## 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.

### MSC:

 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
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### References:

 [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
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