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.

A simple example of a parabolic PDE is the one-dimensional heat equation,

where ${\displaystyle u(t,x)}$ is the temperature at time ${\displaystyle t}$ and at position ${\displaystyle x}$, and ${\displaystyle k}$ is a constant. The symbol ${\displaystyle u_{t}}$ signifies the partial derivative with respect to the time variable ${\displaystyle t}$, and similarly ${\displaystyle u_{xx}}$ is the second partial derivative with respect to ${\displaystyle x}$. For this example, ${\displaystyle t}$ replaced the role of ${\displaystyle y}$ in the equation determining the type.

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