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.

Definition

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

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

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

${\displaystyle B^{2}-AC=0.}$

Usually ${\displaystyle x}$ represents one-dimensional position and ${\displaystyle y}$ 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 ${\displaystyle Ax^{2}+2Bxy+Cy^{2}+Dx+Ey+F=0}$ to define a planar parabola.

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

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

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

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

where

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

denotes the Laplace operator acting on ${\displaystyle u}$. This equation is the prototype of a multi-dimensional parabolic PDE.

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

${\displaystyle u_{t}=-Lu,}$

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

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

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

if the matrix-valued function ${\displaystyle a(x)}$ 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 ${\displaystyle u(x,t)}$, as a function of ${\displaystyle x}$ for a fixed time ${\displaystyle t>0}$, is generally smoother than the initial data ${\displaystyle u(x,0)=u_{0}(x)}$.

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.

Backward parabolic equation

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

An initial-value problem for the backward heat equation,

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

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

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

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.

Examples

• Heat equation
• Mean curvature flow
• Ricci flow

See also

• Hyperbolic partial differential equation
• Elliptic partial differential equation
• Autowave

References

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

• Evans, Lawrence C. (2010) [1998], Partial differential equations (PDF), Graduate Studies in Mathematics, 19 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/019, ISBN 978-0-8218-4974-3, MR 2597943
• "Parabolic partial differential equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Parabolic partial differential equation, numerical methods", Encyclopedia of Mathematics, EMS Press, 2001 [1994]