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An anomalous anisotropic diffusion equation is constructed in which the order of the spatial pseudo-differential operator is generalized to be distributed with a directionally dependent distribution. A time fractional version of this equation is also considered. First, it is proved that the equation is positivity-preserving and properly normalized. Second, the existence of a smooth Green's function solution is proved. Finally, an expression for the diffusive flux density for this new fractional order process is calculated. This approach may find utility in modelling diffusion tensor imaging data in the white matter of the human brain where both the apparent diffusion coefficient and the order of the pseudo-differential operator are anisotropic.

A new anisotropic fractional model of diffusion suitable for applications of diffusion tensor imaging in biological tissues