Matrix approach to discrete fractional calculus II: Partial fractional differential equations

• Published in: Journal of computational physics Vol. 228; no. 8; pp. 3137 - 3153
• Main Authors: Podlubny, Igor, Chechkin, Aleksei, Skovranek, Tomas, Chen, YangQuan, Vinagre Jara, Blas M.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.05.2009; Elsevier
• Subjects: Calculus; Computational techniques; Delay; Derivatives; Differential equations; Differential equations with delays; Diffusion; Discretization; Exact sciences and technology; Fractional diffusion equation; Fractional partial differential equations; Integers; Mathematical analysis; Mathematical methods in physics; Mathematical models; Numerical methods; Physics; 65Z05; Differential equations with delays; Discretization; 65M06; Numerical methods; 65D25; Fractional diffusion equation; Fractional partial differential equations; 91B82; Partial differential equations; Numerical method; Fractional derivative; Calculation methods; Space-time; Numerical solution; Differential equations; S matrix; Diffusion equation; Calculation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.01.014

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 Podlubny’s matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359–386]. 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.