Solution of inverse anomalous mass transfer problems using a hyperbolic space-fractional model and differential evolution

Authors

  • Felipe Augusto Paes de Godoi
  • Fran Sérgio Lobato
  • João Jorge Ribeiro Damasceno

DOI:

https://doi.org/10.55905/oelv21n12-140

Keywords:

fick law, inverse anomalous problem, real experimental data, fractional hyperbolic advection-dispersion model, differential evolution

Abstract

The study of anomalous diffusion phenomenon characterizes an important field of science due to limitations of traditional laws considered to represent the conduction term found in mass and heat transfer models. Mathematically, this phenomenon can be represented by empirical and phenomenological models with different levels of complexity. For the latter, differential models with fractional order have been considered. In addition, based on hyperbolic diffusion theory, this fractional differential model can be represented by a second-order derivative on time. This fractional hyperbolic differential model presents two new parameters (fractional order and time relaxation factor) that should be estimated. For this purpose, an inverse problem considering experimental data needs to be formulated and solved. In this context, the present contribution aims to formulate and solve two inverse anomalous diffusion problems considering a fractional hyperbolic advection-dispersion model to obtain the fractional order and the time relaxation tensor using real experimental data sets. To solve the direct problem the Finite Difference Method is extended for fractional context by using Grünwald-Letnikov Derivative. To solve each inverse problem, the Differential Evolution algorithm is considered as an optimization tool. The obtained results are compared with those found considering the simplification of the hyperbolic space-fractional model. In all analyzed cases, the DE algorithm was able to find good estimates for both parameters and it was demonstrated that the objective function considering the fractional hyperbolic advection-dispersion resulted in lower residual values.

References

Afshari, E.; Sepehrian, B.; Nazari, A. Finite Difference Method for Solving the Space-Time Fractional Wave Equation in the Caputo Form. Fractional Differential Calculus. 10.7153/fdc-05-05, p. 55-63, 2015.

Bevilacqua, L.; Galeão, A. C. N. R.; Costa, F. P.A New Analytical Formulation of Retention Effects on Particle Diffusion Process. Ann. Braz. Acad. Sci., v. 83, 1443–1464, 2011a.

Bevilacqua, L.; Galeão, A. C. N. R.; Costa, F. P. On the Significance of Higher Order Differential Terms in Diffusion Processes, Journal of the Brazilian Society of Mechanical Sciences and Engineering, v. 34, pp. 166-175, 2011b.

Brandani, S.; Jama, M.; Ruthven, D. Diffusion, Self-diffusion and Counter-diffusion of Benzene and p-Xylem in Silicalite. Microporous Mesoporous Mater. v. 35/36, p. 283–300, 2000.

Brociek, R.; Slota, D.; Król, M.; Matula, G.; Kwasny, W. Comparison of Mathematical Models with Fractional Derivative for the Heat Conduction Inverse Problem based on the Measurements of Temperature in Porous Aluminum. International Journal of Heat and Mass Transfer, v. 143, p. 1–14, 2019. 10.1016/j.ijheatmasstransfer.2019.118440.

Cattaneo, C. Sur Une Forme de I’equation de la Chaleur Eliminant le Paradoxe D’une Propagation Instantanee’, Comptes Rendus de l’Académie des Sciences, v. 247, p. 431-433, 1958.

Chang, Al.; Sun, H. G. Time-Space Fractional Derivative Models for CO2Transport in Heterogeneous Media. Fractional Calculus and Applied Analysis, v. 21, n. 1, p. 151–173, 2018. DOI: 10.1515/fca-2018-0010.

Fan, W.; Jiang, X.; Chen, S. Parameter Estimation for the Fractional Fractal Diffusion Model Based on its Numerical Solution. Computers and Mathematics with Applications, v. 71, n. 2, p. 642–651, 10.1016/j.camwa.2015.12.030, 2016.

Gerasimov, D. N. The Nernst–Einstein Equation for An Anomalous Diffusion at Short Spatial Scales. Physica D, v. 419, 132851, p. 1-6,10.1016/j.physd.2021.132851, 2021.

Ghazizadeh, H. R.; Azimi, A.; Maerefat, M. An Inverse Problem to Estimate Relaxation Parameter and Order of Fractionality in Fractional Single-Phase-Lag Heat Equation. International Journal of Heat and Mass Transfer, v. 55, n. 7–8, p. 2095–2101. 10.1016/j.ijheatmasstransfer.2011.12.012, 2012.

