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2 Mathematical Model

[334.3.1] Let \Delta=\sum _{{i=1}}^{d}\partial^{2}/\partial x_{i}^{2} denote the d-dimensional Laplace operator in cartesian coordinates. [334.3.2] Numerous authors postulate a fractional diffusion equation such as

\frac{\partial u}{\partial t}=-(-\Delta)^{{\alpha/2}}u(x,t),\qquad x\in\mathbb{B}\subseteq\mathbb{R}^{d},\ \ t\geq 0, (1)

with 0<\alpha\leq 2 and initial condition

u(x,0)=h(x),\qquad x\in\mathbb{B}\subseteq\mathbb{R}^{d} (2)

[page 335, §0]    for a function u:\mathbb{R}^{d}\to\mathbb{R} as a mathematical model for various physical phenomena (see [10, 15, 14, 28, 29] for examples). [335.0.1] For \alpha=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<\alpha<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 \mathbb{B}=\mathbb{R}^{d} the fractional Laplace operator (-\Delta)^{{\alpha/2}} in eq. (1) may be defined (in the sense of Riesz [23]) as

{\mathcal{F}}\left\{(-\Delta)^{{\alpha/2}}f(x)\right\}(k)=|k|^{\alpha}{\mathcal{F}}\left\{ f(x)\right\}(k), (3)

where {\mathcal{F}}\left\{ f(x)\right\}(k) denotes the Fourier transform of f(x). [335.1.2] A core domain suitable for various extensions are functions f\in\mathcal{S}{(\mathbb{R}^{d})} 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 \mathbb{B}=\mathbb{R}^{d} 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 \mathbb{B}\subset\mathbb{R}^{d} of space. [335.2.3] Thus, practical applications require to consider nonlocal boundary value problems on bounded domains \mathbb{B}\subset\mathbb{R}^{d}.

[335.3.1] Every experiment assumes that the experimental conditions in the region \mathbb{R}^{d}\setminus\mathbb{B} surrounding the region \mathbb{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:\mathbb{R}^{d}\setminus\mathbb{B}\to\mathbb{R} for the unknown u(x,t) such that

u(x,t)=g(x),\qquad x\in\mathbb{R}^{d}\setminus\mathbb{B} (4)

for all times t\geq 0. [335.3.3] The Riesz operator (-\Delta)^{{\alpha/2}} may then be be understood as a Dirichlet form on the space L^{2}(\mathbb{B},\mu) over the bounded set \mathbb{B} equipped with the canonical Borel \sigma-algebra and a \sigma-finite measure \mu, [5].