Category:Parabolic: Difference between revisions

Latest revision as of 15:21, 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 $\sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}} + lower-order terms = 0$ .

If the matrix $A = (a_{i j})_{i j}$ is positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic.

An example is the heat equation: $\frac{\partial u}{\partial t} - Δ u = f$ , where $Δ$ denotes the Laplace operator, $u$ is the unknown, and the function $f$ is given.

Pages in category "Parabolic"

This category contains only the following page.

C

Control of Heat Equation with Actuator Placement

@@ Line 2: / Line 2: @@
 <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 \text{lower-order terms} = 0</math>. </p>
+  <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0</math>.
+</p>
-<p>
+ <p>
-  If <math>A=(a_{ij})_{ij}</math> is positive or negative semidefinite with exact one eigenvalue zero, the  partial differential equation is called parabolic.
+  If the matrix <math>A=(a_{ij})_{ij}</math> is positive or negative semidefinite with exact one eigenvalue zero, the  partial differential equation is called parabolic.
+</p>
+  <p>
   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> is given.