View source for Category:Hyperbolic

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} +\, \text{lower-order terms} = 0</math>. 
</p>

 <p>
 If the matrix <math>A=(a_{ij})_{ij}</math> is indefinite such that <math>n-1</math> eigenvalues have the same sign and the remaining eigenvalue has the other sign, the partial differential equation is called hyperbolic.
</p>
 <p>
 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> is 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 -->

[[Category: Model characterization]] 
[[Category: PDE model]]