A partial differential equation describing the variation in space and time of a physical quantity that is governed by diffusion. The diffusion equation provides a good mathematical model for the variation of temperature through conduction of heat and the propagation of electromagnetic waves in a highly conducting medium. The diffusion equation is a parabolic partial differential equation whose characteristic form relates the first partial derivative of a field with respect to time to its second partial derivatives with respect to spatial coordinates. It is closely related to the wave equation.

∇^{2}E = j ω μ σ E,

where E = electrical field
ω = angular frequency
μ = magnetic permeability
σ = electrical conductivity
∇ = vector differential operator.

A fundamental differential equation obtained by combining the continuity equation, flow law and equation of state. Most of the mathematics of well testing were derived from solutions of this equation, which was originally developed for the study of heat transfer. Fluid flow through porous media is directly analogous to flow of heat through solids. Solutions used in well testing usually assume radial flow and homogenous, isotropic formations.