## Matrix approach to discrete fractional calculus. II: Partial fractional differential equations.(English)Zbl 1160.65308

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:

 65D25 Numerical differentiation 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 91B82 Statistical methods; economic indices and measures 65Z05 Applications to the sciences

Zbl 1030.26011

ma2dfc; Matlab
Full Text:

### References:

 [1] Bulgakov, B.V., Kolebaniya (vibrations), (1954), Gostekhizdat Moscow [2] M. Caputo, Elasticità e dissipazione, Zanichelli, Bologna, 1969. [3] Chechkin, A.; Gonchar, V.; Gorenflo, R.; Korabel, N.; Sokolov, I., Generalized fractional diffusion equations for accelerating subdiffusion and truncated levy flights, Physical review E, 78, (2008), Article 021111 [4] Chechkin, A.; Gonchar, V.; Klafter, J.; Metzler, R.; Tanatarov, L., Stationary states of non-linear oscillators driven by Lévy noise, Chemical physics, 284, 1-2, 233-251, (2002) [5] Chechkin, A.; Gonchar, V.; Szydłowsky, M., Fractional kinetics for relaxation and superdiffusion in magnetic field, Physics of plasmas, 9, 1, 78-88, (2002) [6] Chechkin, A.; Gonchar, V.Y.; Klafter, J.; Metzler, R., Fundamentals of Lévy flight processes, Advances in chemical physics, 133, 439-496, (2006) [7] Chechkin, A.; Gorenflo, R.; Sokolov, I., Retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations, Physical review E, 66, 4, 1-7, (2002), Article 046129 [8] Chechkin, A.; Gorenflo, R.; Sokolov, I., Fractional diffusion in inhomogeneous media, Journal of physics A: mathematical and general, 38, 42, L679-L684, (2005) · Zbl 1082.76097 [9] Chechkin, A.; Gorenflo, R.; Sokolov, I.; Gonchar, V., Distributed order time fractional diffusion equation, Fractional calculus and applied analysis, 6, 3, 259-279, (2003) · Zbl 1089.60046 [10] Cooke, R.G., Infinite matrices and sequence spaces, (1960), Fizmatgiz Moscow · Zbl 0132.28901 [11] del Castillo-Negrete, D., Fractional diffusion models of nonlocal transport, Physics of plasmas, 13, 8, (2006), Article 082308 [12] del Castillo-Negrete, D.; Carreras, B.A.; Lynch, V.E., Front dynamics in reaction – diffusion systems with Lévy flights: a fractional diffusion approach, Physical review letters, 91, 1, (2003), Article 018302 [13] del Castillo-Negrete, D.; Carreras, B.A.; Lynch, V.E., Nondiffusive transport in plasma turbulence: a fractional diffusion approach, Physical review letters, 94, (2005), Article 065003 [14] del Castillo-Negrete, D.; Gonchar, V.; Chechkin, A., Fluctuation-driven directed transport in presence of Lévy flights, Physica A, 27, 6693-6704, (2008) [15] Ervin, V.J.; Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Numerical methods for partial differential equations, 22, 3, 558-576, (2005) · Zbl 1095.65118 [16] Ervin, V.J.; Roop, J.P., Variational solution of fractional advection dispersion equations on bounded domains in $$r^d$$, Numerical methods for partial differential equations, 23, 2, 256-281, (2006) · Zbl 1117.65169 [17] Friedrich, R., Statistics of Lagrangian velocities in turbulent flows, Physical review letters, 90, 8, (2003), Article 084501 [18] Gantmakher, F.R., Theory of matrices, (1988), Nauka Moscow · Zbl 0666.15002 [19] Gorenflo, R.; Abdel-Rehim, E., Discrete models of time-fractional diffusion in a potential well, Fractional calculus and applied analysis, 8, 2, 173-200, (2005) · Zbl 1129.26002 [20] Gorenflo, R.; De Fabritiis, G.; Mainardi, F., Discrete random walk models for symmetric Lévy – feller diffusion processes, Physica A, 269, 79-89, (1999) [21] Gorenflo, R.; Mainardi, F., Random walk models for space-fractional diffusion processes, Fractional calculus and applied analysis, 1, 2, 167-192, (1998) · Zbl 0946.60039 [22] Gorenflo, R.; Mainardi, F., Random walk models approximating symmetric space fractional diffusion processes, Problems in mathematical physics, 121, 120-145, (2001) · Zbl 1007.60082 [23] Gorenflo, R.; Mainardi, F.; Moretti, D.; Pagnini, G.; Paradisi, P., Discrete random walk models for space – time fractional diffusion, Chemical physics, 284, 1-2, 521-541, (2002) [24] Gorenflo, R.; Mainardi, F.; Moretti, D.; Pagnini, G.; Paradisi, P., Fractional diffusion: probability distributions and random walk models, Physica A, 305, 1-2, 106-112, (2002) · Zbl 0986.82037 [25] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear dynamics, 29, 129-143, (2002) · Zbl 1009.82016 [26] Heinsalu, E.; Patriarca, M.; Goychuk, I.; Hanggi, P., Use and abuse of a fractional fokker – planck dynamics for time-dependent driving, Physical review letters, 99, (2007), Article 120602 [27] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with riemann – liouville fractional derivatives, Rheologica acta, 45, 5, 765-771, (2006) [28] () [29] Langlands, T.; Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, Journal of computational physics, 205, 2, 719-736, (2005) · Zbl 1072.65123 [30] Liang, J.; Chen, Y.-Q., Hybrid symbolic and numerical simulation studies of time-fractional order wave-diffusion systems, International journal of control, 79, 11, 1462-1470, (2006) · Zbl 1125.65364 [31] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics, 225, 2, 1533-1552, (2007) · Zbl 1126.65121 [32] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, Journal of computational and applied mathematics, 166, 1, 209-219, (2004) · Zbl 1036.82019 [33] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of the levy – feller advection – dispersion process by random walk and finite difference method, Journal of computational physics, 222, 1, 57-70, (2007) · Zbl 1112.65006 [34] Lu, J.-F.; Hanyga, A., Wave field simulation for heterogeneous porous media with singular memory drag force, Journal of computational physics, 208, 2, 651-674, (2005) · Zbl 1329.76338 [35] Lynch, V.E.; Carreras, B.A.; del Castillo-Negrete, D.; Ferreira-Mejias, K., Numerical methods for the solution of partial differential equations of fractional order, Journal of computational physics, 192, 406-421, (2003) · Zbl 1047.76075 [36] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space – time fractional diffusion equation, Fractional calculus and applied analysis, 4, 2, 153-192, (2001) · Zbl 1054.35156 [37] Mainardi, F.; Mura, A.; Pagnini, G.; Gorenflo, R., Time-fractional diffusion of distributed order, Journal of vibration and control, 14, 9-10, 1267-1290, (2008) · Zbl 1229.35118 [38] Meerschaert, M.; Benson, D.; Bäumer, B., Multidimensional advection and fractional dispersion, Physical review E, 59, 5, 5026-5028, (1999) [39] Meerschaert, M.; Scheffler, H.-P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, Journal of computational physics, 211, 249-261, (2006) · Zbl 1085.65080 [40] Meerschaert, M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion equations, Journal of computational and applied mathematics, 172, 1, 65-77, (2004) · Zbl 1126.76346 [41] Meerschaert, M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Applied numerical mathematics, 56, 80-90, (2006) · Zbl 1086.65087 [42] Meerschaert, M.M.; Scheffler, H.