Category:Elliptic: Difference between revisions

Latest revision as of 15:22, 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 $\sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}} + lower-order terms = 0$ .

If the matrix $A = (a_{i j})_{i j}$ 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 $f$ is given.

Pages in category "Elliptic"

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

C

Controlled Heating

S

Source Inversion

@@ Line 1: / Line 1: @@
 This category contains all control problems which are governed by an elliptic partial differential equation.
 <p>
-A second order linear partial differential equation can be written as
+ A second order linear partial differential equation can be written as
-  <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>.
+  <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0</math>.
+</p>
-  If <math>A=(a_{ij})_{ij}</math> is positive or negative definite, the partial differential equation is called elliptic.
+  <p>
- An example is the Poisson's equation: <math>-\Delta u = f</math>,
+  If the matrix <math>A=(a_{ij})_{ij}</math> is positive or negative definite, the partial differential equation is called elliptic.
-  where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> given.
+</p>
+  <p>
+  An example is the Poisson's equation: <math>-\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.
   </p>