Advertisement

Finite Difference Solution of Diffusion Equation Describing the Flow of Radon Through Soil with Uncertain Parameters

  • T. D. RaoEmail author
  • S. Chakraverty
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)

Abstract

In this paper, an imprecise radon diffusion transport through soil is investigated. As few such researchers have already studied Radon diffusion problems with crisp parameters. Due to various factors, there is a chance of impreciseness to occur in the involved parameters of the model while doing the experiment. So handling a differential equation with imprecise parameters is a challenging task. Accordingly, a second-order radon diffusion equation with imprecise parameters considered as intervals has been studied here. The solution of the considered diffusion equation is modeled by using modified Explicit Finite Difference Method (EFDM) along with parametric concept and for the validation, results are compared with the crisp solutions.

Keywords

Radon Diffusion Radiation Parameters Crisp imprecise Interval. 

Notes

Acknowledgements

The authors are very much thankful for the support and funding given by Board of Research in Nuclear Sciences (BRNS), Mumbai, India.

References

  1. 1.
    Holford DJ, Schery SD, Wilson JL, Phillips FM (1993) Modeling radon transport in dry, cracked soil. J Geophys Res Solid Earth 98(3):567–580CrossRefGoogle Scholar
  2. 2.
    Ames WF (2014) Numerical methods for partial differential equations. Academic PressGoogle Scholar
  3. 3.
    Langtangen HP (1999) Computational partial differential equations: numerical methods and Diffpack programming, vol 1. SpringerGoogle Scholar
  4. 4.
    Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24(3):301–317MathSciNetCrossRefGoogle Scholar
  5. 5.
    Parandin N (2012) Numerical solution of fuzzy differential equations of nth order by Runge Kutta method. Neural Comput Appl 47–355Google Scholar
  6. 6.
    Tapaswini S, Chakraverty S (2014) New analytical method for solving n-th order fuzzy differential equations. Ann Fuzzy Math Inform 8:231–244MathSciNetzbMATHGoogle Scholar
  7. 7.
    Tapaswini S, Chakraverty S (2014) New centre based approach for the solution of n-th order interval differential equation. Reliab Comput 20:25–44MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ren T (2001) Source, level and control of indoor radon. Radiat Prot 21(5):291–297Google Scholar
  9. 9.
    Folkerts KH, Keller G, Muth K (1984) Experimental investigationson diffusion and exhalation of 222Rn and 220Rn from building materials. Radiat Protect Dosim 7:41–44CrossRefGoogle Scholar
  10. 10.
    Escobar VG, Tome FV, Lozano JC (1999) Proceedings for the determination of 222Rn exhalation and effective 226Ra activity in soil samples. Appl Radiat Isot 50(6):1039–1047CrossRefGoogle Scholar
  11. 11.
    Albarracin D, Font Ll, Amgarou K, Domingo C, Fernandez F, Baixeras C (2002) Effect of soil parameters on radon entry into a building by means of the TRANSRAD numerical model. Radiat Prot Dosim 102(4):359–363Google Scholar
  12. 12.
    Varhegyi A, Hakl J, Monnin M, Morin JP, Seidel JL (1992) Experimental study of radon transport in water as test for a transportation microbubble model. J Appl Geophys 29(1):37–46CrossRefGoogle Scholar
  13. 13.
    Jiranek M, Svoboda Z (2009) Transient radon diffusion through radon-proof membranes: a new technique for more precise determination of the radon diffusion coefficient. Build Environ 44(6):1318–1327CrossRefGoogle Scholar
  14. 14.
    Rogers VC, Nielson KK (1991) Multiphase radon generation and transport in porous material. Health Phys 60(6):807–815CrossRefGoogle Scholar
  15. 15.
    Schery SD, Holford DJ, Wilson JL, Phillips FM (1988) The flow and diffusion of radon isotopes in fractured porous media: part 2, semi-infinite media. Radiat Prot Dosim 24(1–4):191–197CrossRefGoogle Scholar
  16. 16.
    Kozak JA, Reeves HW, Lewis BA (2003) Modeling radium and radon transport through soil and vegetation. J Contam Hydrol 66(3):179–200CrossRefGoogle Scholar
  17. 17.
    Renken KJ, Rosenberg T (1995) Laboratory measurements of the transport of radon gas through concrete samples. Health Phys 68(6):800–808CrossRefGoogle Scholar
  18. 18.
    Sasaki T, Gunji Y, Okuda T (2006) Transient-diffusion measurement of radon in Japanese soils from a mathematical viewpoint. J Nucl Sci Technol 43(7):806–810CrossRefGoogle Scholar
  19. 19.
    Dimbylow PJ, Wilkinson P (1985) The numerical solution of the diffusion equation describing the flow of radon through cracks in a concrete slab. Radiat Prot Dosim 11(4):229–236Google Scholar
  20. 20.
    Savovic S et al (2011) Explicit finite difference solution of the diffusion equation describing the flow of radon through soil. Appl Radiat Isot 69(1):237–240CrossRefGoogle Scholar
  21. 21.
    Mikaeilvand N, Khakrangin S (2012) Solving fuzzy partial differential equations by fuzzy two dimensional differential transform method. Neural Comput Appl 21:307–312CrossRefGoogle Scholar
  22. 22.
    Nayak S, Chakraverty S (2015) Numerical solution of uncertain neutron diffusion equation for imprecisely defined homogeneous triangular bare reactor. In: Sadhana—academy proceedings in engineering science (Accepted)Google Scholar
  23. 23.
    Chakraverty S, Nayak S (2015) Fuzzy finite element analysis of multi-group neutron diffusion equation with imprecise parameters. Int J Nucl Energy Sci Technol 9(1)Google Scholar
  24. 24.
    Bede B, Gal SG (2010) Solutions of fuzzy differential equations based on generalized differentiability. Commun Math Anal 9(2):22–41MathSciNetzbMATHGoogle Scholar
  25. 25.
    Moore RE, Baker Kearfott, R, Cloud MJ (2009) Introduction to interval analysis. SiamGoogle Scholar
  26. 26.
    Alefeld G, Herzberger J (2012) Introduction to interval computation. Academic PressGoogle Scholar
  27. 27.
    Behera D, 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):35–49MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tapaswini S, Chakraverty S (2013) Numerical solution of n-th order fuzzy linear differential equations by homotopy perturbation method. Int J Comput Appl 64(6)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

Personalised recommendations