Coupled systems mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Modeling radon diffusion equation in soil pore matrix by using uncertainty based orthogonal polynomials in Galerkin's method

  • Rao, T.D. (Department of Mathematics, National Institute of Technology Rourkela) ;
  • Chakraverty, S. (Department of Mathematics, National Institute of Technology Rourkela)
  • Received : 2017.08.10
  • Accepted : 2017.10.23
  • Published : 2017.12.25
⟨ Previous Next ⟩

Abstract

This paper investigates the approximate solution bounds of radon diffusion equation in soil pore matrix coupled with uncertainty. These problems have been modeled by few researchers by considering the parameters as crisp, which may not give the correct essence of the uncertainty. Here, the interval uncertainties are handled by parametric form and solution of the relevant uncertain diffusion equation is found by using Galerkin's Method. The shape functions are taken as the linear combination of orthogonal polynomials which are generated based on the parametric form of the interval uncertainty. Uncertain bounds are computed and results are compared in special cases viz. with the crisp solution.

Keywords

Acknowledgement

Supported by : Board of Research in Nuclear Sciences (BRNS)

References

  1. Behera, D. and Chakraverty, S. (2015), "New approach to solve fully fuzzy system of linear equations using single and double parametric form of fuzzy numbers", Sadhana, 40(1), 35-49. https://doi.org/10.1007/s12046-014-0295-9
  2. Betts, A.K., Helliker, B. and Berry, J. (2004), "Coupling between CO2, water vapor, temperature, and radon and their fluxes in an idealized equilibrium boundary layer over land", J. Geophys. Res.: Atmosph., 109(D18).
  3. Bhat, R.B. and Chakraverty, S. (2004), Numerical Analysis in Engineering, Alpha Science, Int'l Ltd.
  4. Chakraverty, S., Tapaswini, S. and Behera, D. (2016), Fuzzy Differential Equations and Applications for Engineers and Scientists, CRC Press Taylor and Francis Group, Boca Raton, U.S.A.
  5. Chakraverty, S., Tapaswini, S. and Behera, D. (2016), Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications, John Wiley and Sons.
  6. Dimbylow, P.J. and Wilkinson, P. (1985), "The numerical solution of the diffusion equation describing the flow of radon through cracks in a concrete slab", Radiat. Protect. Dosimetr., 11(4), 229-236.
  7. Dulaiova, H., Camilli, R., Henderson, P.B. and Charette, M.A. (2010), "Coupled radon, methane and nitrate sensors for large-scale assessment of groundwater discharge and non-point source pollution to coastal waters", J. Environ. Radioact., 101(7), 553-563. https://doi.org/10.1016/j.jenvrad.2009.12.004
  8. Escobar, V.G., Tome, F.V. and Lozano, J.C. (1999), "Procedures for the determination of 222 Rn exhalation and effective 226 Ra activity in soil samples", Appl. Radiat. Isotop., 50(6), 1039-1047. https://doi.org/10.1016/S0969-8043(98)00121-3
  9. Alefeld, G. and Herzberger, J. (1983), Introduction to Interval Computations, Academic Press, New York, U.S.A.
  10. Hafez, Y.I. and Awad, E.S. (2016), "Finite element modeling of radon distribution in natural soils of different geophysical regions", Cogen. Phys., 3(1), 1254859.
  11. Hoffmann, T. and Marciniak, A. (2013), "Solving the Poisson equation by an interval difference method of the second order", Computat. Meth. Sci. Technol., 19(1), 13-21. https://doi.org/10.12921/cmst.2013.19.01.13-21
  12. Kozak, J.A., Reeves, H.W. and Lewis, B.A. (2003), "Modeling radium and radon transport through soil and vegetation", J. Contamin. Hydrol., 66(3), 179-200. https://doi.org/10.1016/S0169-7722(03)00032-9
  13. Moore, R.E., Kearfott, R.B. and Cloud, M.J. (2009), Introduction to Interval Analysis, Society for Industrial and Applied Mathematics.
  14. Nazaroff, W.W. (1992), "Radon transport from soil to air. Reviews of geophysics", 30(2), 137-160. https://doi.org/10.1029/92RG00055
  15. Nickel, K.L. (1986), "Using interval methods for the numerical solution of ODE's", J. Appl. Math. Mech., 66(11), 513-523.
  16. Ren, T. (2001), "Source, level and control of indoor radon", Radiat. Prot., 21(5), 291-297.
  17. Rodriguez, J. (1992), "Galerkin's method for ordinary differential equations subject to generalized nonlinear boundary conditions", J. Different. Equat., 97(1), 112-126. https://doi.org/10.1016/0022-0396(92)90086-3
  18. Savovic, S., Djordjevich and Ristic, G. (2011), "Numerical solution of the transport equation describing the radon transport from subsurface soil to buildings", Radiat. Protect. Dosimetr., 150(2), 213-16. https://doi.org/10.1093/rpd/ncr397
  19. Schery, S.D., Holford, D.J., Wilson, J.L. and Phillips, F.M. (1988). "The flow and Diffusion of radon isotopes in fractured porous media part 2, semi-infinite media", Radiat. Protect. Dosimetr., 24(1-4), 191-197. https://doi.org/10.1093/oxfordjournals.rpd.a080268
  20. Stefanini, L. and Bede, B. (2009). "Generalized hukuhara differentiability of interval-valued functions and interval differential equations", Nonlin. Analy. Theor. Meth. Appl., 71(3), 1311-1328. https://doi.org/10.1016/j.na.2008.12.005
  21. Tapaswini, S. and Chakraverty, S. (2013), "Numerical solution of uncertain beam equations using double parametric form of fuzzy numbers", Appl. Comput. Intellig. Soft Comput., 13.
  22. Tapaswini, S. and Chakraverty, S. (2014), New Midpoint-based Approach for the Solution of n-th Order Interval Differential Equations.
  23. Tapaswini, S., Chakraverty, S. and Allahviranloo, T. (2017), "A new approach to nth order fuzzy differential equations", Comput. Math. Model., 28(2), 278-300. https://doi.org/10.1007/s10598-017-9364-3
  24. Van Der Spoel, W.H., Van Der Graaf, E.R. and De Meijer, R.J. (1998), "Combined diffusive and advective transport of radon in a homogeneous column of dry sand", Health Phys., 74(1), 48-63. https://doi.org/10.1097/00004032-199801000-00006
  25. Wrenn, M.E., Rosen, J.C. and Pelt, W.R. (1969), "Steady state solutions for the diffusion equations of radon-222 daughters", Health phys., 16(5), 647-656. https://doi.org/10.1097/00004032-196905000-00012