Diffusion equation diffusivity diagonalization method

Time-Dependent Solutions. In the time-dependent case, the diffusion equations given by Eq. 6.23 are coupled. However, they can be uncoupled by again using the diagonalization method. 

The anisotropic form of Fick s law would seem to complicate the diffusion equation greatly. However, in many cases, a simple method for treating anisotropic diffusion allows the diffusion equation to keep its simple form corresponding to isotropic diffusion. Because Dtj is symmetric, it is always possible to find a linear coordinate transformation that will make the Dij diagonal with real components (the eigenvalues of D). Let elements of such a transformed system be identified by a hat. Then... 

Whereas in the case of the spatially one-dimensional diffusion, the set of difference equations features a tri-diagonal matrix, the system in Eq. (8-68) can be shown to have a block tri-diagonal matrix, which requires the use of special solving methods. 

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