HOME

TheInfoList



OR:

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 Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flows along a te ...
, particle diffusion, and pricing of derivative investment instruments.


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_ + 2Bu_ + Cu_ + 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. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation A x^2 + 2B xy + C y^2 + D x + E y + F = 0 to define a planar
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
. The basic example of a parabolic PDE is the one-dimensional heat equation, :u_t = \alpha\,u_, 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 In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
of u with respect to the time variable t, and similarly u_ 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_ measures how far off the temperature is from satisfying the mean value property of
harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \ ...
. 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+\frac+\frac denotes the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
acting on u. This equation is the prototype of a ''multi-dimensional parabolic'' PDE. 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 \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 In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher) also known as the Kolmogorov–Petrovsky–Piskunov equation (named after Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov), KPP equation or Fis ...
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 Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solut ...
via
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be an ...
.


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 u_ = -\Delta u & \textrm \ \ \Omega \times (0,T), \\ u=0 & \textrm \ \ \partial\Omega \times (0,T), \\ u = f & \textrm \ \ \Omega \times \left \. \end, is equivalent to a final-value problem for the ordinary heat equation, :\begin u_ = \Delta u & \textrm \ \ \Omega \times (0,T), \\ u=0 & \textrm \ \ \partial\Omega \times (0,T), \\ u = f & \textrm \ \ \Omega \times \left \. \end 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.


Examples

* Heat equation * Mean curvature flow *
Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be an ...


See also

* Hyperbolic partial differential equation *
Elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, whe ...
*
Autowave Autowaves are self-supporting non-linear waves in active media (i.e. those that provide distributed energy sources). The term is generally used in processes where the waves carry relatively low energy, which is necessary for synchronization or ...


References


Further reading

* * * * {{DEFAULTSORT:Parabolic Partial Differential Equation *