Category:Elliptic: Difference between revisions

Revision as of 14:26, 24 February 2016

This category contains all control problems which are governed by an elliptic partial differential equation.

A second order linear partial differential equation can be written as $\sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}} + lower - orderterms = 0$ . If $A = (a_{i j})_{i j}$ is positive or negative definite, the partial differential equation is called elliptic. An example is the Poisson's equation: $- Δ u = f$ , where $Δ$ denotes the Laplace operator, $u$ is the unknown, and the function $f$ given.

Pages in category "Elliptic"

The following 2 pages are in this category, out of 2 total.

C

Controlled Heating

S

Source Inversion

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 <p>
 A second order linear partial differential equation can be written as
-  <math>\sum^n_{i,j=1} a_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0</math>.
+  <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0</math>.
-  If <math>A=(a_{i,j})_{i,j}</math> is positive or negative definite, the partial differential equation is called elliptic.
+  If <math>A=(a_{ij})_{ij}</math> is positive or negative definite, the partial differential equation is called elliptic.
   An example is the Poisson's equation: <math>-\Delta u = f</math>,