USING THE BAYESIAN FRAMEWORK FOR INFERENCE
IN FRACTIONAL ADVECTION-DIFFUSION
TRANSPORT SYSTEM

Abstract

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.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 5
Year: 2020

DOI: 10.12732/ijam.v33i5.4

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References

  1. [1] W. Hundsdorfer, J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, Berlin, Heidelberg (2003).
  2. [2] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley and Sons, Inc., New York (2002).
  3. [3] T.E. Graedel, P.J. Crutzen, Atmosphere, Climate and Change, Henry Holt and Company (1997).
  4. [4] K. Aziz, A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers Ltd., London (1979).
  5. [5] I. Glassman, R.A. Yetter, Combustion, Academic Press, London, San Diego, Burlington (2008).
  6. [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. [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. [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. [9] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chemical Physics, 284 (2002), 399-408.
  10. [10] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229-248.
  11. [11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore (2000).
  12. [12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, Elsevier Science Publisher (2006).
  13. [13] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers (2006).
  14. [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. [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. [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. [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. [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. [19] M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130.
  20. [20] S. Havlin, D. Ben-Avraham, Diffusion in disordered media, Advances in Phys., 51 (2002), 187-292.
  21. [21] Y. Luchko, A. Punzi, Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations, Int. J. Geomath., 1 (2011), 257-276.
  22. [22] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Reports, 339 (2000), 1-77.
  23. [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. [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. [25] J.O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd Ed., Springer, New York (1985).
  26. [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. [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. [28] W.R. Gilks, S. Richardson, D.J. Spiegelhaler, Markov Chain Monte Carlo in Practice, Chapman & Hall, CRC Press, Boca Raton (1996).
  29. [29] J. Albert, Bayesian Computation with R, 2nd Ed., Springer, New York (2009).
  30. [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. [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. [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. [33] C.M.A. Pinto, A.R.M. Carvalho, Fractional order model for HIV dynamics, J. Comput. Appl. Math., 312 (2017), 240–256.
  34. [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. [35] D. Wackerly, W. Mendenhall, R.L. Scheaffer, Mathematical Statistics with Applications, 7th Ed., Thomson Brooks/Cole, Belmont, CA (2008).
  36. [36] G. Casella, R.L. Berger, Statistical Inference, 2nd Ed., Duxbury, Belmont, CA (2002).
  37. [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. [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. [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.