Abstract
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.
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References
Abdel-Rehim, E.A.: Explicit approximation solutions and proof of convergence of space-time fractional advection dispersion equations. Appl. Math. 4, 1427–1440 (2013)
Barkai, E.: Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E 63, 046118-1-17 (2001)
Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138 (2000)
Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)
Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)
Chaves, A.S.: A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13–16 (1998)
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)
Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971)
Huang, F., Liu, F.: The time fractional diffusion equation and the advection-dispersion equation. ANZIAM J. 46, 317–330 (2005)
Huang, H., Cao, X.: Numerical method for two dimensional fractional reaction subdiffusion equation. Eur. Phys. J. Spec. Top. 222, 1961–1973 (2013)
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)
Jumarie, G.: A Fokker-Planck equation of fractional order with respect to time. J. Math. Phys. 33, 3536–3542 (1992)
Karatay, I., Bayramoglu, S.R.: An efficient scheme for time fractional advection dispersion equations. Appl. Math. Sci. 6, 4869–4878 (2012)
Kaviany, M.: Principles of Heat Transfer in Porous Media, 2nd edn. Springer, New York (1995)
Kolwankar, K.M., Gangal, A.D.: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 80, 214–217 (1998)
Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Lévy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)
Liu, F., Anh, V., Turner, I., Zhuang, P.: Time-fractional advection-dispersion equation. J. Appl. Math. Comput. 13, 233–245 (2003)
Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)
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)
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)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
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)
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)
Nield, D.D., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)
Nowacki, W.: Dynamical problems of thermodiffusion in solids. Bull. Acad. Polon. Sci., Sér. Sci. Technol. 23, 55–64, 129–135, 257–266 (1974)
Nowacki, W., Olesiak, Z.S.: Thermodiffusion in Solids. Polish Scientific Publishers (PWN), Warsaw (1991) (in Polish)
Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)
Pidstrygach, Ya.S.: Differential equations of thermodiffusion problem in isotropic deformable solid. Dop. Ukrainian Acad. Sci. (2), 169–172 (1961) (in Ukrainian)
Pidstrygach, Ya.S.: Differential equations of the diffusive strain theory of a solid. Dop. Ukrainian Acad. Sci. (3), 336–339 (1963) (in Ukrainian)
Pidstryhach, Ya.S.: Selected Papers. Naukova Dumka, Kyiv (1995) (in Ukrainian and Russian)
Podstrigach, Ya.S.: Theory of diffusive deformation of isotropic continuum. Issues Mech. Real Solid 2, 71–99 (1964) (in Russian)
Podstrigach, Ya.S.: Diffusion theory of inelasticity of metals. J. Appl. Mech. Technol. Phys. (2), 67–72 (1965) (in Russian)
Podstrigach, Ya.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids. Naukova Dumka, Kiev (1985) (in Russian)
Povstenko, Y.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)
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)
Povstenko, Y.: Fundamental solution to three-dimensional diffusion-wave equation and associated diffusive stresses. Chaos, Solitons Fractals 36, 961–972 (2008)
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)
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)
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)
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)
Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Vol. 1: Elementary Functions. Gordon and Breach, Amsterdam (1986)
Risken, H.: The Fokker-Planck Equation. Springer, Berlin (1989)
Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)
Scheidegger, A.E.: The Physics of Flow Through Porous Media, 3rd edn. University of Toronto Press, Toronto (1974)
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)
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)
Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1981)
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)
Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2005)
Zaslavsky, G.M., Edelman, M., Niyazov, B.A.: Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics. Chaos 7, 159–181 (1997)
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)
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)
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)
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Povstenko, Y. (2015). Fractional Advection-Diffusion Equation and Associated Diffusive Stresses. In: Fractional Thermoelasticity. Solid Mechanics and Its Applications, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-15335-3_9
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DOI: https://doi.org/10.1007/978-3-319-15335-3_9
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