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An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. (English) Zbl 1201.80065
Summary: An anomalous diffusion version of a limit Stefan melting problem is posed. In this problem, the governing equation includes a fractional time derivative of order $0 < \beta \leqslant 1$ and a fractional space derivative for the flux of order $0 < \alpha \leqslant 1$. Solution of this fractional Stefan problem predicts that the melt front advance as $s = t^{\gamma}, \gamma = \frac {\beta}{\alpha+1}$. This result is consistent with fractional diffusion theory and through appropriate choice of the order of the time and space derivatives, is able to recover both sub-diffusion and super-diffusion behaviors for the melt front advance.

80A22Stefan problems, phase changes, etc.
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[1] Crank, J.: Free and moving boundary problems, (1984) · Zbl 0547.35001
[2] Juric, D.; Tryggvason, G.: A front tracking method for dendritic solidification, J. comp. Phys. 123, 127-148 (1996) · Zbl 0843.65093 · doi:10.1006/jcph.1996.0011
[3] Chen, S.; Merriman, B.; Osher, S.; Smereka, P.: A simple level set method for solving Stefan problems, J. comp. Phys. 135, 8-29 (1997) · Zbl 0889.65133 · doi:10.1006/jcph.1997.5721
[4] Karma, A.; Rappel, W. -J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. rev. E 57, 4323-4349 (1998) · Zbl 1086.82558 · doi:10.1103/PhysRevE.57.4323
[5] Beckermann, C.; Diepers, H. -J.; Steinbach, I.; Karma, A.; Tong, X.: Modeling melt convection in phase-field simulations of solidification, J. comp. Phys. 154, 468-496 (1999) · Zbl 0960.82015 · doi:10.1006/jcph.1999.6323
[6] Kim, Y. -T.; Goldenfeld, N.; Dantzig, J.: Computation of dendritic microstructures using a level set method, Phys. rev. E 62, 2471-2474 (2000)
[7] Udaykumar, H. S.; Marella, S.; Krishnan, S.: Sharp-interface simulation of dendritic growth with convection: benchmarks, Int. J. Heat mass transfer 46, 2615-2627 (2003) · Zbl 1037.76064 · doi:10.1016/S0017-9310(03)00038-3
[8] Pal, D.; Bhattacharya, J.; Dutta, P.; Chakraborty, S.: An enthalpy model for simulation of dendritic growth, Numer. heat transfer B 50, 59-78 (2006)
[9] Voller, V. R.: An enthalpy method for modeling dendritic growth in a binary alloy, Int. J. Heat mass transfer 51, 823-834 (2008) · Zbl 1132.80315 · doi:10.1016/j.ijheatmasstransfer.2007.04.025
[10] Swenson, J. B.; Voller, V. R.; Paola, C.; Parker, G.; Marr, J. G.: Fluvio-deltaic sedimentation: a generalized Stefan problem, Euro. J. App. math. 11 (2000) · Zbl 0964.80005 · doi:10.1017/S0956792500004198
[11] Marr, J. G.; Swenson, J. B.; Paola, C.; Voller, V. R.: A two-diffusion model of fluvial stratigraphy in closed depositional basins, Basin research 12, 381-398 (2000)
[12] Voller, V. R.; Swenson, J. B.; Paola, C.: An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat mass transfer 47, 5387-5390 (2004) · Zbl 1077.80004 · doi:10.1016/j.ijheatmasstransfer.2004.07.007
[13] Capart, H.; Bellal, M.; Young, D. L.: Self-similar evolution of semi-infinite alluvial channels with moving boundaries, J. sediment. Res. 77, 13-22 (2007)
[14] Lai, S. Y. J.; Capart, H.: Two-diffusion description of hyperpycnal deltas, J. geophys. Res. 112, F03005 (2007)
[15] Lai, S. Y. J.; Capart, H.: Reservoir infill by hyperpycnal deltas over bedrock, Geophys. res. Lett. 36, L08402 (2009)
[16] Lorenzo-Trueba, J.; Voller, V. R.; Muto, T.; Kim, W.; Paola, C.; Swenson, J. B.: A similarity solution for a dual moving boundary problem associated with a coastal-plain depositional system, J. fluid mech. 628, 427-443 (2009) · Zbl 1181.76114 · doi:10.1017/S0022112009006715
[17] Voller, V. R.; Swenson, J. B.; Kim, W.; Paola, C.: An enthalpy method for moving boundary problems on the Earth’s surface, Int. J. Numer. meth. Heat fluid flow 16, 641-654 (2006) · Zbl 1121.86006 · doi:10.1108/09615530610669157
[18] Patnaik, S.; Voller, V. R.; Parker, G.; Frascati, A.: Morphology of a melt front under a condition of spatial varying latent heat, Int. commun. Heat mass transfer 36, 535-538 (2009)
[19] Lorenzo-Trueba, J.; Voller, V. R.: Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation, J. math. Anal. appl. 366 (2010) · Zbl 1183.86002 · doi:10.1016/j.jmaa.2010.01.008
[20] Carslaw, H. S.; Jaeger, J. C.: Conduction of heat in solids, (1959) · Zbl 0029.37801
[21] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[22] Zhang, Y.; Benson, D. A.; Reeves, D. M.: Time and space non-localities underlying fractional-derivative models: distinction and literature review of field applications, Adv. water resources 32, 561-581 (2009)
[23] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: The fractional-order governing equation of Lévy motion, Water resources res. 36, 1413-1423 (2000)
[24] Schumer, R.; Meerschaert, M. M.; Baeumer, B.: Fractional advection -- dispersion equations for modeling transport at the Earth surface, J. geophys. Res. 114 (2009)
[25] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. -P.; Baeumer, B.: Stochastic solution of space -- time-fractional diffusion equations, Phys. rev. E 65 (2002) · Zbl 1244.60080
[26] Chaves, A. S.: A fractional diffusion equation to describe Lévy flights, Phys. lett. A 239, 13-16 (1998) · Zbl 1026.82524 · doi:10.1016/S0375-9601(97)00947-X
[27] Benson, D. A.; Schumer, R.; Meerschaert, M. M.; Wheatcraft, S.: Fractional dispersion, Lévy motion, and the made tracer tests, Transp. porous media 42, 211-240 (2001)
[28] Aoki, Y.; Sen, M.; Paolucci, S.: Approximation of transient temperatures in complex geometries using fractional derivatives, Heat transfer 44, 771-777 (2008)
[29] Jumyi, Liu; Mingyu, Xu: Some exact solutions to Stefan problems with fractional differential equations, J. math. Anal. appl. 351, 536-542 (2009) · Zbl 1163.35043 · doi:10.1016/j.jmaa.2008.10.042
[30] Voller, V. R.; Paola, C.: Can anomalous diffusion describe depositional fluvial profiles?, J. geophys. Res. 115 (2010)
[31] Li, Xicheng; Xu, Mingyu; Wang, Shaowel: Analytical solutions to the moving boundary problems with space -- time-fractional derivatives in drug release devices, J. phys. A: math theor. 40, 12132-12141 (2007) · Zbl 1134.35104 · doi:10.1088/1751-8113/40/40/008
[32] Dantzig, J.; Rappaz, M.: Solidification: methods, microstructure and modeling, (2009) · Zbl 1194.76001
[33] Podlubny, I.: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, some methods of their solution and some of their applications, (1998) · Zbl 0924.34008