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A time-fractional generalised advection equation from a stochastic process

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
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    Cited by:

    1. Chaudhary, Manish & Kumar, Rohit & Singh, Mritunjay Kumar, 2020. "Fractional convection-dispersion equation with conformable derivative approach," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    2. 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).
    3. 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).

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