Sie sind hier: ICP » R. Hilfer » Publikationen

# 2 Mathematical Model

[334.3.1] Let denote the -dimensional Laplace operator in cartesian coordinates. [334.3.2] Numerous authors postulate a fractional diffusion equation such as

 (1)

with and initial condition

 (2)

[page 335, §0]    for a function as a mathematical model for various physical phenomena (see [10, 15, 14, 28, 29] for examples). [335.0.1] For 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 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 the fractional Laplace operator in eq. (1) may be defined (in the sense of Riesz [23]) as

 (3)

where denotes the Fourier transform of . [335.1.2] A core domain suitable for various extensions are functions 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 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 of space. [335.2.3] Thus, practical applications require to consider nonlocal boundary value problems on bounded domains .

[335.3.1] Every experiment assumes that the experimental conditions in the region surrounding the region 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 for the unknown such that

 (4)

for all times . [335.3.3] The Riesz operator may then be be understood as a Dirichlet form on the space over the bounded set equipped with the canonical Borel -algebra and a -finite measure , [5].