Title

The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term

Journal title

Bulletin of the Polish Academy of Sciences Technical Sciences

Yearbook

2021

Volume

69

Issue

6

Affiliation

Błasik, Marek : Institute of Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21, 42-201 Czestochowa, Poland

Authors

Błasik, Marek

Keywords

fractional derivatives and integrals ; integro-differential equations ; numerical methods ; finite difference methods

Divisions of PAS

Nauki Techniczne

Coverage

e138240

Bibliography

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2.  T. Kosztołowicz, K. Dworecki, and S. Mrówczyński, “Measuring subdiffusion parameters,” Phys. Rev. E, vol. 71, p.  041105, 2005.
3.  E. Weeks, J. Urbach, and L. Swinney, “Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example,” Physica D, vol. 97, pp. 291–310, 1996.
4.  T. Solomon, E. Weeks, and H. Swinney, “Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow,” Phys. Rev. Lett., vol. 71, pp. 3975–3979, 1993.
5.  N.E. Humphries, et al., “Environmental context explains Lévy and Brownian movement patterns of marine predators,” Nature, vol. 465, pp. 1066–1069, 2010.
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7.  R. Metzler and J. Klafter, “The random walk:s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep., vol. 339, pp. 1–77, 2000.
8.  R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” J. Phys. A: Math. Gen., vol. 37, pp. 161–208, 2004.
9.  M. Aslefallah, S. Abbasbandy, and E. Shivanian, “Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method,” Eng. Anal. Boundary Elem., vol. 107, pp. 198–207, 2019.
10.  Y. Li and D. Wang, “Improved efficient difference method for the modified anomalous sub-diffusion equation with a nonlinear source term,” Int. J. Comput. Math., vol. 94, pp. 821–840, 2017.
11.  X. Cao, X. Cao, and L. Wen, “The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term,” J. Comput. Appl. Math., vol. 318, pp. 199–210, 2017.
12.  A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006.
13.  E.D. Rainville, Special Functions. New York: The Macmillan Company, 1960.
14.  J.-L. Liu and H.MSrivastava, “Classes of meromorphically multivalent functions associated with the generalized hypergeometric function,” Math. Comput. Modell., vol. 39, pp. 21–34, 2004.
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16.  M. Włodarczyk and A. Zawadzki, “The application of hypergeometric functions to computing fractional order derivatives of sinusoidal functions,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 64, pp. 243–248, 2016.
17.  M. Błasik, “A generalized Crank-Nicolson method for the solution of the subdiffusion equation,” 23rd International Conference on Methods & Models in Automation & Robotics (MMAR), pp.  726–729, 2018.
18.  M. Błasik, “Zagadnienie stefana niecałkowitego rzędu,” Ph.D. dissertation, Politechnika Częstochowska, 2013.
19.  M. Błasik and M. Klimek, “Numerical solution of the one phase 1d fractional stefan problem using the front fixing method,” Math. Methods Appl. Sci., vol. 38, no. 15, pp. 3214–3228, 2015.
20.  K. Diethelm, The Analysis of Fractional Differential Equations. Berlin: Springer-Verlag, 2010.

Date

18.08.2021

Type

Article

Identifier

DOI: 10.24425/bpasts.2021.138240