In some of the preceding chapters, we discussed the origin of partial differential equations (PDEs) by simplifying general PDE systems. For example, we arrived at the damped wave equation for an electric field as a special case of the Maxwell equations for electromagnetic (EM) fields. Also, we obtained Burgers' equation as a special case of the Euler and Navier Stokes equations of fluid mechanics.

While this approach of starting with a general PDE system is always an important first step to consider in developing a PDE model for a physical application, it also has limitations in the sense that the general system may not encompass all of the physical phenomena we wish to include in a mathematical model. For example, the Navier Stokes equations, as we used them, did not include an energy balance so that thermal effects, for example, a temperature field, could not be included in an analysis of a nonisothermal system. Of course, an energy balance could be (and has been) included with the Navier Stokes equations, but this requires some additional analysis. Also, we might be interested in a physical situation that is not reflected in a general PDE system, and we therefore have to derive the relevant PDEs starting from first principles, usually conservation principles in the case of physical applications.

In this appendix we consider the derivation of the equations for anisotropic diffusion, that is, a PDE system for which the diffusivity is a nine-component tensor. The final result is a generalization of the usual diffusion equation that includes directional effects that are significant in certain physical systems. The primary intent is to illustrate a method for the derivation of PDEs to include physical effects and phenomena that might not appear in previously available general PDE systems.