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Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients. (English) Zbl 1185.65146
The authors study a non homogeneous evolution equation with variable coefficients and fractional diffusion term of the form $$d_{+}(x)\frac{\partial^{\alpha}u(x,t)}{\partial_{+}x^{\alpha}}+ d_{-}(x)\frac{\partial^{\alpha}u(x,t)}{\partial_{+}x^{\alpha}},$$ where $1\leq\alpha\leq2,d_{+}(x)\geq0,d_{-}(x)\geq0$. Using a weighted finite difference method and the Grünwald formula for the fractional derivative, the authors obtain an approximate scheme for the initial value problem with the truncation error $ R_{j}^{n}(u)=O(\tau^{2}+h)$ for $r=1/2$ and $R_{j}^{n}(u)=O(\tau+h)$ for $ r\neq1/2,0\leq r\leq 1$, where $r$ is a weighted parameter. Some numerical examples are given.

65M06Finite difference methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35R11Fractional partial differential equations
Full Text: DOI
[1] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: The fractional-order governing equation of Lévy motion, Water resour. Res. 36, No. 6, 1413-1424 (2000)
[2] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: Application of a fractional advection--dispersion equation, Water resour. Res. 36, No. 6, 1403-1412 (2000)
[3] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[4] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[5] Schumer, R.; Benson, D. A.; Meerschaert, M. M.: Multiscaling fractional advection--dispersion equations and their solutions, Water resour. Res. 39, 1022-1032 (2003)
[6] Liu, F.; Anh, V.; Turner, I.: Numerical solution of the space fractional Fokker--plank equation, J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[7] Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for fractional advection--dispersion flow equations, J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[8] Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. Math. 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[9] Meerschaert, M. M.; Scheffler, H. P.; Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equations, J. comput. Phys. 211, 249-261 (2006) · Zbl 1085.65080 · doi:10.1016/j.jcp.2005.05.017
[10] Tadjeran, C.; Meerschaert, M. M.; Scheffler, H.: A second-order accurate numerical approximation for the fractional diffusion equation, J. comput. Phys. 213, 205-213 (2006) · Zbl 1089.65089 · doi:10.1016/j.jcp.2005.08.008
[11] Tadjeran, C.; Meerschaert, M. M.: A second-order accurate numerical approximation for the two-dimensional fractional diffusion equation, J. comput. Phys. 220, 813-823 (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
[12] Fix, G. J.; Roop, J. P.: Least squares finite element solution of a fractional order two-point boundary value problem, Comput. math. Appl. 48, 1017-1033 (2004) · Zbl 1069.65094 · doi:10.1016/j.camwa.2004.10.003
[13] Lynch, V. E.; Carreras, B. A.; Del-Castillo-Negrete, D.; Ferreira-Mejias, K. M.; Hicks, H. R.: Numerical methods for the solution of partial differential equations of fractional order, J. comput. Phys. 192, 406-421 (2003) · Zbl 1047.76075 · doi:10.1016/j.jcp.2003.07.008
[14] Shen, S.; Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, Anziam j. 46, No. E, 871-887 (2005)
[15] Liu, Q.; Liu, F.; Turner, I.; Anh, V.: Approximation of the Lévy--Feller advection--dispersion process by random walk and finite difference method, J. comput. Phys. 222, 57-70 (2007) · Zbl 1112.65006 · doi:10.1016/j.jcp.2006.06.005
[16] Zhang, Y.; Meerschaert, M. M.; Baeumer, B.: Particle tracking for time-fractional diffusion, Phys. rev. E 78, No. 3 (2008) · Zbl 1189.65240
[17] Yuste, S. B.: Weighted average finite difference methods for fractional diffusion equations, J. comput. Phys. 216, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[18] Meerschaert, M. M.; Mortensen, J.; Wheatcraft, S. W.: Fractional vector calculus for fractional advection--dispersion, Physica A 367, 181-190 (2006)