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.
Mathematically, a partial differential equation of the form
is parabolic if it satisfies the condition
- B2 − 4AC = 0.
This definition is analogous to the definition of a planar parabola.
A simple example of a parabolic PDE is the one-dimensional heat equation,
- ut = kuxx,
where u(t,x) is the temperature at time t and at position x, and k is a constant. The symbol ut signifies the partial derivative with respect to the time variable t, and similarly uxx is the second partial derivative with respect to x.
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. [1]
The main generalization of the heat equation is
- ut = Lu,
where L is an elliptic operator. Such a system can be hidden in an equation of the form
if the matrix-valued function a(x) has a kernel of dimension 1.
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Solution
Under broad assumptions, parabolic PDEs as given above have solutions for all x,y and t>0. An equation of the form ut = L(u) 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
See also
Notes
- ^ Indeed, the quantity uxx measures how far off the temperature is from satisfying the mean value property of harmonic functions.
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