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Space-Time-Fractional Advection Diffusion Equation in a Plane

  • Yuriy Povstenko
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 320)

Abstract

The fundamental solution to the Cauchy problem for the space-time-fractional advection diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplace operator is considered in a case of two spatial variables. The solution is obtained using the Laplace integral transform with respect to time t and the double Fourier transform with respect to space variables x and y. Several particular cases of the solution are analyzed in details. Numerical results are illustrated graphically.

Keywords

Fractional calculus advection diffusion equation Caputo derivative Riesz operator 

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References

  1. 1.
    Kaviany, M.: Principles of Heat Transfer in Porous Media, 2nd edn. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  2. 2.
    Povstenko, Y.: Fractional Heat Conduction Equation and Associated Thermal Stress. J. Thermal Stresses 28, 83–102 (2005)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Povstenko, Y.: Thermoelasticity that Uses Fractional Heat Conduction Equation. J. Math. Sci. 162, 296–305 (2009)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Povstenko, Y.: Theory of Thermoelasticity Based on the Space-Time-Fractional Heat Conduction Equation. Phys. Scr. T(136), 014017 (2009)Google Scholar
  5. 5.
    Povstenko, Y.: Theories of Thermal Stresses Based on Space-Time-Fractional Telegraph Equations. Comp. Math. Appl. 64, 3321–3328 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997)CrossRefGoogle Scholar
  7. 7.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  8. 8.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Saichev, A.I., Zaslavsky, G.M.: Fractional Kinetic Equations: Solutions and Applications. Chaos 7, 753–764 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kolwankar, K.M., Gangal, A.D.: Local Fractional Fokker-Planck Equation. Phys. Rev. Lett. 80, 214–217 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    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
  13. 13.
    Yıldırım, A., Koçak, H.: Homotopy Perturbation Method for Solving the Space Time Fractional Advection-Dispersion Equation. Adv. Water Res. 32, 1711–1716 (2009)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Elementary Functions, vol. 1. Gordon and Breach, Amsterdam (1986)Google Scholar
  16. 16.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Special Functions, vol. 2. Gordon and Breach, Amsterdam (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Dlugosz University in CzȩstochowaCzȩstochowaPoland

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