-P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, Journal of computational physics, 211, 1, 249-261, (2006) · Zbl 1085.65080 [43] Metzler, R.; Barkai, E.; Klafter, J., Deriving fractional fokker – planck equations from a generalized master equation, Europhysics letters, 46, 4, 431-436, (1999) [44] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports, 339, 1, 1-77, (2000) · Zbl 0984.82032 [45] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of physics A: mathematical and general, 37, 2, R161-R208, (2004) · Zbl 1075.82018 [46] Milne, W.E., Numerical solution of differential equations, (1953), Wiley New York · Zbl 0007.11305 [47] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004 [48] Ortigueira, M.D., Riesz potential operators and inverses via fractional centred derivatives, International journal of mathematics and mathematical sciences, 1-12, (2006), Article ID 48391 · Zbl 1122.26007 [49] Ortigueira, M.D.; Batista, A.G., On the relation between the fractional Brownian motion and the fractional derivatives, Physics letters A, 372, 958-968, (2008) · Zbl 1217.26016 [50] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [51] Podlubny, I., Matrix approach to discrete fractional calculus, Fractional calculus and applied analysis, 3, 4, 359-386, (2000) · Zbl 1030.26011 [52] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional calculus and applied analysis, 5, 4, 367-386, (2002) · Zbl 1042.26003 [53] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, B. Vinagre Jara, Matrix approach to discretization of ODEs and PDEs of arbitrary real order, November 12, 2008. . · Zbl 1160.65308 [54] Saichev, A.; Zaslavsky, G., Fractional kinetic equations: solutions and applications, Chaos, 7, 4, 753-764, (1997) · Zbl 0933.37029 [55] Scherer, R.; Kalla, S.L.; Boyadjiev, L.; Al-Saqabi, B., Numerical treatment of fractional heat equations, Applied numerical mathematics, 58, 8, 1212-1223, (2008) · Zbl 1143.65105 [56] Sokolov, I.; Klafter, J.; Blumen, A., Fractional kinetics, Physics today, 55, 48-54, (2002) [57] Sokolov, I.M.; Klafter, J., Field-induced dispersion in subdiffusion, Physical review letters, 97, (2006), Article 140602 · Zbl 1142.82364 [58] Stanescu, D.; Kim, D.; Woyczynski, W.A., Numerical study of interacting particles approximation for integro-differential equations, Journal of computational physics, 206, 2, 706-726, (2005) · Zbl 1070.65004 [59] Suprunenko, D.A.; Tyshkevich, R.I., Commutative matrices, (1966), Nauka i Tekhnika Minsk · Zbl 1103.16302 [60] Tadjeran, C.; Meerschaert, M.M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, Journal of computational physics, 220, 2, 813-823, (2007) · Zbl 1113.65124 [61] Tadjeran, C.; Meerschaert, M.M.; Scheffler, H.-P., A second-order accurate numerical approximation for the fractional diffusion equation, Journal of computational physics, 213, 1, 205-213, (2006) · Zbl 1089.65089 [62] Valko, P.P.; Abate, J., Numerical inversion of 2-d Laplace transforms applied to fractional diffusion equation, Applied numerical mathematics, 53, 73-88, (2005) · Zbl 1060.65681 [63] van Loan, C.F., The ubiquitous Kronecker product, Journal of computational and applied mathematics, 123, 85-100, (2000) · Zbl 0966.65039 [64] Weron, A.; Magdziarz, M.; Weron, K., Modeling of subdiffusion in space – time-dependent force fields beyond the fractional fokker – planck equation, Physical review E, 77, (2008), Article 036704 [65] Yong, Z.; Benson, D.A.; Meerschaert, M.M.; Scheffler, H.-P., On using random walks to solve the space-fractional advection – dispersion equations, Journal of statistical physics, 123, 1, 89-110, (2006) · Zbl 1092.82038 [66] Yuan, L.; Agrawal, O.P., A numerical scheme for dynamic systems containing fractional derivatives, Journal of vibration and acoustics, 124, 2, 321-324, (2002) [67] Yuste, S., Weighted average finite difference methods for fractional diffusion equations, Journal of computational physics, 216, 1, 264-274, (2006) · Zbl 1094.65085 [68] Zaslavsky, G., Chaos, fractional kinetics, and anomalous transport, Physics reports, 371, 1, 461-580, (2002) · Zbl 0999.82053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.