Category:Parabolic: Difference between revisions

Revision as of 15:20, 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  $A = (a_{i j})_{i j}$  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

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