Difference between revisions of "Category:Parabolic"

From mintOC
Jump to: navigation, search
Line 1: Line 1:
 
This category contains all control problems which are governed by a parabolic partial differential equation.
 
This category contains all control problems which are governed by a parabolic partial differential equation.
 
+
<p>
 +
A second order linear partial differential equation can be written as
 +
<math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0</math>.
 +
 +
If <math>A=(a_{ij})_{ij}</math> is positive or negative semidefinite with exact one eigenvalue zero, the  partial differential equation is called parabolic.
 +
 +
An example is the heat equation: <math>\frac{\partial u}{\partial t}-\Delta u = f</math>,
 +
where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> given.
 +
</p>
  
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

Revision as of 15:27, 24 February 2016

This category contains all control problems which are governed by a parabolic partial differential equation.

A second order linear partial differential equation can be written as \sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0. If A=(a_{ij})_{ij} is positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic. An example is the heat equation: \frac{\partial u}{\partial t}-\Delta u = f, where \Delta denotes the Laplace operator, u is the unknown, and the function f given.

Pages in category "Parabolic"

This category contains only the following page.