Nonlocal convection-diffusion volume-constrained problems and jump processes

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March 2014, 19(2): 373-389. doi: 10.3934/dcdsb.2014.19.373

Nonlocal convection-diffusion volume-constrained problems and jump processes

Qiang Du ^1, , Zhan Huang ^1, and Richard B. Lehoucq ^2,

1.	Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States
2.	Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, United States

Received June 2013 Revised November 2013 Published February 2014

Abstract
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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.

Keywords: master equation, Nonlocal, convection-diffusion, Monte Carlo., volume-constrained problem, maximum principle.

Mathematics Subject Classification: Primary: 34B10; Secondary: 76R99, 45K05, 45A0.

Citation: Qiang Du, Zhan Huang, Richard B. Lehoucq. Nonlocal convection-diffusion volume-constrained problems and jump processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 373-389. doi: 10.3934/dcdsb.2014.19.373

References:

[1]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010).
[2]	K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes,, Probability Theory and Related Fields, 127* (2003), 89. 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. doi: 10.1615/IntJMultCompEng.2011002402.
[4]	________, Continuous-time random walks on bounded domains,, Physical Review E, 83 (2011).
[5]	N. Burch and R. B. Lehoucq, Computing the Exit-Time for a Symmetric Finite-Range Jump Process,, Technical report SAND 2013-2354J, (2013), 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. 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. 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. doi: 10.1137/110833233.
[9]	L. Ignat and J. Rossi, A nonlocal convection-diffusion equation,, Journal of Functional Analysis, 251* (2007), 399. 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, 18 (2013), 1415. doi: 10.3934/dcdsb.2013.18.1415.

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References:

[1]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010).
[2]	K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes,, Probability Theory and Related Fields, 127* (2003), 89. 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. doi: 10.1615/IntJMultCompEng.2011002402.
[4]	________, Continuous-time random walks on bounded domains,, Physical Review E, 83 (2011).
[5]	N. Burch and R. B. Lehoucq, Computing the Exit-Time for a Symmetric Finite-Range Jump Process,, Technical report SAND 2013-2354J, (2013), 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. 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. 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. doi: 10.1137/110833233.
[9]	L. Ignat and J. Rossi, A nonlocal convection-diffusion equation,, Journal of Functional Analysis, 251* (2007), 399. 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, 18 (2013), 1415. doi: 10.3934/dcdsb.2013.18.1415.

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