Numerical approximations for fractional diffusion equations via splines. (English) Zbl 1228.65153

Sousa, Ercília

Comput. Math. Appl. 62, No. 3, 938-944 (2011).

Summary: A one-dimensional fractional diffusion model is considered, where the usual second order derivative gives place to a fractional derivative of order \(\alpha \), with \(1<\alpha \leq 2\). We consider the Caputo derivative as the space derivative, which is a form of representing the fractional derivative by an integral operator. An implicit numerical method is derived which uses a spline approximation for the Caputo derivative. The consistency and stability of the method are examined and numerical results are presented.

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11	Fractional partial differential equations
26A33	Fractional derivatives and integrals
45K05	Integro-partial differential equations
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

fractional diffusion; integro-differential equations; finite differences

References:

[1]	Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports, 339, 1-77, (2000) · Zbl 0984.82032
[2]	Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport, Physics reports, 371, 461-580, (2002) · Zbl 0999.82053
[3]	Shen, S.; Liu, F., Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, Anziam j., 46, E, C871-C887, (2005) · Zbl 1078.65563
[4]	Tadjeran, C.; Meerschaert, M.M.; Scheffler, H.-P., A second-order accurate numerical approximation for the fractional diffusion equation, Journal of computational physics, 213, 205-213, (2006) · Zbl 1089.65089
[5]	Chen, C.-M.; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of computational physics, 227, 886-897, (2007) · Zbl 1165.65053
[6]	Yuste, S.B.; Acedo, L., An explicit finite difference method and a new von Neumann type stability analysis for fractional diffusion equations, SIAM journal on numerical analysis, 42, 1862-1874, (2005) · Zbl 1119.65379
[7]	Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, Journal of computational physics, 216, 264-274, (2006) · Zbl 1094.65085
[8]	Murio, D.A., Implicit finite difference approximation for time fractional diffusion equations, Computer and mathematics with applications, 56, 1138-1145, (2008) · Zbl 1155.65372
[9]	Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, Journal of computational and applied mathematics, 172, 65-77, (2004) · Zbl 1126.76346
[10]	Sousa, E., Finite difference approximations for a fractional advection diffusion problem, Journal of computational physics, 228, 4038-4054, (2009) · Zbl 1169.65126
[11]	Zhang, X.; Mouchao, Lv.; Crawford, J.W.; Young, I.M., The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: defining the fractional dispersive flux with Caputo derivatives, Advances in water resources, 30, 1205-1217, (2007)
[12]	Huang, Q.; Huang, G.; Zhan, H., A finite element solution for the fractional advection-dispersion equation, Advances in water resources, 31, 1578-1589, (2008)
[13]	Zheng, Y.; Li, C.; Zhao, Z., A note on the finite element method for the space-fractional advection diffusion equation, Computer and mathematics with applications, 59, 1718-1726, (2010) · Zbl 1189.65288
[14]	Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space – time fractional advection-diffusion equation, Applied mathematics and computation, 191, 12-20, (2006) · Zbl 1193.76093
[15]	Shen, S.; Liu, F.; Ahn, V.; Turner, I., The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA journal of applied mathematics, 73, 850-872, (2008) · Zbl 1179.37073
[16]	Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier · Zbl 1092.45003
[17]	Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[18]	Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Science Publishers · Zbl 0818.26003
[19]	E. Sousa, How to approximate the fractional derivative of order \(1 < \alpha \leq 2\), in: Proceedings of the 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, 2010.