Parabolic partial differential equation. 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. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes equation.

Definition 
To define the simplest kind of parabolic PDE, consider a real-valued function  of two independent real variables,  and . 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  and . The PDE is classified as parabolic if the coefficients of the principal part (i.e. the terms containing the second derivatives of ) satisfy the condition

Usually  represents one-dimensional position and  represents time, and the PDE is solved subject to prescribed initial and boundary conditions. Equations with  are termed elliptic while those with  are hyperbolic. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation  to define a planar parabola.

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

where  is the temperature at position  along a thin rod at time  and  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  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.

Noting that  is an elliptic operator suggests a broader definition of a parabolic PDE:

where  is a second-order elliptic operator
(implying that  must be positive;
a case where  is considered below).

A system of partial differential equations for a vector  can also be parabolic.
For example, such a system is hidden in an equation of the form

if the matrix-valued function  has a kernel of dimension 1.

Solution 
Under broad assumptions, an initial/boundary-value problem for a linear parabolic PDE has a solution for all time. The solution , as a function of  for a fixed time , is generally smoother than the initial data .

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.

Backward parabolic equation 
One occasionally encounters a so-called backward parabolic PDE, which takes the form  (note the absence of a minus sign).

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.

See also 
 Elliptic partial differential equation
 Hyperbolic partial differential equation

Notes

References

Further reading