One powerful technique for deriving analytical solutions to stable diffusion equations is separation of variables. This approach assumes that the system's concentration or temperature can be expressed as a product of separate functions, each dependent on a single variable. By substituting this product form into the diffusion equation, we can solve for each individual function and combine them to obtain the overall solution.

For instance, consider the one-dimensional diffusion equation ∂c/∂t = D∂²c/∂x², where c is the concentration, t represents time, x is the spatial coordinate, and D is the diffusion coefficient. Assuming c(x, t) = X(x)T(t), we can separate the variables and derive two ordinary differential equations: X''(x)/X(x) = −λ and T'(t)/T(t) = λD, where λ is a separation constant. By solving these equations independently, we determine X(x) and T(t), and subsequently form the general solution by combining them.