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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, including heat conduction, particle diffusion, and pricing of derivative investment instruments.

Contents

DefinitionEdit

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

 

and this PDE is classified as being parabolic if the coefficients satisfy the condition

 

Usually   represents one-dimensional position and   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   to define a planar parabola.

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

 

where   is the temperature at time   and at position   along a thin rod, and   is a positive constant (the thermal diffusivity). The symbol   signifies the partial derivative of   with respect to the time variable  , and similarly   is the second partial derivative with respect to  . For this example,   plays the role of   in the general second-order linear PDE:  ,  , and the other coefficients are zero.

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.

Parabolic PDEs can also be nonlinear. For example, Fisher's equation is a nonlinear PDE that includes the same diffusion term as the heat equation but incorporates a linear growth term and a nonlinear decay term.

SolutionEdit

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 equationEdit

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,

 

An initial/boundary-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. [1] Moreover, they arise in the pricing problem for certain financial instruments.


ExamplesEdit

See alsoEdit

ReferencesEdit

  1. ^ Taylor, M. E. (1975), "Reflection of singularities of solutions to systems of differential equations", Comm. Pure Appl. Math., 28 (4): 457–478, CiteSeerX 10.1.1.697.9255, doi:10.1002/cpa.3160280403