These methods often make use of variational principles.
However, the finite difference method can be more easliy applied to a lot of classical PDE. In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
One of the most usual problems concern the chemical and thermal diffusion problem, in either steady and unsteady conditions, for which same equations apply. The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time :
$$\frac{\partial T}{\partial t} = \kappa\left(\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2}+\frac{\partial^2T}{\partial z^2}\right)$$