A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes equation.

To define the simplest kind of parabolic PDE, consider a real-valued function ${\displaystyle u(x,y)}$ of two independent real variables, ${\displaystyle x}$ and ${\displaystyle y}$. A second-order, linear, constant-coefficient PDE for ${\displaystyle u}$ takes the form

where the subscripts denote the first- and second-order partial derivatives with respect to ${\displaystyle x}$ and ${\displaystyle y}$. The PDE is classified as parabolic if the coefficients of the principal part (i.e. the terms containing the second derivatives of ${\displaystyle u}$) satisfy the condition

Usually ${\displaystyle x}$ represents one-dimensional position and ${\displaystyle y}$ represents time, and the PDE is solved subject to prescribed initial and boundary conditions. Equations with ${\displaystyle B^{2}-AC<0}$ are termed elliptic while those with ${\displaystyle B^{2}-AC>0}$ are hyperbolic. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation ${\displaystyle Ax^{2}+2Bxy+Cy^{2}+Dx+Ey+F=0}$ to define a planar parabola.

The basic example of a parabolic PDE is the one-dimensional heat equation

where ${\displaystyle u(x,t)}$ is the temperature at position ${\displaystyle x}$ along a thin rod at time ${\displaystyle t}$ and ${\displaystyle \alpha }$ is a positive constant called the thermal diffusivity.

The heat equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity ${\displaystyle u_{xx}}$ measures how far off the temperature is from satisfying the mean value property of harmonic functions.

The concept of a parabolic PDE can be generalized in several ways. For instance, the flow of heat through a material body is governed by the three-dimensional heat equation