Fractional Advection-Diffusion Equation and Associated Diffusive Stresses

  • Yuriy PovstenkoEmail author
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 219)


The theory of diffusive stresses deals with mechanical and diffusive effects in elastic body. The conventional theory is based on the classical Fick law, which relates the matter flux to the concentration gradient. In combination with the balance equation for mass, this law leads to the classical diffusion equation. We study nonlocal generalizations of the diffusive flux governed by the Fick law and of the advection flux associated with the velocity field. The nonlocal constitutive equation with the long-tail power memory kernel results in the fractional advection-diffusion equation. The nonlocal constitutive equation with the middle-tail memory kernel expressed in terms of the Mittag-Leffler function leads to the fractional advection-diffusion equation of the Cattaneo type. The theory of diffusive stresses based on the fractional advection-diffusion equation is formulated. Fundamental solutions to the Cauchy and source problems and associated diffisive stresses are studied. The numerical results are illustrated graphically.


Cauchy Problem Fundamental Solution Source Problem Zero Condition Memory Kernel 
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  1. 1.
    Abdel-Rehim, E.A.: Explicit approximation solutions and proof of convergence of space-time fractional advection dispersion equations. Appl. Math. 4, 1427–1440 (2013)CrossRefGoogle Scholar
  2. 2.
    Barkai, E.: Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E 63, 046118-1-17 (2001)Google Scholar
  3. 3.
    Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138 (2000)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)CrossRefGoogle Scholar
  5. 5.
    Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)CrossRefGoogle Scholar
  6. 6.
    Chaves, A.S.: A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13–16 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cushman, J.H., Ginn, T.R.: Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux. Water Resour. Res. 36, 3763–3766 (2000)Google Scholar
  8. 8.
    Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971)zbMATHGoogle Scholar
  9. 9.
    Huang, F., Liu, F.: The time fractional diffusion equation and the advection-dispersion equation. ANZIAM J. 46, 317–330 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Huang, H., Cao, X.: Numerical method for two dimensional fractional reaction subdiffusion equation. Eur. Phys. J. Spec. Top. 222, 1961–1973 (2013)CrossRefGoogle Scholar
  11. 11.
    Jespersen, S., Metzler, R., Fogedby, H.S.: Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions. Phys. Rev. E 59, 2736–2745 (1999)CrossRefGoogle Scholar
  12. 12.
    Jumarie, G.: A Fokker-Planck equation of fractional order with respect to time. J. Math. Phys. 33, 3536–3542 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Karatay, I., Bayramoglu, S.R.: An efficient scheme for time fractional advection dispersion equations. Appl. Math. Sci. 6, 4869–4878 (2012)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Kaviany, M.: Principles of Heat Transfer in Porous Media, 2nd edn. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kolwankar, K.M., Gangal, A.D.: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 80, 214–217 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Lévy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)CrossRefGoogle Scholar
  17. 17.
    Liu, F., Anh, V., Turner, I., Zhuang, P.: Time-fractional advection-dispersion equation. J. Appl. Math. Comput. 13, 233–245 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Liu, Q., Liu, F., Turner, I., Anh, V.: Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method. J. Comput. Phys. 222, 57–70 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Merdan, M.: Analytical approximate solutions of fractionel convection-diffusion equation with modified Riemann-Liouville derivative by means of fractional variational iteration method. Iranian J. Sci. Technol. A1, 83–92 (2013)MathSciNetGoogle Scholar
  23. 23.
    Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999)CrossRefGoogle Scholar
  24. 24.
    Nield, D.D., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)Google Scholar
  25. 25.
    Nowacki, W.: Dynamical problems of thermodiffusion in solids. Bull. Acad. Polon. Sci., Sér. Sci. Technol. 23, 55–64, 129–135, 257–266 (1974)Google Scholar
  26. 26.
    Nowacki, W., Olesiak, Z.S.: Thermodiffusion in Solids. Polish Scientific Publishers (PWN), Warsaw (1991) (in Polish)Google Scholar
