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, i.a., engineering science, quantum mechanics and financial mathematics.
To define the simplest kind of parabolic PDE, consider a real-valued function
A second-order, linear, constant-coefficient PDE for
takes the form where the subscripts denote the first- and second-order partial derivatives with respect to
The PDE is classified as parabolic if the coefficients of the principal part (i.e. the terms containing the second derivatives of
represents time, and the PDE is solved subject to prescribed initial and boundary conditions.
The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation
The basic example of a parabolic PDE is the one-dimensional heat equation where
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.
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 where denotes the Laplace operator acting on
This equation is the prototype of a multi-dimensional parabolic PDE.
is an elliptic operator suggests a broader definition of a parabolic PDE: where
is a second-order elliptic operator (implying that
A system of partial differential equations for a vector
For example, such a system is hidden in an equation of the form if the matrix-valued function
Under broad assumptions, an initial/boundary-value problem for a linear parabolic PDE has a solution for all time.
For a nonlinear parabolic PDE, a solution of an initial/boundary-value problem might explode in a singularity within a finite amount of time.
It can be difficult to determine whether a solution exists for all time, or to understand the singularities that do arise.
Such interesting questions arise in the solution of the Poincaré conjecture via Ricci flow.
[citation needed] One occasionally encounters a so-called backward parabolic PDE, which takes the form
An initial-value problem for the backward heat equation, is equivalent to a final-value problem for the ordinary heat equation, Similarly to a final-value problem for a parabolic PDE, an initial-value problem for a backward parabolic PDE is usually not well-posed (solutions often grow unbounded in finite time, or even fail to exist).
Nonetheless, these problems are important for the study of the reflection of singularities of solutions to various other PDEs.