Parabolic partial differential equation

A parabolic partial differential equation is a type of second-order partial differential equation (PDE) of the form

that satisfies the condition

This definition is analogous to the definition of a planar parabola.

This form of partial differential equation is used to describe a wide family of problems in science including heat diffusion, ocean acoustic propagation, physical or mathematical systems with a time variable, and processes that behave essentially like heat diffusing through a solid.