The spectral collocation method for efficiently solving PDEs with fractional Laplacian. (English) Zbl 1395.65106
Summary: We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain \(\Omega =[a,b]\). Corresponding matrix representations of \((-\Delta)^{\alpha/2}\) for \(\alpha\in (0,1)\) and \(\alpha\in (1,2)\) 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.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35S11 | Initial-boundary value problems for pseudodifferential operators (MSC2010) |
| 35R11 | Fractional partial differential equations |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
Keywords:
fractional Laplacian; collocation method; space-fractional advection-dispersion equation; fractional differentiation matrixReferences:
| [1] | Caffarelli, LA; Roquejoffre, J-M; Sire, Y, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., (JEMS), 12, 1151-1179, (2010) · Zbl 1221.35453 · doi:10.4171/JEMS/226 |
| [2] | Garroni, A; Müller, S, Γ-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36, 1943-1964, (2005) · Zbl 1094.82008 · doi:10.1137/S003614100343768X |
| [3] | Huang, Q; Huang, G; Zhan, H, A finite element solution for the fractional advection-dispersion equation, Adv. Water Resour., 31, 1578-1589, (2008) · doi:10.1016/j.advwatres.2008.07.002 |
| [4] | Huang, J; Nie, N; Tang, Y, A second order finite difference-spectral method for space fractional diffusion equations, Sci. China Math., 57, 1303-1317, (2014) · Zbl 1305.65185 · doi:10.1007/s11425-013-4716-8 |
| [5] | Huang, Y; Oberman, A, Numerical methods for the fractional Laplacian: a finite difference-quadrature approach, SIAM J. Numer. Anal., 52, 3056-3084, (2014) · Zbl 1316.65071 · doi:10.1137/140954040 |
| [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) · Zbl 1143.35005 · doi:10.1080/03605300701318922 |
| [7] | Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, CA, USA (1999) · Zbl 0924.34008 |
| [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 · Zbl 0973.60001 |
| [9] | Laskin, N, Fractional Schrödinger equations, Phys. Rev. E, 66, 056108, (2002) · doi:10.1103/PhysRevE.66.056108 |
| [10] | Meerschaert, MM; Tadjeran, C, Finite difference approximations for fractional advection dispersion equations, J. Comput. Appl. Math., 2, 65-77, (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033 |
| [11] | Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974) · Zbl 0292.26011 |
| [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) · Zbl 0818.26003 |
| [13] | Schneider, WR; Wyss, W, Fractional diffusion and wave equation, J. Math. Phys., 30, 134-144, (1989) · Zbl 0692.45004 · doi:10.1063/1.528578 |
| [14] | Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006) · Zbl 1234.65005 |
| [15] | Stein, E.M.: Singular integrals differentiability properties of functions. Princeton Math Ser, vol. 30. Princeton University Press, Princeton, NJ (1970) · Zbl 0207.13501 |
| [16] | Tarasov, VE; Zaslavsky, GM, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354, 249-261, (2005) · doi:10.1016/j.physa.2005.02.047 |
| [17] | Wang, K; Wang, H, A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. Water Resour., 34, 810-816, (2011) · doi:10.1016/j.advwatres.2010.11.003 |
| [18] | Zayernouri, M; Karniadakis, GE, Fractional spectral collocation method, SIAM J. Sci. Comput., 36, a40-a62, (2014) · Zbl 1294.65097 · doi:10.1137/130933216 |
| [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) · Zbl 1189.65288 · doi:10.1016/j.camwa.2009.08.071 |
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