IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v342y2019icp224-246.html
   My bibliography  Save this article

Radial basis functions method for solving the fractional diffusion equations

Author

Listed:
  • Saberi Zafarghandi, Fahimeh
  • Mohammadi, Maryam
  • Babolian, Esmail
  • Javadi, Shahnam

Abstract

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. The paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions (RBFs) for the numerical solution of a class of initial-boundary value fractional diffusion equations with variable coefficients on a finite domain. The space fractional derivatives are defined by using Riemann–Liouville fractional derivative. We first provide Riemann–Liouville fractional derivatives for the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. The time-dependent fractional diffusion equation is discretized in space with the RBF collocation method and the remaining system of ordinary differential equations (ODEs) is advanced in time with an ODE method using a method of lines approach. Some numerical results are given in order to demonstrate the efficiency and accuracy of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method. The stability of the linear systems arising from discretizing Riemann–Liouville fractional differential operator with RBFs is also analysed.

Suggested Citation

  • Saberi Zafarghandi, Fahimeh & Mohammadi, Maryam & Babolian, Esmail & Javadi, Shahnam, 2019. "Radial basis functions method for solving the fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 224-246.
    • Handle: RePEc:eee:apmaco:v:342:y:2019:i:c:p:224-246
    DOI: 10.1016/j.amc.2018.08.043
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300318307574
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2018.08.043?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    2. Liu, Q. & Liu, F. & Gu, Y.T. & Zhuang, P. & Chen, J. & Turner, I., 2015. "A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 930-938.
    3. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    4. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    5. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    6. Li, Wei & Li, Can, 2015. "Second-order explicit difference schemes for the space fractional advection diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 446-457.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Abdelouahed Kouibia & Pedro González & Miguel Pasadas & Bassim Mustafa & Hossain Oulad Yakhlef & Loubna Omri, 2024. "Approximation of Bivariate Functions by Generalized Wendland Radial Basis Functions," Mathematics, MDPI, vol. 12(16), pages 1-10, August.
    2. H. Ghafouri & M. Ranjbar & A. Khani, 2020. "The Use of Partial Fractional Form of A-Stable Padé Schemes for the Solution of Fractional Diffusion Equation with Application in Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 695-709, December.

    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. G. Fern'andez-Anaya & L. A. Quezada-T'ellez & B. Nu~nez-Zavala & D. Brun-Battistini, 2019. "Katugampola Generalized Conformal Derivative Approach to Inada Conditions and Solow-Swan Economic Growth Model," Papers 1907.00130, arXiv.org.
    2. Ali Balcı, Mehmet, 2017. "Time fractional capital-induced labor migration model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 91-98.
    3. Valentina V. Tarasova & Vasily E. Tarasov, 2016. "Fractional Dynamics of Natural Growth and Memory Effect in Economics," Papers 1612.09060, arXiv.org, revised Jan 2017.
    4. Álvaro Cartea, 2013. "Derivatives pricing with marked point processes using tick-by-tick data," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 111-123, January.
    5. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Logistic map with memory from economic model," Papers 1712.09092, arXiv.org.
    6. Tarasova, Valentina V. & Tarasov, Vasily E., 2017. "Logistic map with memory from economic model," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 84-91.
    7. Tarasov, Vasily E. & Tarasova, Valentina V., 2018. "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486.
    8. Vasily E. Tarasov & Valentina V. Tarasova, 2019. "Dynamic Keynesian Model of Economic Growth with Memory and Lag," Mathematics, MDPI, vol. 7(2), pages 1-17, February.
    9. Torricelli, Lorenzo, 2020. "Trade duration risk in subdiffusive financial models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    10. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Dynamic intersectoral models with power-law memory," Papers 1712.09087, arXiv.org.
    11. Vasily E. Tarasov, 2019. "On History of Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 7(6), pages 1-28, June.
    12. Jajarmi, Amin & Hajipour, Mojtaba & Baleanu, Dumitru, 2017. "New aspects of the adaptive synchronization and hyperchaos suppression of a financial model," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 285-296.
    13. Tomas Skovranek, 2019. "The Mittag-Leffler Fitting of the Phillips Curve," Mathematics, MDPI, vol. 7(7), pages 1-11, July.
    14. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Concept of dynamic memory in economics," Papers 1712.09088, arXiv.org.
    15. Jean-Philippe Aguilar & Jan Korbel & Yuri Luchko, 2019. "Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations," Mathematics, MDPI, vol. 7(9), pages 1-23, September.
    16. Valentina V. Tarasova & Vasily E. Tarasov, 2016. "Economic Accelerator with Memory: Discrete Time Approach," Papers 1612.07913, arXiv.org, revised Jul 2017.
    17. Marseguerra, Marzio & Zoia, Andrea, 2008. "Pre-asymptotic corrections to fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 2668-2674.
    18. Zheng, Guang-Hui & Zhang, Quan-Guo, 2018. "Solving the backward problem for space-fractional diffusion equation by a fractional Tikhonov regularization method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 148(C), pages 37-47.
    19. 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.
    20. Fan Yang & Ping Fan & Xiao-Xiao Li & Xin-Yi Ma, 2019. "Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source," Mathematics, MDPI, vol. 7(9), pages 1-13, September.

    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:eee:apmaco:v:342:y:2019:i:c:p:224-246. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.