  27. 27.
    Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)CrossRefzbMATHGoogle Scholar
  28. 28.
    Pidstrygach, Ya.S.: Differential equations of thermodiffusion problem in isotropic deformable solid. Dop. Ukrainian Acad. Sci. (2), 169–172 (1961) (in Ukrainian)Google Scholar
  29. 29.
    Pidstrygach, Ya.S.: Differential equations of the diffusive strain theory of a solid. Dop. Ukrainian Acad. Sci. (3), 336–339 (1963) (in Ukrainian)Google Scholar
  30. 30.
    Pidstryhach, Ya.S.: Selected Papers. Naukova Dumka, Kyiv (1995) (in Ukrainian and Russian)Google Scholar
  31. 31.
    Podstrigach, Ya.S.: Theory of diffusive deformation of isotropic continuum. Issues Mech. Real Solid 2, 71–99 (1964) (in Russian)Google Scholar
  32. 32.
    Podstrigach, Ya.S.: Diffusion theory of inelasticity of metals. J. Appl. Mech. Technol. Phys. (2), 67–72 (1965) (in Russian)Google Scholar
  33. 33.
    Podstrigach, Ya.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids. Naukova Dumka, Kiev (1985) (in Russian)Google Scholar
  34. 34.
    Povstenko, Y.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Povstenko, Y.: Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation. Int. J. Eng. Sci. 43, 977–991 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Povstenko, Y.: Fundamental solution to three-dimensional diffusion-wave equation and associated diffusive stresses. Chaos, Solitons Fractals 36, 961–972 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Povstenko Y. Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables. Math. Probl. Eng. 2014, 705364-1-7 (2014)Google Scholar
  38. 38.
    Povstenko, Y.: Theory of diffusive stresses based on the fractional advection-diffusion equation. In: Abi Zeid Daou, R., Xavier, M. (eds.) Fractional Calculus: Applications, pp. 227–242. NOVA Science Publisher, New York (2015)Google Scholar
  39. 39.
    Povstenko, Y.: Space-time-fractional advection diffusion equation in a plane. In: Latawiec, J.K., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in Modelling and Control of Non-integer Order Systems, 6th Conference on Non-integer Order Calculus and Its Applications, Opole, Poland. Lecture Notes in Electrical Engineering, vol. 320, pp. 275–284. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  40. 40.
    Povstenko, Y., Klekot, J.: Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation. J. Appl. Math. Comput. Mech. 13, 95–102 (2014)Google Scholar
  41. 41.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Vol. 1: Elementary Functions. Gordon and Breach, Amsterdam (1986)Google Scholar
  42. 42.
    Risken, H.: The Fokker-Planck Equation. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  43. 43.
    Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Scheidegger, A.E.: The Physics of Flow Through Porous Media, 3rd edn. University of Toronto Press, Toronto (1974)Google Scholar
  45. 45.
    Schneider, W.R.: Fractional diffusion. In: Lima, R., Streit, L, Viela Mendes, R. (eds.) Dynamics and Stochastic Processes, Lecture Notes in Physics, vol. 355, pp. 276–286. Springer, Berlin (1990)Google Scholar
  46. 46.
    Schumer, R., Meerschaet, M.M., Baeumer, B.: Fractional advection-dispersion equations for modeling transport at the Earth surface. J. Geophys. Res. 114, F00A07-1-15 (2009)Google Scholar
  47. 47.
    Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar
  48. 48.
    Yanovsky, V.V., Chechkin, A.V., Schertzer, D., Tur, A.V.: Lévy anomalous diffusion and fractional Fokker-Planck equation. Phys. A 282, 13–34 (2000)CrossRefGoogle Scholar
  49. 49.
    Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2005)zbMATHGoogle Scholar
  50. 50.
    Zaslavsky, G.M., Edelman, M., Niyazov, B.A.: Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics. Chaos 7, 159–181 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Zhang, Y., Benson, D.A., Meerschaert, M.M., Scheffler, H.-P.: On using random walks to solve the space-fractional advection-dispersion equations. J. Stat. Phys. 123, 89–110 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Zhang, Y., Benson, D.A., Meerschaert, M.M., LaBolle E.M.: Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data. Water Resour. Res. 41, W05439-1-14 (2007)Google Scholar
  53. 53.
    Zheng, G.H., Wei, T.: Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation. J. Comput. Appl. Math. 233, 2631–2640 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Długosz UniversityCzęstochowaPoland

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