USING THE BAYESIAN FRAMEWORK FOR INFERENCE
IN FRACTIONAL ADVECTION-DIFFUSION
TRANSPORT SYSTEM
Edward Boone1, Ryad Ghanam2,
Nadeem Malik3, and Joshua Whitlinger4 1 Dept. of Statistical Sci. and Operations Research
Virginia Commonwealth University
Richmond, VA 23284, USA 2 Dept. of Liberal Arts and Sciences
Virginia Commonwealth University in Qatar
Doha, QATAR 3 Dept. of Mechanical Engineering
Texas Tech University
Lubbock, TX 74909, USA 4 Dept. of Statistical Sci. and Operations Research
Virginia Commonwealth University
Richmond, VA 23284, USA
This work shows for the first time the viability of using the Bayesian paradigm for both estimation and hypothesis testing when applied to fractional differential equations. Two distinct fractional differential equation models were explored using simulated data sets to determine the performance of the Bayesian inferential methods across values of α (the fractional order) and σ (the experimental error variance). This inferential paradigm shows promise as it has robust estimation, predictions and provides for hypothesis testing to determine whether a fractional process is warranted by the data. A simulation study, applied to a fractional transport system in porous media, demonstrates the robustness of the estimation and the sensitivity of the hypothesis tests to various levels of α and σ2.
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
References
[1] W. Hundsdorfer, J. Verwer, Numerical Solution of Time-Dependent
Advection-Diffusion-Reaction Equations, Springer-Verlag, Berlin, Heidelberg (2003).
[2] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John
Wiley and Sons, Inc., New York (2002).
[3] T.E. Graedel, P.J. Crutzen, Atmosphere, Climate and Change, Henry Holt
and Company (1997).
[4] K. Aziz, A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers Ltd., London (1979).
[5] I. Glassman, R.A. Yetter, Combustion, Academic Press, London, San
Diego, Burlington (2008).
[6] I. Ali , N.A. Malik, A realistic transport model with pressure-dependent
parameters for gas flow in tight porous media with application to determining shale rock properties, Transport in Porous Media, 124, No 3 (2018),
723-742.
[7] M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Anomalous subdiffusion is a
measure for cytoplasmic crowding in living cells, Biophysical J., 87 (2004),
3518-3524.
[8] W. Chen, H. Sun, X. Zhang, D. Korosak, Anomalous diffusion modeling
by fractal and fractional derivatives, Comp. Math. Appl. 59, No 5 (2010),
1754-1758.
[9] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chemical Physics, 284 (2002), 399-408.
[10] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J.
Math. Anal. Appl., 265 (2002), 229-248.
[11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific
Publishing Company, Singapore (2000).
[12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of
Fractional Differential Equations, North-Holland, Elsevier Science Publisher (2006).
[13] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers (2006).
[14] T.A. Atabong, M.O. Oyesanya, Nonlinear analysis of a fractional reaction
diffusion model for tumour invasion, J. of Mathematics and Computer Sci.
Research 4, No 3 (2011), 136-158.
[15] T. Sandev, R. Metzler, Z. Tomovski, Fractional diffusion equation with a
generalized Riemann-Liouville time fractional derivative, J. Phys. A: Math.
Theor., 44 (2011), Art. 255203, 21 pp.
[16] I. Ali, N.A. Malik, Hilfer fractional advection-diffusion equations with
power-law initial condition: A numerical study using variational iteration
method, Computers and Math. with Appl., 68, No 10 (2014), 1161-1179.
[17] N.A. Malik, R. Ghanam, S. Al-Homidan, Sensitivity of fluid transport
through porous media to the fractional dimension in a fractional Darcy
equation model, Canadian J. Phys., 93 (2015), 1–19.
[18] H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, A new collection
of real world applications of fractional calculus in science and engineering,
Comm. Nonlin. Science and Numer. Simul., 64 (2018), 213-231.
