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The spectral collocation method for efficiently solving PDEs with fractional Laplacian

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Abstract

We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (−△)α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth.

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References

  1. Caffarelli, L.A., Roquejoffre, J.-M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc., (JEMS) 12, 1151–1179 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Garroni, A., Müller, S.: Γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36, 1943–1964 (2005). electronic

    Article  MathSciNet  MATH  Google Scholar 

  3. Huang, Q., Huang, G., Zhan, H.: A finite element solution for the fractional advection-dispersion equation. Adv. Water Resour. 31, 1578–1589 (2008)

    Article  Google Scholar 

  4. Huang, J., Nie, N., Tang, Y.: A second order finite difference-spectral method for space fractional diffusion equations. Sci. China Math. 57(6), 1303–1317 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52(6), 3056–3084 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Imbert, C., Monneau, R., Rouy, E.: Homogenization of first order equations with (u/𝜖)-periodic hamiltonians, II. Application to dislocations dynamics. Comm. Partial Differential Equations 33, 479–516 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, CA, USA (1999)

    MATH  Google Scholar 

  8. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Stud. Adv. Math., vol. 68. Cambridge University Press, Cambridge (1999). translated from the 1990 Japanese original, revised by the author

    Google Scholar 

  9. Laskin, N.: Fractional Schrödinger equations. Phys. Rev. E 66(5), 056108 (2002)

    Article  MathSciNet  Google Scholar 

  10. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection dispersion equations. J. Comput. Appl. Math. 2, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  12. Samko, S.G., Kilbas, A.A, Marichev, O.I: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon, Switzerland (1993)

    MATH  Google Scholar 

  13. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equation. J. Math. Phys. 30(1), 134–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)

    MATH  Google Scholar 

  15. Stein, E.M.: Singular integrals differentiability properties of functions. Princeton Math Ser, vol. 30. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

  16. Tarasov, V.E., Zaslavsky, G.M.: Fractional Ginzburg-Landau equation for fractal media. Physica A 354, 249–261 (2005)

    Article  Google Scholar 

  17. Wang, K., Wang, H.: A fast characteristic finite difference method for fractional advection-diffusion equations. Adv. Water Resour. 34, 810–816 (2011)

    Article  Google Scholar 

  18. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40–A62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zheng, Y., Li, C., Zhao, Z.: A note on the finite element method for the space-fractional advection diffusion equation. Comput. Math. Appl. 59, 1718–1726 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

H. Lu was supported in part by the NSF of China (No. 11601278), the PSF of China (No. 2016M592172) and the NSF of Shandong Province (No. ZR2016AP01). P. W. Bates was supported partly by the NSF DMS-0908348 and DMS-1413060. M. Zhang was supported in part by start-up funds for new faculties at New Mexico Institute of Mining and Technology.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University, Weihai, 264209, China

    Hong Lu

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Peter W. Bates

  3. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China

    Wenping Chen

  4. Department of Mathematics, New Mexico Institution of Mining and Technology, Socorro, NM, 87801, USA

    Mingji Zhang

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  1. Hong Lu
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  2. Peter W. Bates
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  3. Wenping Chen
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  4. Mingji Zhang
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Correspondence to Peter W. Bates.

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Communicated by: Carlos Garcia-Cervera

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Lu, H., Bates, P.W., Chen, W. et al. The spectral collocation method for efficiently solving PDEs with fractional Laplacian. Adv Comput Math 44, 861–878 (2018). https://doi.org/10.1007/s10444-017-9564-6

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  • DOI: https://doi.org/10.1007/s10444-017-9564-6

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