2 Mathematical Model

[334.3.1] Let Δ=∑i=1d∂2/∂⁡xi2 denote the d-dimensional Laplace operator in cartesian coordinates. [334.3.2] Numerous authors postulate a fractional diffusion equation such as

with 0<α≤2 and initial condition

[page 335, §0]    for a function u:Rd→R as a mathematical model for various physical phenomena (see [10, 15, 14, 28, 29] for examples). [335.0.1] For α=2 this becomes the Cauchy problem for the ordinary diffusion equation whose applicability as a mathematical model for physical phenomena has been validated with innumerable experiments. [335.0.2] For 0<α<2 however experimental evidence remains narrowly bounded in space and time scales. [335.0.3] Moreover, theoretical considerations cast fundamental doubts on the applicability of this case to natural phenomena.

[335.1.1] For B=Rd the fractional Laplace operator -Δα/2 in eq. (1) may be defined (in the sense of Riesz [23]) as

where F⁢f⁢x⁢k denotes the Fourier transform of f⁢x. [335.1.2] A core domain suitable for various extensions are functions f∈S⁢Rd from the Schwartz space of smooth functions decreasing rapidly at infinity.

[335.2.1] The implicit idealizing assumption underlying the choice of an unbounded domain B=Rd in eq. (1) is that the boundary is sufficiently far away so that its effects on the observations are negligible. [335.2.2] However, experiments are normally performed inside a bounded laboratory containing a bounded apparatus that occupies a bounded domain B⊂Rd of space. [335.2.3] Thus, practical applications require to consider nonlocal boundary value problems on bounded domains B⊂Rd.

[335.3.1] Every experiment assumes that the experimental conditions in the region Rd∖B surrounding the region B containing the sample can be controlled and reproduced to any desired degree of accuracy. [335.3.2] In the mathematical model this is represented by assuming given boundary data g:Rd∖B→R for the unknown u⁢x,t such that

for all times t≥0. [335.3.3] The Riesz operator -Δα/2 may then be be understood as a Dirichlet form on the space L2⁢B,μ over the bounded set B equipped with the canonical Borel σ-algebra and a σ-finite measure μ, [5].