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Fractional diffusion on bounded domains

Fractional Calculus and Applied Analysis

Defterli, Ozlem; D'Elia, Marta D.; Du, Qiang; Gunzburger, Max D.; Lehoucq, Richard B.; Meerschaert, Mark M.

The mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. This paper discusses the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.

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TYPE Journal Article YEAR 2015
ScopusOSTIDOI

Optimization-based coupling of nonlocal and local diffusion models

Materials Research Society Symposium Proceedings

D'Elia, Marta D.; Bochev, Pavel B.

In this work we introduce an optimization-based method for the coupling of nonlocal and local diffusion problems. Our approach is formulated as a control problem where the states are the solutions of the nonlocal and local equations, the controls are the nonlocal volume constraint and the local boundary condition, and the objective of the optimization is a matching functional for the state variables in the intersection of the nonlocal and local domains. For finite element discretizations we present numerical results in a one-dimensional setting; though preliminary, our tests show the consistency and efficacy of the method, and provide the basis for realistic simulations.

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TYPE Conference Poster YEAR 2015
ScopusOSTIDOI

The exit-time problem for a Markov jump process

European Physical Journal: Special Topics

Burch, N.; D'Elia, Marta D.; Lehoucq, Richard B.

The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. A critical aspect of the analysis is a variational formulation and a recently developed nonlocal vector calculus. This calculus allows us to pose nonlocal backward and forward Kolmogorov equations, the former equation granting the various moments of the exit-time distribution.

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TYPE Journal Article YEAR 2014
ScopusOSTIDOI

The exit-time problem for a Markov jump process

European Physical Journal. A

Burch, N.B.; D'Elia, Marta D.; Lehoucq, Richard B.

The purpose of our paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. A critical aspect of the analysis is a variational formulation and a recently developed nonlocal vector calculus. Furthermore, this calculus allows us to pose nonlocal backward and forward Kolmogorov equations, the former equation granting the various moments of the exit-time distribution.

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TYPE Journal Article YEAR 2014
OSTIDOI
Results 76–95 of 95
Results 76–95 of 95