Nonlocal convection-diffusionvolume-constrained problems and jump processes

Qiang Du; Zhan Huang; Richard B. Lehoucq

doi:10.3934/dcdsb.2014.19.373

Article Contents

2014, Volume 19, Issue 2: 373-389. Doi: 10.3934/dcdsb.2014.19.373

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Nonlocal convection-diffusion volume-constrained problems and jump processes

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

Received: May 31, 2013

Revised: October 31, 2013

Published: February 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 34B10; Secondary: 76R99, 45K05, 45A05.

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