American Institute of Mathematical Sciences

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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

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, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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