Skip to main content
SpringerLink
Log in

Numerical solutions for asymmetric Lévy flights

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Lévy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We consider the formulation of Lévy flights, for 0 < α < 1, in terms of a space-fractional diffusion equation which fundamental solutions are the probability density functions. First, we present how to obtain the governing equation of Lévy motion from the Fourier transform of the jump distribution. Then, we derive a family of implicit numerical methods to determine the numerical solutions and we study their consistency and stability. Although numerical algorithms for the case 1 < α < 2 have been widely discussed, very few works paid attention to the case we discuss here. We present numerical experiments to show the performance of the numerical methods and to highlight the advantages and disadvantages of the different approaches. In the end we determine the numerical solutions of an initial value problem, that considers an approximation of the Dirac delta function as the initial condition, in order to obtain approximations of the probability density functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Albeverio, S., Casati, G., Merlini, D. (eds.): Stochastic processes in classical and quantum systems. Lecture Notes in Physics, vol. 262. Springer, Berlin (1986)

  2. Bai, Z.Z., Lu, K.Y.: Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations. J. Comput. Phys. 404, 109117 (2020)

    Article  MathSciNet  Google Scholar 

  3. Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Springer, Berlin (2016)

    Book  Google Scholar 

  4. Castillo, P., Gómez, S.: Conservative local discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction. Numer. Algorithms 84, 407–425 (2020)

    Article  MathSciNet  Google Scholar 

  5. Fogedby, H.C.: Lévy flights in random environments. Phys. Rev. Lett 73, 2517 (1994)

    Article  Google Scholar 

  6. Huang, Y.C., Lei, S.L.: Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations. Numer. Algorithms 84, 37–62 (2020)

    Article  MathSciNet  Google Scholar 

  7. Jespersen, S., Metzler, R., Fogedby, H.C.: Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions. Phys. Rev. E 59, 2736 (1999)

    Article  Google Scholar 

  8. Cartalade, A., Younsi, A., Néel, M.C.: Multiple-relaxation-time lattice Boltzmann scheme for fractional advection-diffusion equation. Comput. Phys. Commun. 234, 40–54 (2019)

    Article  MathSciNet  Google Scholar 

  9. Diethelm, K., Freed, A. D.: The FracPECE subroutine for the numerical solution of differential equations of fractional order. In: Heinzel, S., Plesser, T. (eds.) “Forschung und wissenschaftliches Rechnen 1998”, pp. 57-71, Göttingen: Gesellschaft für wissenschaftliche Datenverarbeitung (1999)

  10. Gao, T., Duan, J., Li, X., Song, R.: Mean exit time and escape probability for dynamical systems driven by Lévy noises. SIAM J. Sci. Comput 36, A887–A906 (2014)

    Article  Google Scholar 

  11. Guo, S., Mei, L., Zhang, Z., Li, C., Li, M., Wang, Y.: A linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed order time-space fractional nonlinear reaction-diffusion-wave equation: Numerical simulations of Gordon-type solitons. Comput. Phys. Commun. 252, 107144 (2020)

    Article  MathSciNet  Google Scholar 

  12. Górska, K., Penson, K.A.: Lévy stable two-sided distributions: exact and explicit densities for asymmetric case. Phys. Rev. E 83, 061125 (2011)

    Article  Google Scholar 

  13. Hirsch, C.: Numerical computations of internal and external flows: The fundamentals of computational fluid dynamics. Elsevier (2007)

  14. Qu, W., Lei, S.L., Vong, S.W.: Circulant and skew-circulant splitting iteration for fractional advection-diffusion equations. Int. J. Comput. Math 91, 2232–2242 (2014)

    Article  MathSciNet  Google Scholar 

  15. Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)

    Article  MathSciNet  Google Scholar 

  16. Li, C., Zeng, F.: Numerical methods for fractional calculus. CRC Press (2015)

  17. Li, C., Cai, M.: Theory and numerical approximations of fractional integrals and derivatives. SIAM (2019)

  18. Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal 4, 153–192 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-function: Theory and applications. Springer (2009)

  20. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep 339, 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  21. Odibat, Z.: Approximations of fractional integrals and Caputo fractional derivatives. Appl. Math. Comput 178, 527–533 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Padash, A., Chechkin, A., Dybiec, B., Pavlyukevich, I., Shokri, B., Metzler, R.: First-passage properties of asymmetric Lévy flights. J. Phys. A: Math. Theor. 52, 454004 (2019)

    Article  Google Scholar 

  23. Padash, A., Chechkin, A., Dybiec, B., Magdziarz, M., Shokri, B., Metzler, R.: First passage time moments of asymmetric Lévy flights. J. Phys. A: Math. Theor. 53, 275002 (2020)

    Article  Google Scholar 

  24. Penson, K. A., Górska, K.: Exact and explicit probability densities for one-sided Lévy stable distributions. Phys. Rev. Lett. 105, 210604 (2010)

    Article  MathSciNet  Google Scholar 

  25. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  26. Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)

    Article  MathSciNet  Google Scholar 

  27. Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random processes: stochastic models with infinite variance. Chapman and Hall/CRC (1994)

  28. Sousa, E.: Numerical solution of a model for turbulent diffusion. Int. J. Bifurcation Chaos 23, 1350166 (2013)

    Article  MathSciNet  Google Scholar 

  29. Sousa, E.: An explicit high order method for fractional advection diffusion equations. J. Comput. Phys 278, 257–274 (2014)

    Article  MathSciNet  Google Scholar 

  30. Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative. Appl. Numer. Math 90, 22–37 (2015)

    Article  MathSciNet  Google Scholar 

  31. Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math 11, 309–319 (1993)

    Article  MathSciNet  Google Scholar 

  32. Uchaikin, V.V., Zolotarev, V.M.: Chance and Stability. Stable Distributions and their Applications. De Gruyter (1999)

  33. Wang, X., Duan, J., Li, X., Song, R.: Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric Lévy motion. Appl. Math. Comput 337, 618–634 (2018)

    Article  MathSciNet  Google Scholar 

  34. West, B.J., Grigolini, P., Metzler, R., Nonnenmacher, T.F.: Fractional diffusion and Lévy stable processes. Phys. Rev. E 55, 99–106 (1997)

    Article  MathSciNet  Google Scholar 

  35. West, B.J.: Fractional calculus view of complexity: A tutorial. Rev. Mod. Phys 86, 1169 (2014)

    Article  Google Scholar 

  36. Xu, K., Darve, E.: Isogeometric collocation method for the fractional Laplacian in the 2D bounded domain. Comput. Methods Appl. Mech. Eng. 364, 112936 (2020)

    Article  MathSciNet  Google Scholar 

  37. Vázquez, J.L., de Pablo, A., Quirós, F., Rodrí quez, A.: Classical solutions and higher regularity for nonlinear fractional diffusion equations. J. Eur. Math. Soc 19, 1949–1975 (2017)

    Article  MathSciNet  Google Scholar 

  38. Zaky, M.A., Hendy, A.S., Mácias-Dí az, J.E.: Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions. J. Sci. Comput 82 (2020)

  39. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep 371, 461–580 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was partially supported by the Centre for Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. C. Jesus was also supported by FCT, through scholarship PD/BD/142955/2018, under POCH funds, co-financed by the European Social Fund and Portuguese National Funds from MEC.

Author information

Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Carla Jesus & Ercília Sousa

Authors
  1. Carla Jesus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ercília Sousa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ercília Sousa.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jesus, C., Sousa, E. Numerical solutions for asymmetric Lévy flights. Numer Algor 87, 967–999 (2021). https://doi.org/10.1007/s11075-020-00995-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00995-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation