IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v102y2017icp175-183.html
   My bibliography  Save this article

A time-fractional generalised advection equation from a stochastic process

Author

Listed:
  • Angstmann, C.N.
  • Henry, B.I.
  • Jacobs, B.A.
  • McGann, A.V.

Abstract

A generalised advection equation with a time fractional derivative is derived from a continuous time random walk on a one-dimensional lattice, with power law distributed waiting times. We consider walks governed by a two-sided jump density and walks governed by a one-sided jump density. With the two-sided density, the particle can jump in both directions on the lattice, whereas with the one-sided density the particle cannot jump in one of these directions. The master equations describing the evolution of the probability density for the position of the particle are different for each of the jump densities. However in an advective limit both master equations limit to a common generalized advection equation with time fractional derivatives.

Suggested Citation

  • Angstmann, C.N. & Henry, B.I. & Jacobs, B.A. & McGann, A.V., 2017. "A time-fractional generalised advection equation from a stochastic process," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 175-183.
    • Handle: RePEc:eee:chsofr:v:102:y:2017:i:c:p:175-183
    DOI: 10.1016/j.chaos.2017.04.040
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077917301789
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2017.04.040?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. Michelitsch, T.M. & Collet, B.A. & Riascos, A.P. & Nowakowski, A.F. & Nicolleau, F.C.G.A., 2016. "A fractional generalization of the classical lattice dynamics approach," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 43-50.
    2. Alvaro Cartea & Diego del-Castillo-Negrete, 2007. "On the Fluid Limit of the Continuous-Time Random Walk with General Lévy Jump Distribution Functions," Birkbeck Working Papers in Economics and Finance 0708, Birkbeck, Department of Economics, Mathematics & Statistics.
    3. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
    4. Angstmann, C.N. & Henry, B.I. & McGann, A.V., 2016. "A fractional-order infectivity SIR model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 86-93.
    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. Michelitsch, Thomas M. & Polito, Federico & Riascos, Alejandro P., 2021. "On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    2. Darvishi, M.T. & Najafi, Mohammad & Wazwaz, Abdul-Majid, 2021. "Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    3. Chaudhary, Manish & Kumar, Rohit & Singh, Mritunjay Kumar, 2020. "Fractional convection-dispersion equation with conformable derivative approach," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).

    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. Edgardo Alvarez & Carlos Lizama, 2020. "The Super-Diffusive Singular Perturbation Problem," Mathematics, MDPI, vol. 8(3), pages 1-14, March.
    2. Feng, Libo & Liu, Fawang & Anh, Vo V., 2023. "Galerkin finite element method for a two-dimensional tempered time–space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 517-537.
    3. Soma Dhar & Lipi B. Mahanta & Kishore Kumar Das, 2019. "Formulation Of The Simple Markovian Model Using Fractional Calculus Approach And Its Application To Analysis Of Queue Behaviour Of Severe Patients," Statistics in Transition New Series, Polish Statistical Association, vol. 20(1), pages 117-129, March.
    4. Saif Eddin Jabari & Nikolaos M. Freris & Deepthi Mary Dilip, 2020. "Sparse Travel Time Estimation from Streaming Data," Transportation Science, INFORMS, vol. 54(1), pages 1-20, January.
    5. Katarzyna Górska & Andrzej Horzela, 2021. "Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character," Mathematics, MDPI, vol. 9(5), pages 1-13, February.
    6. Byun, Jong Hyuk & Jung, Il Hyo, 2021. "Phase-specific cancer-immune model considering acquired resistance to therapeutic agents," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    7. Zaheer Masood & Muhammad Asif Zahoor Raja & Naveed Ishtiaq Chaudhary & Khalid Mehmood Cheema & Ahmad H. Milyani, 2021. "Fractional Dynamics of Stuxnet Virus Propagation in Industrial Control Systems," Mathematics, MDPI, vol. 9(17), pages 1-27, September.
    8. Goswami, Koushik, 2021. "Work fluctuations in a generalized Gaussian active bath," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    9. Virginia Kiryakova, 2021. "A Guide to Special Functions in Fractional Calculus," Mathematics, MDPI, vol. 9(1), pages 1-40, January.
    10. DAŞBAŞI, Bahatdin, 2020. "Stability analysis of the hiv model through incommensurate fractional-order nonlinear system," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    11. Murat A. Sultanov & Durdimurod K. Durdiev & Askar A. Rahmonov, 2021. "Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation," Mathematics, MDPI, vol. 9(17), pages 1-12, August.
    12. Vieira, N. & Ferreira, M. & Rodrigues, M.M., 2022. "Time-fractional telegraph equation with ψ-Hilfer derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    13. Iomin, A. & Zaburdaev, V. & Pfohl, T., 2016. "Reaction front propagation of actin polymerization in a comb-reaction system," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 115-122.
    14. Sánchez, Ewin, 2019. "Burr type-XII as a superstatistical stationary distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 516(C), pages 443-446.
    15. Serkan Araci & Gauhar Rahman & Abdul Ghaffar & Azeema & Kottakkaran Sooppy Nisar, 2019. "Fractional Calculus of Extended Mittag-Leffler Function and Its Applications to Statistical Distribution," Mathematics, MDPI, vol. 7(3), pages 1-14, March.
    16. Charles K. Amponsah & Tomasz J. Kozubowski & Anna K. Panorska, 2021. "A general stochastic model for bivariate episodes driven by a gamma sequence," Journal of Statistical Distributions and Applications, Springer, vol. 8(1), pages 1-31, December.
    17. Iomin, A., 2016. "Quantum continuous time random walk in nonlinear Schrödinger equation with disorder," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 64-70.
    18. Tarasov, Vasily E., 2023. "Nonlocal statistical mechanics: General fractional Liouville equations and their solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    19. Najma Ahmed & Nehad Ali Shah & Farman Ali & Dumitru Vieru & F.D. Zaman, 2021. "Analytical Solutions of the Fractional Mathematical Model for the Concentration of Tumor Cells for Constant Killing Rate," Mathematics, MDPI, vol. 9(10), pages 1-14, May.
    20. Hainaut, Donatien, 2021. "Lévy interest rate models with a long memory," LIDAM Discussion Papers ISBA 2021020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    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:102:y:2017:i:c:p:175-183. 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.