[19] M. Caputo, Diffusion of fluids in porous media with memory, Geothermics,
28 (1999), 113-130.
[20] S. Havlin, D. Ben-Avraham, Diffusion in disordered media, Advances in
Phys., 51 (2002), 187-292.
[21] Y. Luchko, A. Punzi, Modeling anomalous heat transport in geothermal
reservoirs via fractional diffusion equations, Int. J. Geomath., 1 (2011),
257-276.
[22] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a
fractional dynamics approach, Phys. Reports, 339 (2000), 1-77.
[23] M. Caputo, Diffusion with space memory modelled with distributed order
space fractional differential equations, Annals of Geophys., 2, No 46 (2003),
223-234.
[24] N.A. Malik, I. Ali, B. Chanane, Numerical solutions of non-linear fractional transport models in unconventional hydrocarbon reservoirs using
variational iteration method, Proc. 5th Intern. Conf. on Porous Media
and its Applications in Science and Engineering, Kona Hawaii (2014).
[25] J.O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd Ed.,
Springer, New York (1985).
[26] A. Gelman, J.B. Carlin, H.S. Stern, D.B. Dunson, A. Vehtari, D.B. Rubin,
Bayesian Data Analysis, 3rd Ed., CRC Press, Boca Raton, FL (2013).
[27] L. Jing, N. Ou N, Bayesian inference using intermediate distribution based
on coarse multiscale model for time fractional diffusion equations, Multiscale Model. Simul., 16, No 1 (2018), 327–355.
[28] W.R. Gilks, S. Richardson, D.J. Spiegelhaler, Markov Chain Monte Carlo
in Practice, Chapman & Hall, CRC Press, Boca Raton (1996).
[29] J. Albert, Bayesian Computation with R, 2nd Ed., Springer, New York
(2009).
[30] J.F. Reverey, J.H. Jeon, H. Bao, M. Leippe, R. Metzler, C. Selhuber-Unkel,
Superdiffusion dominates intracellular particle motion in the supercrowded
space of pathogenic acanthamoeba castellanii, Sci. Rep., 5 (2015), Art.
11690.
[31] W. Deng, X. Wu, W. Wang, Mean exit time and escape probability for
the anomalous processes with the tempered power-law waiting times, EPL
117 (2017), Art. 10009.
[32] P.D. Mandic, T.B. Scekara, M.P. Lazarevic, Dominant pole placement with
fractional order PID controllers: D-decomposition approach, ISA Trans.
67 (2017), 76–86.
[33] C.M.A. Pinto, A.R.M. Carvalho, Fractional order model for HIV dynamics,
J. Comput. Appl. Math., 312 (2017), 240–256.
[34] T. Bayes, R. Price, An essay towards solving a Problem in the Doctrine
of Chance (By the late Rev. Mr. Bayes, Communicated by Mr. Price in a
letter to John Canton), A.M.F.R.S. Philos. Trans. of Royal Soc. London,
53 (1763), 370-418.
[35] D. Wackerly, W. Mendenhall, R.L. Scheaffer, Mathematical Statistics with
Applications, 7th Ed., Thomson Brooks/Cole, Belmont, CA (2008).
[36] G. Casella, R.L. Berger, Statistical Inference, 2nd Ed., Duxbury, Belmont,
CA (2002).
[37] R.E. Kass, L. Wasserman, A reference Bayesian test for nested hypothesis
and its relationship to the Schwarz criterion, J. of the Amer. Stat. Assoc.,
90, No 431 (1995), 928-934.
[38] J.A. Hoeting, D. Madigan, A.E. Raftery, C.T. Volinsky, Bayesian model
averaging: A tutorial, Statistical Sci., 14, No 4 (1999), 382-417.
[39] R. Ospina, SLP. Ferrari, A general class of zero-or-one inflated beta regression models, Computat. Stat. & Data Analysis, 56, No 6 (2012), 1609-1623.