Parabolic partial differential equation

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Parabolic partial differential equation

A parabolic partial differential equation is a type of second-order partial differential equation, describing a wide family of problems in science including heat diffusion and stock option pricing. These problems, also known as evolution problems, describe physical or mathematical systems with a time variable, and which behave essentially like heat diffusing through a medium like a metal plate.

The most fundamental parabolic PDE is the heat equation:

:$u_t = u_\left\{xx\right\} + u_\left\{yy\right\}$ ,

where $u\left(t,x,y\right)$ is the temperature at time $t$ and at position $\left(x,y\right)$. The symbol $u_t$ signifies the partial derivative with respect to the time variable $t$, and similarly $u_\left\{xx\right\}$ and $u_\left\{yy\right\}$ are second partial derivatives with respect to $x$ and $y.$

In plain English, this equation says that the temperature at a given time and point will rise or fall at a rate proportional to the difference between the temperature at that point and the average temperature near that point. [Indeed, the quantity $u_\left\{xx\right\}+u_\left\{yy\right\}$ measures how far off the temperature is from satisfying the mean value property of harmonic functions.]

The main generalization of the heat equation is

:$u_t = Lu$,

where $L$ is an elliptic operator. Such a system can be hidden in an equation of the form

:$abla cdot \left(a\left(x\right) abla u\left(x\right)\right) + b\left(x\right)^T abla u\left(x\right) + cu\left(x\right) = f\left(x\right)$

if the matrix-valued function $a\left(x\right)$ has a kernel of dimension 1.

olution

Under broad assumptions, parabolic PDEs as given above have solutions for all "x,y" and "t">0. An equation of the form $u_t = L\left(u\right)$ is considered to be parabolic if "L" is a (possibly nonlinear) function of "u" and its first and second derivatives, with some further conditions on "L". With such a nonlinear parabolic differential equation, solutions exist for a short time but may explode in a singularity in a finite amount of time. Hence, the difficulty is in determining solutions for all time, or more generally studying the singularities that arise. This is in general quite difficult, as in the Solution of the Poincaré conjecture via Ricci flow.

Examples

* Heat equation
* Mean curvature flow
* Ricci flow

*Hyperbolic partial differential equation
*Elliptic partial differential equation

Notes

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