Solution of Fourth Order Diffusion Equations and Analysis Using the Second Moment

Abstract:

The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.

• Periodical: Defect and Diffusion Forum (Volume 399)
• Pages: 10-20
• DOI: https://doi.org/10.4028/www.scientific.net/DDF.399.10
• Online since: February 2020
• Authors: Jader Lugon Junior*, João Flávio Vieira Vasconcellos, Diego Campos Knupp, Gisele Moraes Marinho, Luiz Bevilacqua, Antônio José da Silva Neto
• Keywords: Anomalous Diffusion, Bi-Flux Diffusion, Differential Equations, Mean Square Displacement, Retention

References

[1] A. Fick, Uber diffusion, Ann. Pys. 170 n. 1 (1855) 59–86.

[2] J. Klafter, I. M. Sokolov, Anomalous diffusion spreads its wings, Physics World, 18 n. 8, (2005) 29–32.

DOI: 10.1088/2058-7058/18/8/33

[3] C. Nicholson, E. Syková, Extracellular space structure revealed by diffusion analysis, Trends in Neuroscience, 21 (1998) 207–215.

DOI: 10.1016/s0166-2236(98)01261-2

[4] L. A. Richards, Cappilary conduction of liquids through porous mediums. Physics, 1 n. 5 (1931) 318–333.

DOI: 10.1063/1.1745010

[5] Benson DA, The fractional advection-dispersion equation: development and applications, Ph.D. thesis, University of Nevada, USA (1998).

[6] Moradi G., B. Mehdinejadiani, Modelling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection-dispersion equation, Soil & Water Research 13 (2018) 18-28.

DOI: 10.17221/245/2016-swr

[7] J. Blackledge, Application of the fractional diffusion equation for predicting market behavior, International Journal of Applied Mathematics, 40 n. 3 (2010) 130–158.

[8] O. S. Iyiola, F. D. Zaman, A fractional diffusion equation model for cancer tumor, AIP Advances, 4 n. 10 (2014) 107-121.

DOI: 10.1063/1.4898331

[9] A. McNabb, P. Foster, A new analysis of the diffusion of hydrogen in iron and ferritic steels, Transactions of the Metallurgical Society of AIME, 227 (1963) 618-627.

[10] H. Atsumi, Hydrogen bulk retention in graphite and kinetics of diffusion, Journal of Nuclear Materials (2002) 1466–1470.

DOI: 10.1016/s0022-3115(02)01069-3

[11] M. V. D'Angelo, E. Fontana, R. Chertcoff, M. Rosen, Retention phenomena in non-Newtonian fluids flow, Physica A: Statistical Mechanics and its Applications, 327 n. 1, (2003) 44-48.

DOI: 10.1016/s0378-4371(03)00436-9

[12] L. Bevilacqua, A. C. N. R. Galeão, F.P. Costa, A new analytical formulation of retention effects on particle diffusion process, An Acad Bras Cienc. 83 (2011) 1443-1464.

DOI: 10.1590/s0001-37652011005000033

[13] L. Bevilacqua, A. C. N. R. Galeão, J. G. Simas, A. P. R. Doce, A new theory for anomalous diffusion with a bimodal flux distribution, J. Brazilian Soc. Mech Sci Eng. 35 n.4 (2013) 1-10.

DOI: 10.1007/s40430-013-0041-y

[14] L. G. Silva, D. C. Knupp, L. Bevilacqua, A. C. N. R. Galeão, A. J. Silva Neto, Inverse problem in anomalous diffusion with uncertainty propagation, Computer Assisted Methods in Engineering and Science, 21 (2014) 245–255.

DOI: 10.1007/978-3-319-96433-1_9

[15] D. C. Knupp, L. G. Silva, L. Bevilacqua, A. C. N. R. Galeão, A. J. Silva Neto, Inverse Analysis of a New Anomalous Diffusion Model Employing Maximum Likelihood and Bayesian Estimation, in: A. J. Silva Neto, O. Lannes Santiago, G. N. Silva, Mathematical Modeling and Computational Intelligence in Engineering Applications, Springer International Publishing, Switzerland, 2016, pp.89-104.

