001). · Zbl 1054.35156
[19] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York: John Wiley, New York · Zbl 0789.26002
[20] K.B. Oldham and J. Spanier,The Fractional Calculus, Academic Press, 1974. · Zbl 0292.26011
[21] I. Podlubny,Fractional Differential Equations, Academic Press, 1999. · Zbl 0924.34008
[22] Saichev, A.; Zaslavsky, G., Fractional kinetic equations: solutions and applications, Chaos, 7, 753-764 (1997) · Zbl 0933.37029 · doi:10.1063/1.166272
[23] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Newark, N J: Gordon and Breach, Newark, N J · Zbl 0818.26003
[24] Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578
[25] Wyss, W., The fractional diffusion equation, J. Math. Phys., 27, 2782-2785 (1986) · Zbl 0632.35031 · doi:10.1063/1.527251
[26] Wyss, W., The fractional Black-Scholes equation, Fractional Calculus Appl. Anal., 3, 51-61 (2000) · Zbl 1058.91045
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.