The spectral collocation method for efficiently solving PDEs with fractional Laplacian
- PDF / 802,009 Bytes
- 18 Pages / 439.642 x 666.49 pts Page_size
- 59 Downloads / 261 Views
The spectral collocation method for efficiently solving PDEs with fractional Laplacian Hong Lu1 · Peter W. Bates2 Mingji Zhang4
· Wenping Chen3 ·
Received: 28 April 2016 / Accepted: 18 September 2017 © Springer Science+Business Media, LLC 2017
Abstract We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain = [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. Keywords Fractional Laplacian · Collocation method · Space-fractional advection-dispersion equation · Fractional differentiation matrix Communicated by: Carlos Garcia-Cervera Peter W. Bates
[email protected] Hong Lu [email protected] Wenping Chen [email protected] Mingji Zhang [email protected]; [email protected] 1
School of Mathematics and Statistics, Shandong University, Weihai 264209, China
2
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3
School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
4
Department of Mathematics, New Mexico Institution of Mining and Technology, Socorro, NM 87801, USA
H. Lu et al.
Mathematics Subject Classification (2010) 65M70 · 35S11 · 35R11
1 Introduction The fractional Laplacian appears in diverse areas including mathematical finances, physics, biological modeling and so on. It has been the subject of great interest in recent years (for example, in crystals [2], in dislocations processes of mechanical systems [6], in anomalous diffusion [11] and in combustion theory [1]). In the theory of probability, the fractional Laplacian is the infinitesimal generator of an α-stable L´evy process, which has a nice probabilistic interpretation (see [8]). The partial differential equations involving fractional powers of the Laplacian arise in recent decades, such as fractional diffusion equation [13], fractional Ginzburg-Landau equation [16], fractional Schr¨odinger equation [9] and so on. Alhough the fractional Laplacian is applied widely, its nonlocal property (long-range spatial dependence) makes it difficult to handle. Because of the nonlocal property, it is not trivial to extend existing numerical methods for integer-order differential equations to their corresponding fractional differential counterparts. The fractional Laplacian operator (−)α/2 (0 < α < 2) is defined differently on the entire space Rn and on a bounded domain. On Rn , it can be defined by two equivalent forms: As a pseudodifferential operator with symbol |ξ |α ([15]), that is, one definition is given through the Fourier transform F , F ((−)α/2 u)(ξ ) = |ξ |α F (u)(ξ ).
Another is given through a sin
Data Loading...
