Difference between revisions of "Category:Hyperbolic"
From mintOC
FelixMueller (Talk | contribs) |
FelixMueller (Talk | contribs) |
||
| Line 1: | Line 1: | ||
This category contains all control problems which are governed by a hyperbolic partial differential equation. | This category contains all control problems which are governed by a hyperbolic partial differential equation. | ||
| − | + | <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>. | ||
| + | |||
| + | If <math>A=(a_{ij})_{ij}</math> is indefinite such that <math>n-1</math> eigenvalues have the same sign and the remaining eigenvalue the other sign, the partial differential equation is called hyperbolic. | ||
| + | |||
| + | An example is the wave equation: <math>\frac{\partial^2 u}{\partial t^2}-\Delta u = f</math>, | ||
| + | where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> given. | ||
| + | </p> | ||
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> | <!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> | ||
Revision as of 15:29, 24 February 2016
This category contains all control problems which are governed by a hyperbolic partial differential equation.
A second order linear partial differential equation can be written as
.
If 
is indefinite such that 
eigenvalues have the same sign and the remaining eigenvalue the other sign, the partial differential equation is called hyperbolic.
An example is the wave equation: 
,
where 
denotes the Laplace operator, 
is the unknown, and the function 
given.
Pages in category "Hyperbolic"
This category contains only the following page.
