Diffusion and the self-measurability
Applied and Computational Mechanics, Vol. 3, No. 1, pp. 51 - 62
M. Holeček (corresponding author:  holecek@kme.zcu.cz )
Abstract:

The familiar diffusion equation, ∂g/∂t = DΔg, is studied by using the patially averaged quantities. A non-local relation, so-called the self-measurability condition, fulfilled by this equation is obtained. We define a broad class of diffusion equations defined by some “diffusion inequality”, ∂g/∂t · Δg ≥ 0, and show that it is equivalent to the self-measurability condition. It allows formulating the diffusion inequality in a non-local form. That represents an essential generalization of the diffusion problem in the case when the field g(x, t) is not smooth. We derive a general differential equation for averaged quantities coming from the self-measurability condition.

Keywords:

diffusion, spatial averaging, nonlocal thermomechanics

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