Category:Parabolic: Difference between revisions

Revision as of 15:19, 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 $\sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}} + lower - orderterms = 0$ . </p

If $A = (a_{i j})_{i j}$ 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"

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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 \textrm{ lower-order terms} = 0</math>.
+  <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0</math>. </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.