\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Nonlocal convection-diffusion volume-constrained problems and jump processes

Abstract Related Papers Cited by
  • We introduce the Cauchy and time-dependent volume-constrained problems associated with a linear nonlocal convection-diffusion equation. These problems are shown to be well-posed and correspond to conventional convection-diffusion equations as the region of nonlocality vanishes. The problems also share a number of features such as the maximum principle, conservation and dispersion relations, all of which are consistent with their corresponding local counterparts. Moreover, these problems are the master equations for a class of finite activity Lévy-type processes with nonsymmetric Lévy measure. Monte Carlo simulations and finite difference schemes are applied to these nonlocal problems, to show the effects of time, kernel, nonlocality and different volume-constraints.
    Mathematics Subject Classification: Primary: 34B10; Secondary: 76R99, 45K05, 45A05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010.

    [2]

    K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes, Probability Theory and Related Fields, 127 (2003), 89-152.doi: 10.1007/s00440-003-0275-1.

    [3]

    N. Burch and R. Lehoucq, Classical, nonlocal, and fractional diffusion equations on bounded domains, International Journal for Multiscale Computational Engineering, 9 (2011), 661-674.doi: 10.1615/IntJMultCompEng.2011002402.

    [4]

    ________, Continuous-time random walks on bounded domains, Physical Review E, 83 (2011), p. 012105.

    [5]

    N. Burch and R. B. Lehoucq, Computing the Exit-Time for a Symmetric Finite-Range Jump Process, Technical report SAND 2013-2354J, Sandia National Laboratories, 2013.

    [6]

    Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM review, 54 (2012), 667-696.doi: 10.1137/110833294.

    [7]

    Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Mathematical Models and Methods in Applied Sciences, 23 (2013), 493-540.doi: 10.1142/S0218202512500546.

    [8]

    Q. Du, J. Kamm, R. Lehoucq and M. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM Journal on Applied Mathematics, 72 (2012), 464-487.doi: 10.1137/110833233.

    [9]

    L. Ignat and J. Rossi, A nonlocal convection-diffusion equation, Journal of Functional Analysis, 251 (2007), 399-437.doi: 10.1016/j.jfa.2007.07.013.

    [10]

    T. Mengesha and Q. Du, Analysis of a scalar nonlocal peridynamic model with a sign changing kernel, Disc. Cont. Dyn. Sys, B, 18 (2013), 1415-1437.doi: 10.3934/dcdsb.2013.18.1415.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(312) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return