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Zbl 1160.65308
Podlubny, Igor; Chechkin, Aleksei; Skovranek, Tomas; Chen, Yangquan; Vinagre Jara, Blas M.
Matrix approach to discrete fractional calculus. II: Partial fractional differential equations. (English)
[J] J. Comput. Phys. 228, No. 8, 3137-3153 (2009). ISSN 0021-9991

Summary: A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of the first author's matrix approach [Fract. Calc. Appl. Anal. 3, No. 4, 359--386 (2000; Zbl 1030.26011)]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.

MSC 2000:: *65D25 Numerical differentiation
65M06 Finite difference methods (IVP of PDE)
91B82 Statistical methods in economics
65Z05 Applications to physics

Keywords: fractional partial differential equations; differential equations with delays; fractional diffusion equation; numerical methods; discretization

Citations: Zbl 1030.26011

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