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

Definition

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

Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0,

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

B^{2}-AC=0.

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

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

u_{t}=\alpha \,u_{xx},

where $u(x,t)$ is the temperature at time $t$ and at position $x$ along a thin rod, and $\alpha$ is a positive constant (the thermal diffusivity). The symbol $u_{t}$ signifies the partial derivative of $u$ with respect to the time variable $t$ , and similarly $u_{xx}$ is the second partial derivative with respect to $x$ . For this example, $t$ plays the role of $y$ in the general second-order linear PDE: $A=\alpha$ , $E=-1$ , 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 $u_{xx}$ 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,

u_{t}=\alpha \,\Delta u,

where

\Delta u:={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}

denotes the Laplace operator acting on $u$ . This equation is the prototype of a multi-dimensional parabolic PDE.^[1]

Noting that $-\Delta$ is an elliptic operator suggests a broader definition of a parabolic PDE:

u_{t}=-Lu,

where $L$ is a second-order elliptic operator (implying that $L$ must be positive; a case where $u_{t}=+Lu$ is considered below).

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

\nabla \cdot (a(x)\nabla u(x))+b(x)^{\text{T}}\nabla u(x)+cu(x)=f(x)

if the matrix-valued function $a(x)$ 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.

Solution

Under broad assumptions, an initial/boundary-value problem for a linear parabolic PDE has a solution for all time. The solution $u(x,t)$ , as a function of $x$ for a fixed time $t>0$ , is generally smoother than the initial data $u(x,0)=u_{0}(x)$ .

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

One occasionally encounters a so-called backward parabolic PDE, which takes the form $u_{t}=Lu$ (note the absence of a minus sign).

An initial-value problem for the backward heat equation,

{\begin{cases}u_{t}=-\Delta u&{\textrm {on}}\ \ \Omega \times (0,T),\\u=0&{\textrm {on}}\ \ \partial \Omega \times (0,T),\\u=f&{\textrm {on}}\ \ \Omega \times \left\{0\right\}.\end{cases}}

is equivalent to a final-value problem for the ordinary heat equation,

{\begin{cases}u_{t}=\Delta u&{\textrm {on}}\ \ \Omega \times (0,T),\\u=0&{\textrm {on}}\ \ \partial \Omega \times (0,T),\\u=f&{\textrm {on}}\ \ \Omega \times \left\{T\right\}.\end{cases}}

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.^[2]

Notes

^ Zauderer 2006, p. 139.
^ Taylor 1975.

References

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.