Gomez, H.; Colominas, I.; Navarrina, F.; París, J.; Casteleiro, M. A Hyperbolic Theory for Advection-Diffusion Problems: Mathematical Foundations and Numerical Modeling. Archives of Computational Methods in Engineering, v. 17(2), p. 191-211. 10.1007/s11831-010-9042-5, 2010.

Greer, J. B.; Bertozzi, A. L.; Sapiro, G. Fourth Order Partial Differential Equations on General Geometries. Journal of Computational Physics, v. 216, n. 1, p. 216-246. 10.1016/j.jcp.2005.11.031, 2006.

Jia, J.; Wang, H. A Fast Finite Volume Method for Conservative Space–Time Fractional Diffusion Equations Discretized on Space–Time Locally Refined Meshes, Computers & Mathematics with Applications, v. 78, 10.1016/j.camwa.2019.04.003, p. 1345-1356, 2019.

Jia, J.; Wang, H. A Fast Finite Volume Method for Conservative Space-Fractional Diffusion Equations in Convex Domains. Journal of Computational Physics, v. 310, p. 63-84, 10.1016/j.apm.2013.10.007, 2016.

Kheybari, S.; Darvishi, M. T.; Hashemi, M. S. A Semi-Analytical Approach to Caputo Type Time-Fractional Modified Anomalous Sub-Diffusion Equations. Applied Numerical Mathematics, v. 158, p. 103-122,10.1016/j.apnum.2020.07.023, 2020.

Kumar, P.; Kumar, D.; Rai, K. N. A Mathematical Model for Hyperbolic Space-Fractional Bioheat Transfer During Thermal Therapy. Procedia Engineering, v. 127, p. 56-62. 10.1016/j.proeng.2015.11.329, 2015.

Li, X.; Zhang, Y.; Reeves, D. M.; Zheng, C. Fractional-Derivative Models for Non-Fickian Transport in a Single Fracture and its Extension. Journal of Hydrology, v. 590, 125396, p. 1-16, 10.1016/j.jhydrol.2020.125396, 2020.

Lima, W. J.; Lobato, F. S.; Arouca, F. O. Solution of Inverse Anomalous Diffusion Problems using Empirical and Phenomenological Models. Heat and Mass Transfer, v. 55(11), p. 3053-3063, 2019. 10.1007/s00231-019-02642-w, 2020.

Liu, F.; Zhuang, P.; Turner, I.; Burrage, K.; Anh, V. A New Fractional Finite Volume Method for Solving the Fractional Diffusion Equation. Applied Mathematical Modelling, v. 38(15-16), p. 3871-3878, https://doi.org/10.1016/j.apm.2013.10.007, 2014.

Mirzaee, F.; Bimesl, S. A New Approach to Numerical Solution of Second-Order Linear Hyperbolic Partial Differential Equations Arising from Physics and Engineering. Results in Physics, v. 3, p. 241-247,10.1016/j.rinp.2013.10.002, 2013.

Mozafarifard, M.; Mortazavinejad, S. M.; Toghraie, D. Numerical Simulation of Fractional Non-Fourier Heat Transfer in Thin Metal Films Under Short-Pulse Laser. International Communications in Heat and Mass Transfer, v. 115, 104607, p. 1-9,10.1016/j.icheatmasstransfer.2020.104607, 2020.

Neuman, S. P.; Tartakovsky, D. M. Perspective on Theories of Non-Fickian Transport in Heterogeneous Media. Advances in Water Resources. v. 32, p. 670-680. 10.1016/j.advwatres.2008.08.005, 2009.

Nowamooz, A.; Radilla, G.; Fourar, M.; Berkowitz, B. Non-fickian Transport in Transparent Replicas of Rough-Walled Rock Fractures. Transport in Porous Media, v. 98, n. 3, p. 651–682,10.1007/s11242-013-0165-7, 2013.

Ordóñez-Miranda, J.; Alvarado-Gil, J. J. Thermal Wave Oscillations and Thermal Relaxation Time Determination in a Hyperbolic Heat Transport Model. International Journal of Thermal Sciences, v. 48, n. 11, p. 2053–2062,10.1016/j.ijthermalsci.2009.03.008, 2009.

Paradisi, P.; Cesari, R.; Mainardi, F.; Maurizi, A.; Tampieri, F. A Generalized Fick’s Law to Describe Non-Local Transport Effects. Physics and Chemistry of the Earth, Part B: Hydrology, Oceans and Atmosphere, v. 26, n. 4, p. 275-279,10.1016/S1464-1909(01)00006-5, 2001.

Pel, L.; Kopinga, K; Brocken, H. Moisture Transport in Porous Building Materials, HERON, v. 41, p. 95-105, 1996.

Podlubny, I. Fractional Differential Equations, Academic Press, San Diego, 1999.

Press, W. H.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 3rd Edition, 1235 pages, ISBN-13 ‏ : ‎ 978-0521880688, 2007.

Silva, L. G. Direct and Inverse Problems in Anomalous Diffusion Process (In Portuguese), Postgraduate Program in Computational Modeling at the State University of Rio de Janeiro, UERJ, Nova Friburgo-RJ, Brazil. 112 pages, 2016.

Storn, R.; Price, K. Differential Evolution: A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces. International Computer Science Institute, v. 12, p. 1-16, 1995.

Qiao, C.; Xu, Y.; Zhao, W.; Qian, J.; Wu, Y.; Sun, H. Fractional Derivative Modeling on Solute Non-Fickian Transport in a Single Vertical Fracture. Frontiers in Physics, v. 8, 1-9. DOI: 10.3389/fphy.2020.00378, 2020.

Salehi, Y.; Darvishi, M. T.; Schiesser, W. E. Numerical Solution of Space Fractional Diffusion Equation by the Method of Lines and Splines. Applied Mathematics and Computation, v. 336, p. 465-480,10.1016/j.amc.2018.04.053, 2018.

Schwarzwälder, M. C.; Myers, T. G.; Hennessy, M. G. The One-Dimensional Stefan Problem with Non-Fourier Heat Conduction. International Journal of Thermal Sciences, v. 150, 106210, p. 1-11,10.1016/j.ijthermalsci.2019.106210, 2020.

Sun, H. G.; Wang,Y.; Jiazhong, Q.; Zhang, Y.; Zhou, D. An Investigation on Fractional Derivative Model in Characterizing Sodium Chloride Transport in a Single Fracture. The European Physical Journal Plus, p. 134-440, 10.1140/epjp/i2019-12954-9, 2019.

Tzou, D. Y. A Unified Field Approach for Heat Conduction from Macro-to-Micro-Scales. Journal of Heat Transfer, v. 117, n. 1, p. 8-16,10.1115/1.2822329, 1995.

Vernotte, P. Les Paradoxes de La Theorie Continue de L'equation de La Chaleur. Comptes Rendus, v. 246, p. 3154-3155, 1958.

Wang, X.; Qi, H.; Yang, X.; Xu, H. Analysis of the Time-Space Fractional Bioheat Transfer Equation for Biological Tissues During Laser Irradiation. International Journal of Heat and Mass Transfer, v. 177, 121555, p. 1-15,10.1016/j.ijheatmasstransfer.2021.121555, 2021.

Yan, X.; Qian, J.; Lei, Ma; Mu, Wang; Hu., A. Non-Fickian Solute Transport in a Single Fracture of Marble Parallel Plate. Geofluids, v. 2018, Article ID 7418140, 9 pages, 10.1155/2018/7418140, p. 1-9, 2018.

Yazdani, A.; Mojahed, N.; Babaei, A.; Cendon, E. Using Finite Volume-Element Method for Solving Space Fractional Advection-Dispersion Equation. Progress in Fractional Differentiation and Applications, 10.18576/pfda/060106, p. 55-66, 2020.

Zhao, Y.; Chen, J.; Yang, M.; Liu, Y.; Song, Y. A Rapid Method for the Measurement and Estimation of CO2Diffusivity in Liquid Hydrocarbon-Saturated Porous Media Using MRI. Magnetic Resonance Imaging, v. 34, n. 4, p. 437–441,10.1016/j.mri.2015.12.024, 2016.

Zhokh, A. A. Size-Controlled Non-Fickian Diffusion in a Combined Micro- and Mesoporous Material. Chemical Physics, v. 520, p. 27-31,10.1016/j.chemphys.2018.12.013, 2019.

Downloads

Published

2023-12-15

How to Cite

de Godoi, F. A. P., Lobato, F. S., & Damasceno, J. J. R. (2023). Solution of inverse anomalous mass transfer problems using a hyperbolic space-fractional model and differential evolution. OBSERVATÓRIO DE LA ECONOMÍA LATINOAMERICANA, 21(12), 26050–26075. https://doi.org/10.55905/oelv21n12-140

Issue

Section

Articles

Most read articles by the same author(s)