Parabolic partial differential equation

• Nonlinear parabolic differential equations
• Theoretical results
• Numerical Methods
• Applications

Parabolic partial differential equations are a special class of partial differential equations ( PDE ) of second or higher order, which are used in the description of a wide range of scientific problems for use. These are so-called evolution problems, in which a "time variable" appears and the development of the "time" is described by a first-order derivative. The solutions of parabolic differential equations often behave like the solutions of the heat equation, which describes the conduction of heat in solids or the diffusion in liquids and gases.

Generalizing the heat equation, we obtain the important class of linear parabolic PDE of second order. These can be found except in the case of heat conduction in addition application for example in the calculation of the propagation of sound in the sea or the development of stock options (Black-Scholes model). Below are just a parabolic second order differential equations are considered.

• 2.1 Semilinear parabolic differential equations in dimensions
• 2.2 Initial and boundary values

Definition in the linear case

Parabolic differential equation in two dimensions

The general linear partial differential equation of second order with two variables

Is parabolic at the point ( x, y), when the coefficients in the functions of the derivatives of the highest point (x, y) of the condition

. meet This means that the determinant of coefficient matrix

At the point ( x, y) takes the value 0. The origin of the name comes from the parabolic analogy of the above coefficients condition to the general conic equation. Analog classifications exist for elliptic and hyperbolic differential equations, see Partial Differential Equation # Classification according to basic type.

Parabolic differential equation in n dimensions

A generalization to several variables is the linear partial differential equation of second order

In generalization of the two-dimensional case is defined as the differential equation as a parabolic point if the coefficient matrix is positive semidefinite and singular. This means that all the eigenvalues ​​of the matrix coefficients are non-negative and an eigenvalue disappears.

Time-dependent notation

Semilinear parabolic differential equations in dimensions

In the last section the abstract classification was declared as parabolic differential equation. In many applications, the singular direction of the coefficient matrix has the meaning of time. Then the solution is a function of time t, and n position variable dependent. Since the type classification depends only on the coefficients of the highest derivatives, you can just allow non-linear dependencies in the lower derivatives. With coefficient functions and a function, the equation

A semi- parabolic linear differential equation is, if the matrix of coefficients is everywhere positive definite. The simplest case occurs when the identity matrix, then the main part of this equation precisely the Laplace operator. Important classes of parabolic differential equations are

• The heat equation ( S.U. )
• Reaction-diffusion equations, where the function does not depend on the gradient,
• Convection-diffusion equations, such as the compressible Navier -Stokes equations

Initial and boundary values

Usually one considers parabolic differential equations according to their structure into "space" - and " time " variable as a combined initial and boundary value problem. If the solution is sought within a spatial area for times, you are giving the initial values ​​at the time by a function

Before, the boundary values ​​on the boundary of the geographic area to be for hours by a function (or its first spatial derivative )

Specified.

Examples

A simple example of a parabolic differential equation is the heat equation in one space dimension:

Here, the temperature at the location is to the time constant refers to the thermal conductivity.

Numerical methods for parabolic initial-boundary value problems

If the domain of definition of the equation does not change with time, represents the parabolic initial-boundary value problem in the time direction t an initial value problem and local direction of a boundary value problem for an elliptic differential equation dar. In the numerical treatment can these two problems iw separated tackle. There are two approaches:

• Line method (English MOL = method- of- lines): one discretized first in place by employing standard methods for elliptic boundary value problems, such as the finite difference method or finite element methods. This yields an ordinary initial value problem of very large dimension for the degrees of freedom of the discretization. But this is a stiff initial value problem and should be solved with implicit or linearly implicit methods such as Rosenbrock - Wanner methods or BDF methods. The advantage of this approach is that one can use the above standard method for the time integration. The disadvantage is that the spatial discretization is fixed, and therefore local, time-dependent refinements are not possible.
• Rothe method: We first discretized in time with one of the just -mentioned methods for stiff initial value problems. This gives at each time step an elliptic boundary value problem for the current solution in the field. To solve this boundary value problem can now, for example, Finite element methods are used with adaptive lattice matching. The programming is much more expensive than using the method of lines.

A simple numerical method for parabolic problems is the Crank- Nicolson method. This uses one hand for the spatial discretization, the finite difference method with a fixed grid and time discretization as the implicit trapezoidal method.