IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i8p1838-d1122107.html
   My bibliography  Save this article

A Second-Order Accurate Numerical Approximation for a Two-Sided Space-Fractional Diffusion Equation

Author

Listed:
  • Taohua Liu

    (School of Science, Shaoyang University, Shaoyang 422000, China)

  • Xiucao Yin

    (School of Science, Shaoyang University, Shaoyang 422000, China)

  • Yinghao Chen

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China
    Eastern Institute for Advanced Study, Yongriver Institute of Technology, Ningbo 315201, China)

  • Muzhou Hou

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China)

Abstract

In this paper, we investigate a practical numerical method for solving a one-dimensional two-sided space-fractional diffusion equation with variable coefficients in a finite domain, which is based on the classical Crank-Nicolson (CN) method combined with Richardson extrapolation. Second-order exact numerical estimates in time and space are obtained. The unconditional stability and convergence of the method are tested. Two numerical examples are also presented and compared with the exact solution.

Suggested Citation

  • Taohua Liu & Xiucao Yin & Yinghao Chen & Muzhou Hou, 2023. "A Second-Order Accurate Numerical Approximation for a Two-Sided Space-Fractional Diffusion Equation," Mathematics, MDPI, vol. 11(8), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1838-:d:1122107
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/8/1838/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/8/1838/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Li, Shengyue & Zhou, Zhaojie, 2019. "Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 331-347.
    2. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    3. Feng, L.B. & Zhuang, P. & Liu, F. & Turner, I., 2015. "Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 52-65.
    4. Ebru Ozbilge & Fatma Kanca & Emre Özbilge, 2022. "Inverse Problem for a Time Fractional Parabolic Equation with Nonlocal Boundary Conditions," Mathematics, MDPI, vol. 10(9), pages 1-8, April.
    5. James W. Kirchner & Xiahong Feng & Colin Neal, 2000. "Fractal stream chemistry and its implications for contaminant transport in catchments," Nature, Nature, vol. 403(6769), pages 524-527, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhang, Jingyuan, 2018. "A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 209-227.
    2. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    3. Qu, Wei & Li, Zhi, 2021. "Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    4. Treena Basu, 2015. "A Fast O ( N log N ) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation," Mathematics, MDPI, vol. 3(4), pages 1-13, October.
    5. Yin, Baoli & Liu, Yang & Li, Hong, 2020. "A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations," Applied Mathematics and Computation, Elsevier, vol. 368(C).
    6. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    7. Scalas, Enrico & Kaizoji, Taisei & Kirchler, Michael & Huber, Jürgen & Tedeschi, Alessandra, 2006. "Waiting times between orders and trades in double-auction markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 463-471.
    8. Wael W. Mohammed & Mohammed Alshammari & Clemente Cesarano & Sultan Albadrani & M. El-Morshedy, 2022. "Brownian Motion Effects on the Stabilization of Stochastic Solutions to Fractional Diffusion Equations with Polynomials," Mathematics, MDPI, vol. 10(9), pages 1-9, April.
    9. Asma Alharbi & Rafik Guefaifia & Salah Boulaaras, 2020. "A New Proof of the Existence of Nonzero Weak Solutions of Impulsive Fractional Boundary Value Problems," Mathematics, MDPI, vol. 8(5), pages 1-16, May.
    10. Schumer, Rina & Baeumer, Boris & Meerschaert, Mark M., 2011. "Extremal behavior of a coupled continuous time random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(3), pages 505-511.
    11. Langlands, T.A.M., 2006. "Solution of a modified fractional diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 136-144.
    12. Hamid, M. & Usman, M. & Haq, R.U. & Wang, W., 2020. "A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 551(C).
    13. Staccioli, Jacopo & Napoletano, Mauro, 2021. "An agent-based model of intra-day financial markets dynamics," Journal of Economic Behavior & Organization, Elsevier, vol. 182(C), pages 331-348.
    14. D’Amico, Guglielmo & Janssen, Jacques & Manca, Raimondo, 2009. "European and American options: The semi-Markov case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(15), pages 3181-3194.
    15. Przemyslaw Repetowicz & Peter Richmond, 2004. "Pricing of options on stocks driven by multi-dimensional operator stable Levy processes," Papers math-ph/0412071, arXiv.org, revised Feb 2005.
    16. Jiang, Zhi-Qiang & Chen, Wei & Zhou, Wei-Xing, 2009. "Detrended fluctuation analysis of intertrade durations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 433-440.
    17. Enrico Scalas & Rudolf Gorenflo & Hugh Luckock & Francesco Mainardi & Maurizio Mantelli & Marco Raberto, 2004. "Anomalous waiting times in high-frequency financial data," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 695-702.
    18. Plamen Ch Ivanov & Ainslie Yuen & Pandelis Perakakis, 2014. "Impact of Stock Market Structure on Intertrade Time and Price Dynamics," PLOS ONE, Public Library of Science, vol. 9(4), pages 1-14, April.
    19. Berardi, Luca & Serva, Maurizio, 2005. "Time and foreign exchange markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 353(C), pages 403-412.
    20. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1838-:d:1122107. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.