The numerical methods for solving of the one-dimensional anomalous reaction-diffusion equation
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Silesian University of Technology, Department of Mathematics Applications and Methods for Artificial Intelligence, Poland
Submission date: 2023-12-27
Final revision date: 2024-02-12
Acceptance date: 2024-02-19
Online publication date: 2024-04-13
Publication date: 2024-04-30
Corresponding author
Marek Błasik   

Department of Mathematics Applications and Methods for Artificial Intelligence, Silesian University of Technology, Poland
Journal of Theoretical and Applied Mechanics 2024;62(2):365-376
This paper presents numerical methods for solving the one-dimensional fractional reaction- -diffusion equation with the fractional Caputo derivative. The proposed methods are based on transformation of the fractional differential equation to an equivalent form of a integro- -differential equation. The paper proposes an improvement of the existing implicit method, and a new explicit method. Stability and convergence tests of the methods were also con- ducted.
Błasik M., 2021, The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term, Bulletin of the Polish Academy of Sciences. Technical Sciences, 69, e138240.
Coronel-Escamilla A., Gómez-Aguilar J.F.,Torres L., Escobar-Jimenéz R.F., 2018, A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A, 491, 406-424.
Diethelm K., 2010, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin.
Gu X.M., Sun H.W., Zhao Y.L., Zheng X., 2021, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Applied Mathematics Letters, 120, 107270.
Haq S.., Ali I., Sooppy Nisar K., 2021, A computational study of two-dimensional reaction-diffusion Brusselator system with applications in chemical processes, Alexandria Engineering Journal, 60, 4381-4392.
Humphries N.E., Queiroz N., Dyer J.R.M., Pade N.G., Musyl M.K. et al., 2010, Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature, 465, 1066-1069.
Kilbas A.A., Srivastava H.M., Trujillo J.J., 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Kosztołowicz T., Dworecki K., Mrówczyński S., 2005a, How to measure subdiffusion parameters, Physical Review Letters, 94, 170602.
Kosztołowicz T., Dworecki K., Mrówczyński S., 2005b, Measuring subdiffusion parameters, Physical Review E, 71, 041105.
Liu Y., Du Y., Li H., Li J., He S., 2015, A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Computers and Mathematics with Applications, 70, 2474-2492.
Metzler R., Klafter J., 2000, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1-77.
Metzler R., Klafter J., 2004, 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, 161-208.
Owolabi K.M., Atangana A., Akgul A., 2020, Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model, Alexandria Engineering Journal, 59, 2477-2490.
Podlubny I., 1999, Fractional Differential Equations, Academic Press, San Diego.
Pradip R., Prasad Goura V.M.K., 2023, An efficient numerical scheme and its stability analysis for a time-fractional reaction diffusion model, Journal of Computational and Applied Mathematics, 422, 114918.
Saad K.M., Gómez-Aguilar J.F., 2018, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A, 509, 703-716.
Sandip M., Srinivasan N., 2023, Analytical and numerical solutions of time-fractional advection-diffusion-reaction equation, Applied Numerical Mathematics, 185, 549-570.
Saxena R.K., Mathai A.M., Haubold H.J., 2015, Computational solutions of unified fractional reaction-diffusion equations with composite fractional time derivative, Communications in Nonlinear Science and Numerical Simulation, 27, 1-11.
Solomon T.H., Weeks E.R., Swinney H.L., 1993, Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow, Physical Review Letters, 71, 3975-3979.
Weeks E.R., Urbach J.S., Swinney L., 1996, Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example, Physica D: Nonlinear Phenomena, 97, 291-310.
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