DOI: 10.1007/978-3-319-38869-4_7

[16] L. G. Silva, D. C. Knupp, A. C. N. R. Galeão, A. J. Silva Neto, Inverse Problem of an Anomalous Diffusion Model Employing Lightning Optimization Algorithm, in: G. M. Platt, X. – S. Yang, A. J. Silva Neto. Computational Intelligence, Optimization and Inverse Problems with Applications in Engineering, Springer Nature, Switzerland, 2019, pp.185-200.

DOI: 10.1007/978-3-319-96433-1_9

[17] J. F. V. Vasconcellos, G. M. Marinho, J. H. Zanni, Numerical analysis of the anomalous diffusion equation with a bimodal flux, Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 33 3–4 (2017) 242-249. (in Portuguese).

DOI: 10.1016/j.rimni.2016.05.001

[18] L. G. Silva, D. C. Knupp, L. Bevilacqua, A. C. N. R. Galeão, A. J. Silva Neto, Formulation and solution of an Inverse Anomalous Diffusion Problem with Stochastic Techniques, Ciência e Natura, 36, (2014) 82–96. (in Portuguese).

[19] M. Jiang, L. Bevilacqua, A. J. Silva Neto, A. C. N. R. Galeão, J. Zhu, Bi-Flux Theory Applied to the Dispersion of Particle in Anisotropic Substratum, Applied Mathematical Modeling, 64, (2018) 121-134.

DOI: 10.1016/j.apm.2018.07.022

[20] M. Jiang, The fourth order diffusion model for a bi-flux mass transfer. Ph.D. Thesis, Federal University of Rio de Janeiro, Brazil, (2017).

[21] J. Lugon Jr., P. P. G. W. Rodrigues, L. Bevilacqua, D. C. Knupp, J. F. V. Vasconcellos, A. J. Silva Neto, Study of the secondary diffusion coefficient in problems of anomalous diffusion, Proceedings of the XXXVIII Congresso Nacional de Matemática Aplicada e Computacional – CNMAC, 6 n.2 (2018), Campinas, Brazil. (in Portuguese).

[22] L. VLAHOS, H. ISLIKER, Normal and anomalous diffusion: A tutorial. arXiv: 0805.0419, (2008).

[23] E. K Lenzi, L. C. Malacarne, R. S. Mendes, I. T. Pedron, Anomalous diffusion, nonlinear fractional Fokker–Planck equation and solutions, Physica A: Statistical Mechanics and its Applications, 319 (2003) 245-252.

DOI: 10.1016/s0378-4371(02)01495-4

[24] R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2017). URL https://www.R-project.org/.

[25] J. Lugon Jr., P. P. G. W. Rodrigues, L. Bevilacqua, G. M. Marinho, D. C. Knupp, J. F. V. Vasconcellos, A. J. Silva Neto, AdvDif4: Solving 1D Advection Bi-Flux Diffusion Equation, 2018). https://CRAN.R-project.org/package=AdvDif4.

[26] Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, (2018).

[27] A. C. N. R. Galeão, L. Bevilacqua, S. L. Delphin, F. S. P. Costa, A Numerical Solution of the Diffusion-Retention Problem in Homogeneous Media, Proceedings of the 13th Congress of Thermal Science and Engineering, Uberlândia, Brazil (2010).

[28] J. D. Murray, Mathematical Biology, 3. ed, Springer-Verlag, New York, (2008).

[29] F. D. Moura Neto, and A. J. Silva Neto, An Introduction to Inverse Problems with Applications, Springer-Verlag, Berlin Heidelberg, (2013).

[30] C. Hok Eab, S. C. Lim, Accelerating and retarding anomalous diffusion. Journal of Physics A: Mathematical and Theorical, 45, 14, p.17, (2012).

[31] S. H. Rakotonasy, Modèle fractionnaire pour la sous-diffusion: version stochastique et edp. Ph.D. thesis, Université d'Avignon, France, (2012).