The similarity group and anomalous diffusion equations

A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diff usion on fractal objects. Although all are constructed to correctly reprodu ce the basic subdiffusive property of this phenomenon, using similarity met hods it becomes very clear that this is far from sufficient to confirm thei r validity. The similarity group that they all have in common is the natura l basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natu ral space in which to compare these solutions both with one another and wit h data from numerical experiments on fractals. It also reduces the differen tial equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equa tions cannot agree even qualitatively with either each other or the data, s uggesting that a new approach is needed.