Parabolic Partial Differential Equation
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A parabolic partial differential equation is a type of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
(PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction,
particle diffusion In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 fo ...
, 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 descript ...
. 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 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, : \fr ...
. 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 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 Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
; 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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
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 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 ana ...
.


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 The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that: # a solution exists, # the sol ...
(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 ana ...


See also

*
Hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
* Elliptic partial differential equation * Autowave


References


Further reading

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