An Algorithm for the Numerical Solutions of the Time-Space Fractional Reaction-Diffusion-Drift Equation

The paper is devoted to the construction and program implementation of the computational algorithm for modeling a process of diffusion-drift nature based on the fractional diffusion approach. The mathematical model is formulated as an initial-boundary value problem for the time-space fractional diffusion-drift equation in a limited domain. Time and space fractional derivatives are considered in the sense of Caputo and Riemann – Liouville, respectively. A modified implicit finite-difference scheme is constructed. The concept of the considered mathematical problem provides an example of a deterministic model of the charging process of dielectric materials. An application program has been developed that implements the constructed numerical algorithm. The results were verified using the example of solving a test problem.

1. Uchajkin V.V. Metod drobnyh proizvodnyh. Ul'yanovsk: Izd-vo «Artishok», 2008. 512 p. (In Russ.).
2. Deng W., Hou R., Wang W., Xu P. Modeling Anomalous Diffusion. From Statistics to Mathematics. Singapore: World Scientific, 2020. 268 p.
3. Evangelista L.R., Lenzi E.K. Fractional diffusion equations and anomalous diffusion. Cambridge: Cambridge University Press, 2018. 345 p.
4. Samko S.G., Kilbas A.A., Marichev O.I. Fractional integrals and derivatives: theory and applications. New York: Gordon and Breach, 1993. 1016 p.
5. Vasilyev V.V., Simak L.O. Drobnoe ischislenie i approksimatsionnye metody v modeli-rovanii dinamicheskikh sistem. Kiev: NAN Ukraine, 2008, 256 p. (In Russ.).
6. Scherera R., Kallab S.L., Tangc Y., Huang J. The Grünwald – Letnikov method for frac-tional differential equations. Computers & Mathematics with Applications. 2011. Vol. 62. pp. 902–917. DOI: 10.1016/j.camwa.2011.03.054
7. Tadjeran С., Meerschaert M.M. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. Journal of Computational Physics. 2007. Vol. 220. pp. 813–823. DOI: 10.1016/j.jcp.2006.05.030
8. Meerschaert M.M., Tadjeran С. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 2004. Vol. 172, no. 1, pp. 65–77. DOI: 10.1016/j.cam.2004.01.033
9. Cao J., Li C. Finite difference scheme for the time-space fractional diffusion equations. Open Physics. 2013. Vol. 11. pp. 1440–1456. DOI: 10.2478/s11534-013-0261-x
10. Zhang F., Gao X., Xie Z. Difference numerical solutions for time-space fractional advection diffusion equation. Bound Value Probl. 2019. Vol. 14. pp. 1–11. DOI:10.1186/s13661-019-1120-5
11. Moroz L.I., Maslovskaya A.G. CHislennoe modelirovanie processa anomal'noj diffuzii na osnove skhemy povyshennogo poryadka tochnosti. Matematicheskoe modelirovanie,2020. Vol. 32, no. 10, pp. 62–76. DOI: 10.20948/mm-2020-10-05. (In Russ.).
12. Rau E.I., Evstafyeva E.N., Andrianov M.V. Mekhanizmy zaryadki dielektrikov pri ikh obluchenii elektronnymi puchkami srednikh energiy. Fizika tverdogo tela. 2007. Vol. 50. no. 4. pp. 599–607. (In Russ.).
13. Chezganov D.S., Kuznetsov D.K., Shur V.Ya. Simulation of spatial distribution of electric field after electron beam irradiation of MgO-doped LiNbO3 covered by resist layer. Ferroelectrics, 2016. Vol. 496, pp.70–78. DOI: 10.1080/00150193.2016.1157436
14. Maslovskaya A.G., Pavelchuk A.V. Simulation of delay reaction-drift-diffusion system applied to charging effects in electron-irradiated dielectrics. Proc. of IOP Conf. Series: Journal of Physics: Conf. Series, 2019, pp. 012009 (6). DOI: 10.1088/1742-6596/1163/1/012009
15. Moroz L.I., Maslovskaya A.G. Hybrid stochastic fractal-based approach to modeling the switching kinetics of ferroelectrics in the injection mode. Mathematical Models and Computer Simulations, 2020. Vol. 12, pp.348–356. DOI:10.1134/S0234087919090077
16. Mejlanov R.P., Sadykov S.A. Fraktal'naya model' kinetiki pereklyucheniya polyarizacii v segnetoelektrikah. Zhurnal tekhnicheskoj fiziki,1999. Vol. 69, pp. 128–129. (In Russ.).
17. Galiyarova N.M. Fractal dielectric response of multidomain ferroelectrics from the irreversible thermodynamics standpoint. Ferroelectrics, 1999. Vol. 222, pp. 381–387. DOI: 10.1080/00150199908014841
18. Ducharne B., Sebald G., Guyomar D. Time fractional derivative for frequency effect in ferroelectrics. 18th IEEE International Symposium on the Applications of Ferroelectrics, 2009, pp.1–4. DOI: 10.1109/ISAF.2009.5307619
19. Asghari Y., Eslami M., Rezazadeh H. Soliton solutions for the time‑fractional nonlinear diferential‑diference equation with conformable derivatives in the ferroelectric materials. Optical and Quantum Electronics. 2023. Vol. 55. pp. 289–230. DOI: 10.1007/s11082-022-04497-8
20. Brizickij R.V., Maksimova N.N., Maslovskaya A.G. Teoreticheskij analiz i chislennaya realizaciya stacionarnoj diffuzionno-drejfovoj modeli zaryadki polyarnyh dielektrikov. Matematicheskaya fizika, 2022 Vol. 62, pp. 1696–1706. DOI:10.31857/S0044466922100039(In Russ.).
21. Samarskij A.A., Vabishchevich P.N. Chislennye metody resheniya zadach konvekciidiffuzii. M: Knizhnyj dom «LIBROKOM», 2015. 248 p. (In Russ.).