How to search?
advanced search

Search results

Filters

  • Journals
  • Authors
  • Keywords
  • Date
  • Type

Search results

Number of results: 1
items per page: 25 50 75
Sort by:

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

Marek Błasik
Bulletin of the Polish Academy of Sciences Technical Sciences | 2021 | 69 | 6 | e138240 | DOI: 10.24425/bpasts.2021.138240
Keywords fractional derivatives and integrals integro-differential equations numerical methods finite difference methods
Download PDF Download RIS Download Bibtex

Abstract

In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.
Go to article

Bibliography

  1.  T. Kosztołowicz, K. Dworecki, and S. Mrówczyński, “How to measure subdiffusion parameters,” Phys. Rev. Lett., vol. 94, p.  170602, 2005, doi: 10.1016/j.tins.2004.10.007.
  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.
  6.  U. Siedlecka, “Heat conduction in a finite medium using the fractional single-phase-lag model,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 67, pp. 402–407, 2019.
  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.
  15.  Y.L. Luke, “Inequalities for generalized hypergeometric functions,” J. Approximation Theory, vol. 5, pp. 41–65, 1972.
  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.
Go to article

Authors and Affiliations

Marek Błasik
1
  1. Institute of Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21, 42-201 Czestochowa, Poland

This page uses 'cookies'. Learn more