4,211 research outputs found
A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws
A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The nonlocal calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations
The exit-time problem for a Markov jump process
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.Comment: 15 pages, 7 figure
Fractional Vector Calculus and Fractional Maxwell's Equations
The theory of derivatives and integrals of non-integer order goes back to
Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional
vector calculus (FVC) has only 10 years. The main approaches to formulate a
FVC, which are used in the physics during the past few years, will be briefly
described in this paper. We solve some problems of consistent formulations of
FVC by using a fractional generalization of the Fundamental Theorem of
Calculus. We define the differential and integral vector operations. The
fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of
these theorems are realized for simplest regions. A fractional generalization
of exterior differential calculus of differential forms is discussed.
Fractional nonlocal Maxwell's equations and the corresponding fractional wave
equations are considered.Comment: 42 pages, LaTe
- …