Parabolic partial differential equation

Summary

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 and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and Black–Scholes equation.

Definition

edit

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[1]

 

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.[2]

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

edit

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.[citation needed]

Backward parabolic equation

edit

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.[3]

See also

edit

Notes

edit
  1. ^ Zauderer 2006, p. 124.
  2. ^ Zauderer 2006, p. 139.
  3. ^ Taylor 1975.

References

edit
  • Taylor, Michael E. (1975). "Reflection of singularities of solutions to systems of differential equations". Communications on Pure and Applied Mathematics. 28 (4): 457–478. CiteSeerX 10.1.1.697.9255. doi:10.1002/cpa.3160280403. ISSN 0010-3640.
  • Zauderer, Erich (2006). Partial Differential Equations of Applied Mathematics. Hoboken, N.J: Wiley-Interscience. ISBN 978-0-471-69073-3. OCLC 70158521.

Further reading

edit