International Scientific Journal

Authors of this Paper

External Links


In this paper, the Laplace transform method is used to solve the advection-diffusion equation having source or sink term with initial and boundary conditions. The solution profile of normalized field variable for both conservative and non-conservative systems are calculated numerically using the Bellman method and the results are presented through graphs for different particular cases. A comparison of the numerical solution with the existing analytical solution for standard order conservative system clearly exhibits that the method is effective and reliable. The important part of the study is the graphical presentations of the effect of the reaction term on the solution profile for the non-conservative case in the fractional order as well as standard order system. The salient feature of the article is the exhibition of stochastic nature of the considered fractional order model.
PAPER REVISED: 2017-11-13
PAPER ACCEPTED: 2017-12-20
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S309 - S316]
  1. Havlin, S., Ben- Avraham, D., Diffusion in disordered media, Advances in Physics 51 (2002), pp. 187-292
  2. Lee, B. P., Renormalization group calculation for the reaction kA to OE, J. Phys. A 27 (1994), pp. 2633-2652
  3. Crank, J., The mathematics of Diffusion, Oxford Univ. Press, London, 1956.
  4. Hilfer, R., Anton, L., Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. 848-851
  5. Hilfer, R., Exact solutions for a class of fractal time random walks, Fractals 3 (1995), pp. 211-216
  6. Gorenflo, R., et al., Discrete random walk models for space-time fractional diffusion, Chem. Phys. 284 (2002), pp. 512-541
  7. Metzler, R., et al., Anomalous Diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett. 82 (1999), pp. 3563-3567
  8. Metzler, R., Klafter, J., The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), pp. 1-77
  9. Langlands, T. A. M., Henry, B.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Physics. A 205 (2005), pp. 719-736
  10. Yuste, S.B., Lindenberg, K., Subdiffusion-Limited A+A Reaction, Phys. Rev. Lett. 87 (2001), pp. 118301-118304
  11. Weiss, G.H., Aspects and applications of the Random Walk, North Holland, Amsterdam, 1994
  12. Hughes, B.D., Random Walks and Random Environments, Clarendon Press, Oxford, 1995
  13. Henry, B.I., Wearne, S.L., Fractional reaction-diffusion, Phys. A. 276 (2000), pp. 448-455
  14. Chen, C.M., et al., A Fourier method for the fractional diffusion equation describing subdiffusion, J. Compt. Phys. 227 (2007), pp. 886-897
  15. Schot, A., et al., Fractional diffusion equation with an absorbent term and a linear external force: Exact solution, Phys. Lett. A. 366 (2007), pp. 346-350
  16. Zahran, M.A., On the derivation of fractional diffusion equation with an absorbent term and a linear external force, Appl. Math. Model. 33 (2009), pp. 3088-3092
  17. Angulo, J.M., et al., Fractional diffusion and fractional heat equation, Adv. Appl. Prob. 32 (2000), pp. 1077-1099
  18. Pezat, S., Zabczyk, J., Nonlinear stochastic wave and heat equations, Probab. Theory Reltal. Fields 116 (2000), pp. 421-443
  19. Schneider, W.R., Wyss, W., Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), pp. 134-144
  20. Yu, R., Zhang, H., New function of Mittag- Leffler type and its application in the fractional diffusion-wave equation, Chaos. Solit. Fract. 30 (2006), pp. 946-955
  21. Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett. 9 (1996), pp. 23-28
  22. Mainardi, F., et al., The fundamental solution of the space- time fractional diffusion equation, Frac. Calc. Appl. Anal. 4 (2001), pp. 153-192
  23. Anh, V.V., Leonenko, N.N, Harmonic analysis of random fractional diffusion-wave equations, Appl. Math. Comput. 141 (2003), pp. 77-85
  24. Sierociuk, D., et al., Diffusion process modeling by using fractional-order models, Applied Math. Comput. 257 (2015), pp. 2-11
  25. Ervin, V.J., et al., Regularity of the Solution to 1-D Fractional Order Diffusion Equations, arXiv: 1608.00128
  26. Cui, M., A high-order compact exponential scheme for the fractional convection-diffusion equation, J. of Comput. and Appl. Math. 255 (2014), pp. 404-416
  27. Zheng, G.H., Wei, T., Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation, J. of Comput. and Appl. Math. 233 (2010), pp. 2631-2640
  28. Das, S., A note on fractional diffusion equations, Chaos, Solitons and Fractals 42 (2009), pp. 2074-2079
  29. Tripathi, D, et al., Influence of slip condition on peristaltic transport of a viscoelastic fluid with fractional Burger's model, Thermal Science 15 (2011), pp. 501-515
  30. Walther, É., et al., Lattice Boltzmann method and diffusion in materials with large diffusivity ratios, Thermal Science, Accepted (2017)
  31. Davies, B., Martin, B., Numerical inversion of the Laplace transform: a survey and comparison of methods, J. Comput. Phys. 33 (1979), pp. 1-32
  32. Piessens, R., Huysmany, R., Algorithm 619: automatic numerical inversion of the Laplace transform, ACM Trans. Math. Soft. 10 (1984), pp. 348-353
  33. Weeks, W.T., Numerical inversion of the Laplace transform using Laguerre functions, J. ACM 13 (1966), pp. 419-429
  34. Lyness, J.N., Giunta, G., A modification of the Weeks method for numerical inversion of the Laplace transform, Math. Comput. 47 (1986), pp. 313-322
  35. Piessens, R., Branders, M., Numerical inversion of the Laplace transform using generalized Laguerre polynomials, Proc. IEE 118 (1971), pp. 1517-1522
  36. McWhirter, J.G., Pike, E.R., On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A: Math. Gen. 11 (1978), pp. 1729-1745
  37. Bellman, R., et al., A numerical inversion of the Laplace Transform, The Rand Corporation, RM-3513-ARPA (1963)
  38. Ueda, S., On some numerical inversion Methods of the Laplace Transform, Bulletin of the Education Faculty, Shizuoka University, Natural science series, 38 (1988), pp. 97-105
  39. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego. CA, 1999
  40. Gorenflo, R., Mainradi, F., Essentials of fractional calculus. Preprint submitted to Maphysto Centre, Preliminary version 2000.
  41. Oldham, K, and Spanier, J., The fractional calculus, Academic Press. New York, London, 1974.

© 2019 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence