IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v52y2013icp30-35.html
   My bibliography  Save this article

Fokker–Planck equation on fractal curves

Author

Listed:
  • Satin, Seema E.
  • Parvate, Abhay
  • Gangal, A.D.

Abstract

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.

Suggested Citation

  • Satin, Seema E. & Parvate, Abhay & Gangal, A.D., 2013. "Fokker–Planck equation on fractal curves," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 30-35.
    • Handle: RePEc:eee:chsofr:v:52:y:2013:i:c:p:30-35
    DOI: 10.1016/j.chaos.2013.03.013
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077913000659
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2013.03.013?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. Metzler, Ralf & Barkai, Eli & Klafter, Joseph, 1999. "Anomalous transport in disordered systems under the influence of external fields," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 266(1), pages 343-350.
    2. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
    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. Golmankhaneh, Alireza K. & Tunc, Cemil, 2017. "On the Lipschitz condition in the fractal calculus," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 140-147.
    2. Satin, Seema & Gangal, A.D., 2019. "Random walk and broad distributions on fractal curves," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 17-23.
    3. Khalili Golmankhaneh, Alireza & Ontiveros, Lilián Aurora Ochoa, 2023. "Fractal calculus approach to diffusion on fractal combs," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

    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. Viacheslav V. Saenko & Vladislav N. Kovalnogov & Ruslan V. Fedorov & Dmitry A. Generalov & Ekaterina V. Tsvetova, 2022. "Numerical Method for Solving of the Anomalous Diffusion Equation Based on a Local Estimate of the Monte Carlo Method," Mathematics, MDPI, vol. 10(3), pages 1-19, February.
    2. Fernando Alcántara-López & Carlos Fuentes & Rodolfo G. Camacho-Velázquez & Fernando Brambila-Paz & Carlos Chávez, 2022. "Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs," Energies, MDPI, vol. 15(13), pages 1-11, July.
    3. Caicedo, Alejandro & Cuevas, Claudio & Mateus, Éder & Viana, Arlúcio, 2021. "Global solutions for a strongly coupled fractional reaction-diffusion system in Marcinkiewicz spaces," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    4. Razminia, Kambiz & Razminia, Abolhassan & Baleanu, Dumitru, 2019. "Fractal-fractional modelling of partially penetrating wells," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 135-142.
    5. Saenko, Viacheslav V., 2016. "The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 765-782.
    6. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    7. Pavlos, G.P. & Karakatsanis, L.P. & Iliopoulos, A.C. & Pavlos, E.G. & Xenakis, M.N. & Clark, Peter & Duke, Jamie & Monos, D.S., 2015. "Measuring complexity, nonextensivity and chaos in the DNA sequence of the Major Histocompatibility Complex," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 188-209.
    8. Duan, Jun-Sheng & Wang, Zhong & Liu, Yu-Lu & Qiu, Xiang, 2013. "Eigenvalue problems for fractional ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 46(C), pages 46-53.
    9. Lin, Guoxing, 2017. "Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 277-288.
    10. Qureshi, Sania & Bonyah, Ebenezer & Shaikh, Asif Ali, 2019. "Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    11. Lenzi, E.K. & Mendes, R.S. & Gonçalves, G. & Lenzi, M.K. & da Silva, L.R., 2006. "Fractional diffusion equation and Green function approach: Exact solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 360(2), pages 215-226.
    12. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    13. Razminia, Kambiz & Razminia, Abolhassan & Torres, Delfim F.M., 2015. "Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 374-380.
    14. Roscani, Sabrina D. & Bollati, Julieta & Tarzia, Domingo A., 2018. "A new mathematical formulation for a phase change problem with a memory flux," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 340-347.
    15. Halley Gomes & Arlúcio Viana, 2021. "Existence, symmetries, and asymptotic properties of global solutions for a fractional diffusion equation with gradient nonlinearity," Partial Differential Equations and Applications, Springer, vol. 2(1), pages 1-30, February.
    16. Claudia A. Pérez-Pinacho & Cristina Verde, 2022. "A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model," Mathematics, MDPI, vol. 10(5), pages 1-13, February.
    17. Vyacheslav Svetukhin, 2021. "Nucleation Controlled by Non-Fickian Fractional Diffusion," Mathematics, MDPI, vol. 9(7), pages 1-11, March.
    18. Dmitry Zhukov & Konstantin Otradnov & Vladimir Kalinin, 2024. "Fractional-Differential Models of the Time Series Evolution of Socio-Dynamic Processes with Possible Self-Organization and Memory," Mathematics, MDPI, vol. 12(3), pages 1-19, February.
    19. Essex, Christopher & Schulzky, Christian & Franz, Astrid & Hoffmann, Karl Heinz, 2000. "Tsallis and Rényi entropies in fractional diffusion and entropy production," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 299-308.
    20. Afzaal Mubashir Hayat & Muhammad Bilal Riaz & Muhammad Abbas & Moataz Alosaimi & Adil Jhangeer & Tahir Nazir, 2024. "Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function," Mathematics, MDPI, vol. 12(13), pages 1-23, July.

    More about this item

    Statistics

    Access and download statistics

    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:chsofr:v:52:y:2013:i:c:p:30-35. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.