Thus by letting we obtain the anisotropic diffusion equations.

Compared to the simple diffusion equation, the term proportional to the velocity field represents the effect of advection.

Green's functions are also a useful tool in solving wave equations and diffusion equations.

By taking the divergence, we obtain the new diffusion equation:

In a similar manner, the new diffusion equation:

Since the magnetization is a vector, there are 3 diffusion equations, one for each dimension.

It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion.

Earlier, such terms were introduced in the Maxwell-Stefan diffusion equation.

The heat equation and similar diffusion equations are useful tools to study random phenomena.

The governing diffusion equation can be solved by a 3D random walk algorithm.