Numerical simulation of an anomalous diffusion process based on the higher-order accurate scheme. (Russian. English summary) Zbl 1456.60008
Summary: The paper is devoted to the development and program implementation of a computational algorithm for modeling a process of anomalous diffusion. The mathematical model is formulated as an initial-boundary value problem for a nonlinear fractional order partial differential equation. An implicit finite-difference scheme based on an increased accuracy order approximation for the Caputo derivative is constructed. An application program was designed to perform computer simulation of the anomalous diffusion process. The numerical analysis of the accuracy of approximate solutions is conducted using a test-problem. The results of computational experiments are presented on the example of the modeling of a fractal nonlinear dynamic reaction-diffusion system.
MSC:
| 60-08 | Computational methods for problems pertaining to probability theory |
| 60K50 | Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
Keywords:
anomalous diffusion equation; reaction-diffusion process; Caputo fractional derivative; implicit finite difference schemeReferences:
| [1] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Theory and Application, Gordon & Breach Sci. Publ., Switzerland, Philadelphia, Pa, USA, 1993, 976 pp. · Zbl 0818.26003 |
| [2] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Fractional differential equations, 198, Acad. Press Inc., San Diego, CA, USA, 1999, 340 pp. · Zbl 0924.34008 |
| [3] | V. V. Uchaikin, Metod drobnykh proizvodnykh, Artishok, Ulianovsk, 2008, 512 pp. |
| [4] | W. Chen, H. Sun, X. Zhang, D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives”, Computers & Math. with Applicat., 59:5 (2010), 1754-1758 · Zbl 1189.35355 · doi:10.1016/j.camwa.2009.08.020 |
| [5] | X. Liang, F. Gao, C. Zhou et al, “An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type”, Advances in Difference Equation, 25 (2018), 1-11 · Zbl 1445.35305 |
| [6] | A. N. Bogolyubov, A. A. Koblikov, D. D. Smirnova, N. E. Shapkina, “Matematicheskoe modelirovanie sred s vremennoj dispersiej pri pomoshchi drobnogo differentsirovaniya”, Matem. model, 25:12 (2013), 50-64 · Zbl 1356.35233 |
| [7] | I. Petras, Fractional-order nonlinear systems. Modeling, analysis and simulation, Springer-Verlag, Berlin-Heidelberg, 2011, 218 pp. · Zbl 1228.34002 |
| [8] | R. Scherera, S. L. Kallab, Y. Tangc, J. Huang, “The Grünwald-Letnikov method for fractional differential equations”, Computers & Math. with Appl., 62 (2011), 902-917 · Zbl 1228.65121 · doi:10.1016/j.camwa.2011.03.054 |
| [9] | E. Sousa, “How to approximate the fractional derivative of order \(1<\alpha\leqslant 2\)”, International journal of bifurcation and chaos, 22:4 (2012), 1-6 · Zbl 1258.26006 |
| [10] | K. Diethelm, N. J. Ford, A. D. Freed, Yu. Luchko, “Algorithms for the fractional calculus: a selection of numerical method”, Comp. Methods Appl. Mech. Eng., 194 (2005), 743-773 · Zbl 1119.65352 · doi:10.1016/j.cma.2004.06.006 |
| [11] | L. Xiaoting, H. Sun, Y. Zhang, Z.-J. Fu, “A scale-dependent finite difference approximation for time fractional differential equation”, Computational Mechanics, 63 (2019), 429-442 · Zbl 1467.76038 |
| [12] | U. Ali, F. A. Abdullah, A. I. Ismail, “Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation”, J. of Interpolation and Approximation in Sci. Comp., 2017:2 (2017), 18-29 |
| [13] | B. R. Sontakke, A. S. Shelke, “Approximate scheme for time fractional diffusion equation and its applications”, Global J. of Pure and Applied Math., 13:8 (2017), 4333-4345 |
| [14] | I. Podlubny, “Matrix approach to discrete fractional calculus II: partial differential equations”, Journal of Computational Physics, 3 (2000), 359-386 · Zbl 1030.26011 |
| [15] | B. J. Szekeres; F. Izsák, “A finite difference method for fractional diffusion equations with Neumann boundary conditions”, Open Mathematics, 13:1 (2015), 553-561 · Zbl 1327.65215 |
| [16] | A. A. Petukhov, D. L. Reviznikov, “Algoritmy chislennogo resheniia drobno-differentsialnykh uravnenii”, Vestnik MAI, 16 (2009), 228-234 |
| [17] | C. Li, R. Wu, H. Ding, “High-order approximation to Caputo derivative and Caputo-type advection-diffusion equations”, Comm. in Appl. & Industrial Math., 6:2 (2015), 1-33 |
| [18] | Y. Dimitrov, “Three-point compact approximation for the Caputo fractional derivative”, Communications on Applied Mathematics and Computation, 31:4 (2017), 413-442 · Zbl 1399.26010 |
| [19] | H. Ding, “A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation”, Applied Numerical Mathematics, 135 (2019), 30-46 · Zbl 1404.65085 |
| [20] | L. I. Moroz, A. G. Maslovskaia, “Drobno-differentsialnaia model protsessa teploprovodnosti segnetoelectricheskikh materialov v usloviiakh intensivnogo nagreva”, Matematika i mathematicheskoe modelirovanie, 2 (2019), 29-47 |
| [21] | W. Deng, “Short memory principle and a predictor-corrector approach for fractional differentional equations”, J. of Comp. and Applied Mathematics, 206 (2007), 174-188 · Zbl 1121.65128 · doi:10.1016/j.cam.2006.06.008 |
| [22] | R. P. Meilanov, S. A. Sadykov, “Fractal model for polarization switching kinetics in ferroelectric crystals”, Technical Physics, 44 (1999), 595-596 |
| [23] | Z. Bin, Model for coupled ferroelectric hysteresis using time fractional operators: Application to innovative energy harvesting, diss. … doctor of science, Lyon, 2014, 95 pp. |
| [24] | L. I. Moroz, A. G. Maslovskaya, “Hybrid stochastic fractal-based approach to modelling ferroelectrics switching kinetics in injection mode”, MM&CS, 12:3 (2020), 348-356 · Zbl 1441.93296 |
| [25] | K. M. Rabe, C. Ahn, J. M. Triscone, Physics of ferroelectrics: a modern perspective, Springer, Berlin, 2007, 388 pp. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
