Recover the solute concentration from source measurement and boundary data

References

• Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000;339:1–77.
   Web of Science ®Google Scholar
• Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198, Mathematics in science and engineering. San Diego (CA): Academic Press; 1999.
   Google Scholar
• Scalas E, Gorenflo R, Mainardi F. Fractional calculus and continuous-time finance. Phys. A. 2000;284:376–384.
   Web of Science ®Google Scholar
• Jiang X, Xu M, Qi H. The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes. Nonlinear Anal. Real World Appl. 2010;11:262–269. Available from: http://dx.doi.org/10.1016/j.nonrwa.2008.10.057
   Web of Science ®Google Scholar
• Nourdin I. Selected aspects of fractional Brownian motion. Vol. 4, Bocconi and springer series. Milan: Springer; 2012. Available from: http://dx.doi.org/10.1007/978-88-470-2823-4
   Google Scholar
• Benson DA, Wheatcraft SW, Meerschaert MM. The fractional-order governing equation of Lévy motion. Water Resour. Res. 2000;36:1413–1423.
   Web of Science ®Google Scholar
• Sokolov IM, Klafter J. From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos. 2005;15:1–7.
   Web of Science ®Google Scholar
• Li XC, Xu MY, Jiang XY. Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition. Appl. Math. Comput. 2009;208:434–439.
   Web of Science ®Google Scholar
• Scherer R, Kalla SL, Boyadjiev L, Al-Saqabi B. Numerical treatment of fractional heat equations. Appl. Numer. Math. 2008;58:1212–1223.
   Web of Science ®Google Scholar
• Zhang P, Liu F. Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 2006;22:87–99.
   Google Scholar
• Lin YM, Xu CJ. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 2007;225:1533–1552.
   Web of Science ®Google Scholar
• Shen S, Liu F, Anh V, Turner I. Detailed analysis of a conservative difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 2006;22:1–19.
   Google Scholar
• Wyss W. The fractional diffusion equation. J. Math. Phys. 1986;27:2782–2785.
   Web of Science ®Google Scholar
• Metzler R, Klafter J. Boundary value problems for fractional diffusion equations. Phys. A. 2000;278:107–125.
   Web of Science ®Google Scholar
• Gorenflo R, Mainardi F. Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 2009;229:400–415.
   Web of Science ®Google Scholar
• Zhang Y, Meerschaert MM, Baeumer B. Particle tracking for time-fractional diffusion. Phys. Rev. E. 2008;78:1–7.
   Web of Science ®Google Scholar
• Mainardi F, Pagnini G, Saxena RK. Fox H functions in fractional diffusion. J. Comput. Appl. Math. 2005;178:321–331.
   Web of Science ®Google Scholar
• Mainardi F, Pagnini G, Gorenflo R. Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 2007;187:295–305.
   Web of Science ®Google Scholar
• Murio DA. Time fractional IHCP with Caputo fractional derivatives. Comput. Math. Appl. 2008;56:2371–2381.
   Web of Science ®Google Scholar
• Murio DA. Stable numerical solution of a fractional-diffusion inverse heat conduction problem. Comput. Math. Appl. 2007;53:1492–1501.
   Web of Science ®Google Scholar
• Murio DA. Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional ihcp. Inverse Probl. Sci. Eng. 2009;17:229–243.
   Web of Science ®Google Scholar
• Zheng GH, Wei T. A new regularization method for a Cauchy problem of the time fractional diffusion equation. Adv. Comput. Math. 2012;36:377–398. Available from: http://dx.doi.org/10.1007/s10444-011-9206-3
   Web of Science ®Google Scholar
• Zheng GH, Wei T. Two regularization methods for solving a Riesz–Feller space-fractional backward diffusion problem. Inverse Probl. 2010;26:115017, 22. Available from: http://dx.doi.org/10.1088/0266-5611/26/11/115017
   Web of Science ®Google Scholar
• Qian Z, Fu CL. Regularization strategies for a two-dimensional inverse heat conduction problem. Inverse Probl. 2007;23:1053–1068.
   Web of Science ®Google Scholar
• Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375, Mathematics and its applications. Dordrecht: Kluwer Academic Publishers Group; 1996. Available from: http://dx.doi.org/10.1007/978-94-009-1740-8
   Google Scholar
• Murio DA. Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 2008;56:1138–1145.
   Web of Science ®Google Scholar