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Finite element method for drifted space fractional tempered diffusion equation

  • Ayan ChakrabortyEmail author
  • B. V. Rathish Kumar
Original Research

Abstract

Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in different field of science have been reconsidered in this terms like diffusion wave equations, Schr\(\ddot{o}\)dinger equation and so on. In the present paper, a time dependent tempered fractional diffusion equation of order \(\gamma \in (0,1)\) with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank–Nicolson discretization is used in the time direction. By implementing finite element approximation a priori space–time estimate has been derived and we proved that the convergent order is \(\mathcal {O}(h^2+\varDelta t ^2)\) where h is the space step size and \(\varDelta t\) is the time difference. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations.

Keywords

Tempered fractional Stability b-spline finite element Error estimates Gronwall’s lemma 

Mathematics Subject Classification

65N30 65J10 65N15 

Notes

Acknowledgements

The authors would like to express deep gratitude to Cem Celik and Melda Duman for helpful suggestions and discussions. The authors are also grateful to the reviewers for their careful reading and intuitive implications.

References

  1. 1.
    Meng, Q.-J., Ding, D., Sheng, Q.: Preconditioned iterative methods for fractional diffusion models in finance. Numer. Methods Partial Differ. Equ. 31(5), 1382–1395 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Political Econ. 81(3), 637–654 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gorenflo, R., et al.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29(1–4), 129–143 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baeumer, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233(10), 2438–2448 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cartea, A., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A Stat. Mech. Appl. 374(2), 749–763 (2007)Google Scholar
  6. 6.
    Jin, B., et al.: Error analysis of a finite element method for the space-fractional parabolic equation. SIAM J. Numer. Anal. 52(5), 2272–2294 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Arqub, O.A., Odibat, Z., Al-Smadi, M.: Numerical solutions of time-fractional partial integrodifferential equations of Robin functions types in Hilbert space with error bounds and error estimates. Nonlinear Dyn. 94(3), 1819–1834 (2018)CrossRefGoogle Scholar
  9. 9.
    Arqub, O.A., Maayah, B.: Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator. Chaos Solitons Fractals 117, 117–124 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Arqub, O.A., Al Smadi, M.: Numerical algorithm for solving time? Fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions. Numer. Methods Partial Differ. Equ. 34(5), 1577–1597 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Al-Smadi, M., Arqub, O.A.: Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Appl. Math. Comput. 342, 280–294 (2019)MathSciNetGoogle Scholar
  12. 12.
    Arqub, O.A.: Solutions of time? Fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer. Methods Partial Differ. Equ. 34(5), 1759–1780 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Arqub, O.A., Al-Smadi, M.: Atangana–Baleanu fractional approach to the solutions of Bagley-Torvik and Painlevé equations in Hilbert space. Chaos Solitons Fractals 117, 161–167 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Weng, Z., Zhai, S., Feng, X.: A Fourier spectral method for fractional-in-space Cahn–Hilliard equation. Appl. Math. Model. 42, 462–477 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gajda, J., Magdziarz, M.: Fractional Fokker–Planck equation with tempered \(\alpha \)-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 82(1), 011117 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Balint, A.M., Balint, S.: In classical mechanics objectivity lost when Riemann-Liouwille or Caputo fractional order derivatives are used. arXiv preprint arXiv:1806.04186 (2018)
  17. 17.
    Ding, H., Li, C., Chen, Y.Q.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shen, S., Liu, F., Anh, V.: Numerical approximations and solution techniques for the space–time Riesz–Caputo fractional advection-diffusion equation. Numer. Algorithms 56(3), 383–403 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Çelik, C., Duman, M.: Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231(4), 1743–1750 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, X., Deng, W.: Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations. arXiv preprint arXiv:1706.02826 (2017)
  21. 21.
    Çelik, C., Duman, M.: Finite element method for a symmetric tempered fractional diffusion equation. Appl. Numer. Math. 120, 270–286 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100 (1987)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)CrossRefGoogle Scholar
  24. 24.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)CrossRefGoogle Scholar
  25. 25.
    Evans, L.C.: Partial differential equations. American Mathematical Society (2010)Google Scholar
  26. 26.
    Quarteroni, A., Valli, A.: Introduction. Springer, Berlin (1994)Google Scholar
  27. 27.
    Chen, M. et al.: A fast multigrid finite element method for the time-dependent tempered fractional problem. arXiv preprint arXiv:1711.08209 (2017)

Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  1. 1.Faculty Building, 526Indian Institutue of Technology KanpurKanpurIndia
  2. 2.Research Scholar, 555IIT KanpurKanpurIndia

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