Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter December 28, 2013

The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

  • Robin Stern EMAIL logo , Frederic Effenberger , Horst Fichtner and Tobias Schäfer

Abstract

The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

[1] J.M. Chambers, C.L. Mallows, B.W. Stuck, A method for simulating stable random variables. J. of the American Statistical Association 71 (1976), 340–344. http://dx.doi.org/10.1080/01621459.1976.1048034410.1080/01621459.1976.10480344Search in Google Scholar

[2] I. Eliazar, J. Klafter, Anomalous is ubiquitous. Annals of Physics 326 (2011), 2517–2531. http://dx.doi.org/10.1016/j.aop.2011.07.00610.1016/j.aop.2011.07.006Search in Google Scholar

[3] R. García-García, A. Rosso, G. Schehr, Lévy flights on the half line. Phys. Rev. E. 86 (2012), # 011101. 10.1103/PhysRevE.86.011101Search in Google Scholar PubMed

[4] R. Gorenflo, F. Mainardi, Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1, No 2 (1998), 167–192. Search in Google Scholar

[5] R. Gorenflo, F. Mainardi, Approximation of Lévy-Feller diffusion by random walk. J. for Analysis and its Applications 18 (1999), 231–246. Search in Google Scholar

[6] M. Hahn, K. Kobayashi, S. Umarov, SDEs driven by a timechanged Lévy process and their associated time-fractional order pseudodifferential equations. J. of Theoretical Probability 25 (2012), 262–279. http://dx.doi.org/10.1007/s10959-010-0289-410.1007/s10959-010-0289-4Search in Google Scholar

[7] M. Hahn, S. Umarov, Fractional Fokker-Planck-Kolmogorov type equations and their associated stochastic differential equations. Frac. Calc. Appl. Anal. 14, No 1 (2011), 56–79; DOI: 10.2478/s13540-011-0005-9; http://link.springer.com/article/10.2478/s13540-011-0005-9. Search in Google Scholar

[8] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific Publishing Company Inc. (2011). http://dx.doi.org/10.1142/807210.1142/8072Search in Google Scholar

[9] R. Hilfer, Threefold Introduction to Fractional Derivatives. In: Anomalous Transport, Wiley-VCH Verlag GmbH & Co. KGaA (2008), 17–73. http://dx.doi.org/10.1002/9783527622979.ch210.1002/9783527622979.ch2Search in Google Scholar

[10] F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells. Reports on Progress in Physics 76 (2013), # 046602. 10.1088/0034-4885/76/4/046602Search in Google Scholar PubMed

[11] M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (I). Fract. Calc. Appl. Anal. 8, No 3 (2005), 323–341; at http://www.math.bas.bg/~fcaa. Search in Google Scholar

[12] M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) — with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal. 9, No 4 (2006), 333–349; at http://www.math.bas.bg/~fcaa. Search in Google Scholar

[13] S. Jespersen, R. Metzler, H.C. Fogedby, Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions. Phys. Rev. E 59 (1999), 2736–2745. http://dx.doi.org/10.1103/PhysRevE.59.273610.1103/PhysRevE.59.2736Search in Google Scholar

[14] G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional brownian motion. Appl. Math. Lett. 18 (2005), 817–826. http://dx.doi.org/10.1016/j.aml.2004.09.01210.1016/j.aml.2004.09.012Search in Google Scholar

[15] A. Kopp, I. Büsching, R.D. Strauss, M.S. Potgieter, A stochastic differential equation code for multidimensional Fokker-Planck type problems. Computer Physics Communications 183 (2012), 530–542. http://dx.doi.org/10.1016/j.cpc.2011.11.01410.1016/j.cpc.2011.11.014Search in Google Scholar

[16] N. Krepysheva, L. di Pietro, M.C. Néel, Space-fractional advectiondiffusion and reflective boundary condition. Phys. Rev. E 73 (2006), # 021104. 10.1103/PhysRevE.73.021104Search in Google Scholar

[17] M. Magdziarz, A. Weron, Competition between subdiffusion and Lévy flights: A Monte Carlo approach. Phys. Rev. E 75 (2007), # 056702. 10.1103/PhysRevE.75.056702Search in Google Scholar

[18] F. Mainardi, G. Pagnini, R.K. Saxena, Fox H functions in fractional diffusion. J. of Comput. and Appl. Mathematics 178 (2005), 321–331. http://dx.doi.org/10.1016/j.cam.2004.08.00610.1016/j.cam.2004.08.006Search in Google Scholar

[19] M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations. J. of Comput. and Appl. Mathematics 172 (2004), 65–77. http://dx.doi.org/10.1016/j.cam.2004.01.03310.1016/j.cam.2004.01.033Search in Google Scholar

[20] M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56 (2006), 80–90. http://dx.doi.org/10.1016/j.apnum.2005.02.00810.1016/j.apnum.2005.02.008Search in Google Scholar

[21] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77. http://dx.doi.org/10.1016/S0370-1573(00)00070-310.1016/S0370-1573(00)00070-3Search in Google Scholar

[22] K. Oldham, J. Spanier, The Fractional Calculus (Theory and Applications of Differentiation and Integration to Arbitrary Order). Academic Press New York (1974). Search in Google Scholar

[23] D. Perrone, R.O. Dendy, I. Furno, R. Sanchez, G. Zimbardo, A. Bovet, A. Fasoli, K. Gustafson, S. Perri, P. Ricci, F. Valentini, Nonclassical transport and particle-field Coupling: From laboratory plasmas to the solar wind. Space Sci. Rev. 178 (2013), 233–270. http://dx.doi.org/10.1007/s11214-013-9966-910.1007/s11214-013-9966-9Search in Google Scholar

[24] I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, Elsevier Science & Acad. Press, N. York etc. (1999). Search in Google Scholar

[25] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 367–386; at http://www.math.bas.bg/~fcaa. Search in Google Scholar

[26] E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation. Comp. Math. Appl. 64 (2012), 3141–3152. http://dx.doi.org/10.1016/j.camwa.2012.03.00210.1016/j.camwa.2012.03.002Search in Google Scholar

[27] C. Tadjeran, M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. of Computational Physics 220 (2007), 813–823. http://dx.doi.org/10.1016/j.jcp.2006.05.03010.1016/j.jcp.2006.05.030Search in Google Scholar

[28] V. Volpert, Y. Nec, A. Nepomnyashchy, Fronts in anomalous diffusionreaction systems. Philos. Trans. A Math. Phys. Eng. Sci. 371 (2013), # 20120, 179. 10.1098/rsta.2012.0179Search in Google Scholar PubMed

Published Online: 2013-12-28
Published in Print: 2014-3-1

© 2014 Diogenes Co., Sofia

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 19.3.2024 from https://www.degruyter.com/document/doi/10.2478/s13540-014-0161-9/html
Scroll to top button